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+{"text":"\\section{Introduction}\nIsometric actions of Lie groups on Riemannian manifolds have been widely studied in the literature, as they constitute a powerful tool applied to deal with several interesting problems in both mathematics and physics, yielding important geometric and topological consequences. They appear in all branches of science where symmetries preserving length and angle measures play a role. In this paper we propose a generalization of the classical setting of isometric actions by studying a notion of isometric action of a categorified version of a Lie group \\cite{BD,BS} on a categorified version of a Riemannian manifold \\cite{dHF,dHF2,GGHR,PPT}. More precisely, we are interested in studying isometric actions of Lie $2$-groups on Riemannian groupoids. \n\nOn the one hand, as is explicitly mentioned by Baez and Lauda in \\cite{BD}, the notion of Lie 2-group goes back to Brown and Spencer in \\cite{BS} where it became clear that classical group theory is just the beginning of a larger subject that sometimes is called higher-dimensional group theory. In many contexts where we are tempted in using groups to tackle certain symmetries-involved problems, it turns out actually to be more natural to use a richer kind of structure where, in addition to group elements describing symmetries, we also have isomorphisms between these, thus describing symmetries between symmetries. On the other hand, Riemannian groupoids recently appeared into the picture as a differentiable model allowing to perform Riemannian geometry techniques over more general singular spaces than orbifolds or leaf spaces of singular foliations. Since the seminal works \\cite{dHF,dHF2}, the notion of Riemannian groupoid (stack) defined and studied therein has been applied to satisfactorily extend several theories where Riemannian manifolds have played an important role. For instance, some of the recent contributions in which Riemannian groupoid metrics have been used as a tool to describe topological and geometric feutures of Lie groupoids and their differentiable stacks are the following.\n\\begin{itemize}\n\\item Resolutions of proper Riemannian groupoids were introduced in \\cite{PTW} in order to obtain a desingularization of their underlying differentiable stacks via a successive blow-up construction.\n\\item A theory of stacky geodesics on Riemannian stacks was developed in \\cite{dHdM} allowing to establish a stacky version\nof the Hopf--Rinow Theorem.\n\\item The problem of understanding invariant linearization of proper Lie groupoids was addressed in \\cite{dHdM2} where the authors fixed and extended previous results in the literature as well as provided a sufficient criterion that uses compatible complete metrics and covers the case of proper group actions.\n\\item Recently, a notion of Morse Lie groupoid morphism was introduced in \\cite{OV}, allowing to extend the main results of classical Morse theory to the context of Lie groupoids. It is worth mentioning that the notion of isometric 2-action we will introduce in this paper was used in \\cite{OV} as one of the fundamental ingredients needed to construct an equivariant double Morse--Bott complex which computes the equivariant cohomology of a $2$-action as defined in \\cite{OBT}. \n\\end{itemize}\nThe paper is organized as follows. In Section \\ref{S:2} we briefly introduce the necessary terminology and notions about Riemannian Lie groupoids and Lie $2$-groups we will be dealing with along the paper. In Section \\ref{S:3} we define isometric Lie $2$-group actions on Riemannian groupoids and exhibit some of their immediate properties. We provide sufficient conditions to ensure that Riemannian groupoid metrics that are invariant by the action of a Lie $2$-group exist and use the nice ideas implemented to give a proof of the groupoid linearization theorem in \\cite{dHF} to describe an equivariant weakly groupoid linearization result. Based in the classical constructions we state versions of both the Slice Theorem and the Equivariant Tubular Neighborhood Theorem for proper and free Lie $2$-group actions. We also quickly explain a way to define a ``2-group orbit type topological stratification'' for the Lie groupoid we are working with, define orthogonal Lie $2$-groups and describe their infinitesimal counterpart, exhibit a few examples and provide some interesting applications using principal connection warpings and Cheeger deformations. Finally, in Section \\ref{S:4} we present a description of the Lie $2$-group of strong (weak) groupoid isometries of a Lie groupoid equipped with a $0$-metric and determine a model for its Lie $2$-algebra of strong (weak) Killing multiplicative vector fields. We apply some of the results of this section to the case in which we are equipped with an isometric Lie 2-group action. It is important to mention that our infinitesimal descriptions are motivated by those results developed in \\cite{OW} in order to describe the Lie $2$-algebra structure that the set of multiplicative vector fields on a Lie groupoid has. In particular, we show that the Lie $2$-algebra of weak Killing multiplicative vector fields is Morita invariant, thus yielding a good notion of geometric Killing vector field on a quotient Riemannian stack. This notion of geometric Killing vector field recovers the classical notions of Killing vector field on both a Riemannian manifold and a Riemannian orbifold as defined for instance in \\cite{BZ} as well as the notion of transverse Killing vector field on a regular Riemannian foliation as defined in \\cite[p. 84]{Mo}. We end the section by proving that the algebra of geometric Killing vector fields on a quotient Riemannian stack that is represented by a Riemannian foliation groupoid has finite dimension.\n\\medskip\n\n{\\bf Acknowledgments:} We would like to tank Mateus de Melo and Cristian Ortiz for several enlightened discussions, comments, and suggestions which allowed us to improve this work. Herrera was supported by the Brazilian Federal Foundation CAPES - Finance Code 001. Valencia was supported by Grant 2020\/07704-7 Sao Paulo Research Foundation - FAPESP.\n\n\\section{Preliminaries}\\label{S:2}\nIn this short section we briefly introduce the basics on Lie groupoids we shall be using throughout. For specific details the reader is recommended to visit for instance \\cite{BD,BS,dH,Ma,dHF}. \n\nA \\emph{Lie groupoid} $X_1\\rightrightarrows X_0$ consists of a manifold $X_0$ of objects and a manifold $X_1$ of arrows, two surjective submersions $s_X,t_X:X_1\\to X_0$ respectively indicating the source and the target of the arrows, and a smooth associative composition $m_X:X_2\\to X_1$ over the set of composable arrows $X_2=X_1\\times_{X_0} X_1$, admitting unit $u_X:X_0\\to X_1$ and inverse $i_X:X_1\\to X_1$, subject\nto the usual groupoid axioms. The collection of maps mentioned above are called \\emph{structural maps} of the Lie groupoid $X_1\\rightrightarrows X_0$. We shall drop up the sub-index notation for the structural maps only if there is no risk of confusion.\n\nSpecial instances of Lie groupoids are given by manifolds, Lie groups, Lie group actions, surjective submersions, foliations, pseudogroups, principal bundles, vector bundles, among others \\cite{dH, Ma}.\n\nLet $X_1\\rightrightarrows X_0$ be a Lie groupoid. For each $x\\in X_0$, its \\emph{isotropy group} $X_x:=s_X^{-1}(x)\\cap t_X^{-1}(x)$ is a Lie group and an embedded submanifold in $X_1$. There is an equivalence relation in $X_0$ defined by $x\\sim y$ if there exists $p\\in X_1$ with $s_X(p)=x$ and $t_X(p)=y$. The corresponding equivalence class of $x\\in X_0$ is denoted by $\\mathcal{O}_x\\subseteq X_0$ and called the \\emph{orbit} of $x$. The previous equivalence relation defines a quotient space $X_0\/X_1$ called the \\emph{orbit space} of $X_0\\rightrightarrows X_1$.  This space equipped with the quotient topology is in general a \\emph{singular space}, that is, it does not carry a differentiable structure making the quotient projection $X_0\\to X_0\/X_1$ a surjective submersion.\n\nA \\emph{Lie groupoid morphism} between two Lie groupoids $X_1\\rightrightarrows X_0$ and $X_1'\\rightrightarrows X_0'$ is a pair $\\phi:=(\\phi^1,\\phi^0)$ where $\\phi^1:X_1\\to X_1'$ and $\\phi^0:X_0\\to X_0'$ are smooth maps commuting with both source and target maps and preserving the composition maps.\n\nFinally, we recall that the \\emph{Lie algebroid} associated to the Lie groupoid $X_1\\rightrightarrows X_0$ is defined to be the vector bundle $A_X:=\\textnormal{ker}(ds_X)|_{X_0}\\subset TX_1$ with anchor map $\\rho:A_X\\to TX_0$ obtained by restricting $dt_X:TX_1\\to TX_0$ to $A_X$.\n\n\n{\\bf Riemannian groupoids:} The notion of Riemannian metric on a Lie groupoid that we will be working with along the paper was introduced in \\cite{dHF} (see also \\cite{GGHR,PPT}). Such a notion is compatible with the groupoid composition so that it plays an important role in several parts of our work. Because of our purposes we mainly focus in using both $1$-metrics and $0$-metrics which were initially defined in \\cite{GGHR} and \\cite{PPT}, respectively, but they were recovered later in \\cite{dHF} as particular cases of a more general notion. We start by recalling that a submersion $\\pi:(E,\\eta^E)\\to B$ with $(E,\\eta^E)$ a Riemannian manifold is said to be \\emph{Riemannian} if the fibers of it are equidistant (transverse condition). In this case the base $B$ gets an induced metric $\\eta^B:=\\pi_\\ast \\eta^E$ for which the linear map $d\\pi(e)=(\\textnormal{ker}(d\\pi(e)))^\\perp\\to T_{\\pi(e)}B$ is an isometry for all $e\\in E$. If $(\\eta^{E})^\\ast$ denotes the dual metric associated to $\\eta^{E}$ then the condition for a Riemannian submersion can be rephrased as follows. For all $e\\in E$ the map  $d\\pi(e)^\\ast: T_{\\pi(e)}^\\ast B \\to \\textnormal{ker}(d\\pi(e))^\\circ$ is an isometry, where $\\textnormal{ker}(d\\pi(e))^\\circ$ denotes the annihilator of the vectors tangent to the fiber.\n\nIt is well known that given a Lie groupoid $X_1\\rightrightarrows X_0$ every pair of composable arrows in $X_2$ may be identified with an element in the space of commutative triangles so that it admits an action of $S_3$ determined by permuting the vertices of such triangles. In these terms, a \\emph{Riemannian groupoid} is a pair $(X_1\\rightrightarrows X_0,\\eta)$ where $X_1\\rightrightarrows X_0$ is a Lie groupoid and $\\eta=\\eta^{(2)}$ is a Riemannian metric on $X_2$ that is invariant by the $S_3$-action and transverse to the composition map $m_X:X_2\\to X_1$. The metric $\\eta^{(2)}$ induces metrics $\\eta^{(1)}=((\\pi_2)_X)_\\ast \\eta^{(2)}=(m_X)_\\ast \\eta^{(2)}=((\\pi_1)_X)_\\ast \\eta^{(2)}$ on $X_1$ and $\\eta^{(0)}=(s_X)_\\ast \\eta^{(1)}=(t_X)_\\ast \\eta^{(1)}$ on $X_0$ such that $(\\pi_2)_X, m_X, (\\pi_1)_X:X_2\\to X_1$ and $s_X,t_X:X_1\\to X_0$ are Riemannian submersions and $i_X:X_1\\to X_1$ is an isometry. This is because the $S_3$-action permutes these face maps.\n\nThe metric $\\eta^{(j)}$, for $j=2,1,0$, is called a $j$-\\emph{metric}. It is important to mention that every proper groupoid can be endowed with a $2$-metric (more generally, an $n$-metric as defined below) and if a Lie groupoid admits a $2$-metric then it is weakly linearizable around any saturated submanifold. For more details visit \\cite{dHF}.\n\n\n{\\bf Lie 2-groups:} A \\emph{Lie 2-group} is a Lie groupoid in the category of Lie groups. In other words, a Lie 2-group is a Lie groupoid $G_1\\rightrightarrows G_0$ where both $G_1$ and $G_0$ are Lie groups and the structural maps of $G_1\\rightrightarrows G_0$ are Lie group homomorphisms \\cite{BD,BS}.  We will denote by $\\ast$ the composition of arrows in $G_1\\rightrightarrows G_0$ and by $\\cdot$ the product of arrows in $G_1$. For instance, the fact that the multiplication is a morphism of groups amounts to the identity:\n\\begin{equation}\\label{MultiProduct}\n\t(g_1\\ast g_2)\\cdot(g_1'\\ast g_2')=(g_1\\cdot g_1')\\ast(g_2\\cdot g_2'),\\quad \\forall (g_1,g_2),(g_1',g_2')\\in G_2,\n\\end{equation}\nwhere $G_2$ is the space of pairs of composable arrows of $G_1\\rightrightarrows G_0$.\n\nIt is well known that there are several alternative ways to think of Lie 2-groups. One of them is described in terms of crossed modules. Recall that a \\emph{crossed module of Lie groups} is a quadruple $(G,H,\\rho,\\alpha)$ consisting of a Lie group homomorphism $\\rho:H\\to G$ together with an action $\\alpha$ of $G$ on $H$, $(g,h)\\mapsto g h$, by Lie group automorphisms such that:\n\\[ \\rho(g h)=g \\rho(h) g^{-1},\\quad \\rho(h)h'=h h' h^{-1}, \\quad g\\in G,\\ h,h'\\in H. \\]\n\nTo such a crossed module one associates a Lie $2$-group with objects the Lie group $G_0:=G$, arrows the semi-direct product $G_1:=H\\rtimes G$ so that:\n\\[ (h_1,g_1)\\cdot (h_2,g_2):=(h_1 (g_1h_2), g_1 g_2), \\]\nand structure maps:\n\\[ s_G(h,g)=g,\\quad t_G(h,g)=\\rho(h)g, \\quad (h_1,\\rho(h_2)g_2)\\ast (h_2,g_2)=(h_1h_2,g_2). \\]\nConversely, any Lie 2-group $G_1 \\rightrightarrows G_0$ has an associated crossed module of Lie groups $(G,H,\\rho,\\alpha)$, where $G=G_0$, $H=\\textnormal{ker}(s_G)$, $\\rho:=t_G|_H:H\\to G$ and $G$ acts on $H$ by conjugation via the identity bisection: $g h:=1_g\\cdot h\\cdot 1_{g^{-1}}$. This defines an equivalence of categories between the category of Lie 2-groups and the category of crossed modules of Lie groups \\cite{BD}.\n\nA \\emph{Lie 2-algebra} is a Lie groupoid in the category of Lie algebras. More concretely, a Lie 2-algebra is a Lie groupoid $\\mathfrak{g}_1\\rightrightarrows \\mathfrak{g}_0$ where both $\\mathfrak{g}_1$ and $\\mathfrak{g}_0$ are Lie algebras and the structural maps of $\\mathfrak{g}_1\\rightrightarrows \\mathfrak{g}_0$ are Lie algebra homomorphisms. As expected, the Lie functor provides a one-to-one correspondence between Lie 2-groups and Lie 2-algebras.\n\nThe infinitesimal object associated to a crossed module of Lie groups is the so-called \\emph{crossed module of Lie algebras}. This is a quadruple $(\\mathfrak{g},\\mathfrak{h},\\partial,\\mathcal{L})$ where $\\mathfrak{g}$ and $\\mathfrak{h}$ are Lie algebras and $\\partial:\\mathfrak{h}\\to\\mathfrak{g}$ and $\\mathcal{L}:\\mathfrak{g}\\to\\textnormal{Der}(\\mathfrak{h})$ are Lie algebra homomorphisms verifying\n$$\\partial(\\mathcal{L}_xy)=[x,\\partial(y)]_\\mathfrak{g}\\quad\\textnormal{and}\\quad \\mathcal{L}_{\\partial(x)}y=[x,y]_\\mathfrak{h}.$$ \n\nA Lie $2$-algebra $\\mathfrak{g}_1\\rightrightarrows \\mathfrak{g}_0$ has an associated crossed module given by the data $\\mathfrak{h}=\\textnormal{ker}(s)$, $\\mathfrak{g}=\\mathfrak{g}_0$, $\\partial=t|_\\mathfrak{h}$ and $\\mathcal{L}_x=\\textnormal{ad}^{1}_{u(x)}$ for all $x\\in \\mathfrak{g}$. Conversely, a crossed module of Lie algebras $(\\mathfrak{g},\\mathfrak{h},\\partial,\\mathcal{L})$ has an associated Lie $2$-algebra which in turn is given by the data $\\mathfrak{g}_1=\\mathfrak{h}\\rtimes \\mathfrak{g}$ with Lie algebra structure provided by the semi-direct product with respect to $\\mathcal{L}$, $\\mathfrak{g}_1=\\mathfrak{g}$, and structural maps\n$$s(x,y)=y,\\quad t(x,y)=\\partial(x)+y,\\quad u(y)=(0,y),\\quad i(x,y)=(-x,y+\\partial(x)), $$\n$$m((x',y+\\partial(x)),(x,y))=(x+x',y).$$\n\nA \\emph{morphism} of crossed modules  $f:(\\mathfrak{g},\\mathfrak{h},\\partial,\\mathcal{L})\\to (\\mathfrak{g}',\\mathfrak{h}',\\partial',\\mathcal{L}')$ consists of two Lie algebra homomorphisms $f_0:\\mathfrak{g}\\to \\mathfrak{g}'$ and $f_1:\\mathfrak{h}\\to \\mathfrak{h}'$ such that $f_1\\circ \\partial =\\partial'\\circ f_0$ and $f_0(\\mathcal{L}_xy)=\\mathcal{L}'_{f_1(x)}(f_0(y))$. This induces a pair of Lie algebra homomorphisms $\\textnormal{ker}(\\partial)\\to \\textnormal{ker}(\\partial')$ and $\\textnormal{coker}(\\partial)\\to \\textnormal{coker}(\\partial')$. A morphism of crossed modules is a \\emph{quasi-isomorphism} if both of these Lie algebra morphisms are isomorphisms. The \\emph{derived category} of crossed modules of Lie algebras is defined to be the localization of the category of crossed modules of Lie algebras obtained by inverting all quasi-isomorphisms.\n\n\\section{Isometric Lie 2-group actions}\\label{S:3}\nIn the sequel we shall denote by $G_1\\rightrightarrows G_0$ a Lie 2-group and by $(X_1\\rightrightarrows X_0,\\eta)$ a Riemannian groupoid. A \\emph{left 2-action} of $G_1\\rightrightarrows G_0$ on $X_1\\rightrightarrows X_0$ is defined to be a Lie groupoid morphism $\\theta=(\\theta^1,\\theta^0):(G_1\\times X_1\\rightrightarrows G_0\\times X_0)\\to(X_1\\rightrightarrows X_0)$ such that both maps $\\theta^1$ and $\\theta^0$ are usual left Lie group actions. Here $G_1\\times X_1\\rightrightarrows G_0\\times X_0$ denotes the product Lie groupoid.  Note that given a left 2-action of $G_1\\rightrightarrows G_0$ on $X_1\\rightrightarrows X_0$ we immediately get that the structural maps of $X_1\\rightrightarrows X_0$ are equivariant with respect the structural maps of $G_1\\rightrightarrows G_0$. More precisely, if $p\\in X_1, x\\in X_0$ and $g\\in G_1, g_0\\in G_0$ then we obtain that\n\\begin{equation}\\label{Equivariant}\ns_X(gp)=s_G(g)s_X(p),\\quad t_X(gp)=t_G(g)t_X(p),\\quad u_X(g_0x)=u_G(g_0)u_X(x).\n\\end{equation}\n\n\nMoreover, we have that the action on arrows is \\emph{multiplicative}, meaning that for pairs of composable arrows $(p,q)\\in X_2$ and $(g,h)\\in G_2$ the following formula holds true\n\\begin{equation}\\label{MultAction}\n\t(p*q)(g*h)=(pg)*(qh).\n\\end{equation} \nHere we are denoting by $m_X(p,q)=p*q$ and $m_G(g,h)=g*h$. \n\\begin{definition}\nA $2$-action of $G_1\\rightrightarrows G_0$ on $(X_1\\rightrightarrows X_0,\\eta)$ is said to be \\emph{isometric} if $G_1$ acts by isometries on $(X_1,\\eta^{(1)})$.\n\\end{definition}\nAn immediate consequence of the previous definition is the following.\n\\begin{lemma}\\label{Rmk1}\nThe action $G_0$ on $(X_0,\\eta^{(0)})$ is also by isometries.\n\\end{lemma}\n\\begin{proof}\nLet $v,w\\in T_{x}X_0$ and $g_0\\in G_0$. It is clear that there are $\\tilde{v},\\tilde{w}\\in \\textnormal{ker}(d(s_X)_p)^{\\perp_{\\eta^{(1)}}}$ with $s_X(p)=x$ such that $d(s_X)_p(\\tilde{v})=v$ and $d(s_X)_p(\\tilde{w})=w$ and there is $g\\in G_1$ such that $s_G(g)=g_0$. Therefore,\n\\begin{eqnarray*}\n(\\theta^0_{g_0})_x^\\ast\\eta^{(0)}(v,w) & = & \\eta^{(0)}_{g_0x}(d(\\theta^0_{g_0})_x(v),d(\\theta^0_{g_0})_x(w))=\\eta^{(0)}_{g_0x}(d(\\theta^0_{g_0}\\circ s_X)_p(\\tilde{v}),d(\\theta^0_{g_0}\\circ s_X)_p(\\tilde{w}))\\\\\n\\star& = & \\eta^{(0)}_{g_0x}(d(s_X\\circ \\theta^1_{g})_p(\\tilde{v}),d(s_X\\circ \\theta^1_{g})_p(\\tilde{w}))=\\eta^{(1)}_{gp}(d(\\theta^1_{g})_p(\\tilde{v}),d(\\theta^1_{g})_p(\\tilde{w}))\\\\\n& = & \\eta^{(1)}_{p}(\\tilde{v},\\tilde{w})=\\eta^{(0)}_{x}(v,w).\n\\end{eqnarray*}\nIn the equality $\\star$ above we used the fact that $s_X$ is a Riemannian submersion and that the action $\\theta^1$ preserves the horizontal distribution $\\textnormal{ker}(ds_X)^{\\perp_{\\eta^{(1)}}}$. Note that this computation does not depend on the choice of $\\tilde{v},\\tilde{w},p$ and $g$. Furthermore, we may obtain the same conclusion by choosing $\\eta^{(0)}=(t_X)_\\ast \\eta^{(1)}$ instead of $\\eta^{(0)}=(s_X)_\\ast \\eta^{(1)}$. \n\\end{proof}\n\n\\begin{remark}\\label{Rmk2}\nIt is clear that $G_2$ is also a Lie group with the structure induced from the direct product $G_1\\times G_1$. With this in mind it is simple to see that the $2$-action $\\theta$ indices a canonical left action $\\theta^2$ of $G_2$ on $X_2$. We could have defined the notion of isometric $2$-action by requiring that $G_2$ acts on $(X_2,\\eta^{(2)})$ by isometries. If we do so then, just as we did in Lemma \\ref{Rmk1}, we can prove that the action of $G_1$ on $(X_1,\\eta^{(1)})$ is by isometries and, in turn, the action of $G_0$ on $(X_0,\\eta^{(0)})$ will be also by isometries. This is because $\\eta^{(1)}=(\\pi_{2,X})_\\ast \\eta^{(2)}=(m_X)_\\ast \\eta^{(2)}=(\\pi_{1,X})_\\ast \\eta^{(2)}$ and $\\pi_{2,X}, m_X, \\pi_{1,X}:X_2\\to X_1$ are Riemannian submersions. More generally, if $\\eta^{(n)}$ is an $n$-metric on $X_n$ in the sense of \\cite{dHF} and the induced left action $\\theta^n$ of $G_n$ on $(X_n,\\eta^{(n)})$ is by isometries then the action $\\theta^k$ of $G_k$ on $(X_k,\\eta^{(k)})$ will be by isometries for all $0\\leq k\\leq n-1$. Conversely, if we suppose that $G_0$ and $G_1$ act by isometries on $(X_0,\\eta^{(0)})$ and $(X_1,\\eta^{(1)})$, respectively, then  the induced action of $G_n$ on $(X_n,\\eta^{(n)})$ is by isometries for all $n\\geq 2$. Last assertion follows by using the formula\n$$\\eta^{(2)}=(\\pi_1)_X^\\ast \\eta^{(1)}+(\\pi_2)_X^\\ast \\eta^{(1)}-s_X^\\ast \\eta^{(0)}\\circ (d(\\pi_1)_X\\oplus d(\\pi_1)_X),$$\nwhich is derived from Remark 2.5 from \\cite{dHF} once the $n$-metric is chosen.\n\\end{remark}\nFor the purposes of this paper we only work with both $1$-metrics and $0$-metrics unless otherwise stated. Most of the notions we will introduce below shall have an analogous statement if the simplicial approach from Remark \\ref{Rmk2} is considered.\n\n\\begin{remark}\\label{Rmk3}\nThere is a different formulation for the notion of $2$-action in terms of crossed modules \\cite{hsz}. Namely, an action $\\beta=(\\beta^0, \\beta^1, \\beta^2)$ of a crossed module of Lie groups $(G,H,\\rho,\\alpha)$ on $X_1\\rightrightarrows X_0$ consists of three smooth actions $\\beta^0:G\\times X_0\\to X_0$, $\\beta^1:G\\times X_1\\to X_1$, and $\\beta^{2}:H\\times X_1\\to X_1$ satisfying the compatibility conditions (6.3.8)\u2013(6.3.12) from \\cite[Def. 6.3.4]{hsz}. Furthermore, from  \\cite[Lem. 6.3.13]{hsz} if $\\theta=(\\theta^1,\\theta^2)$ is a $2$-action of $G_1\\rightrightarrows G_0$ on $X_1\\rightrightarrows X_0$ then $\\beta^0=\\theta^0$, $\\beta^1=\\theta^0\\circ (u\\times \\textnormal{id}_{X_1})$, and $\\beta^2=\\theta^1|_H$ define an action of the associated crossed module on $X_1\\rightrightarrows X_0$. Conversely, given an action $\\beta=(\\beta^0, \\beta^1, \\beta^2)$ of a crossed module $(G,H,\\rho,\\alpha)$ on $X_1\\rightrightarrows X_0$ then $\\theta^0=\\beta^0$ and $\\theta^1_{(h,g)}=\\beta^2_{h}\\circ \\beta^1_g $ for all  $(h,g)\\in H\\rtimes G$, defines a $2$-action of the associated Lie $2$-group.\n\\end{remark}\nMotivated by what we just mentioned we set up the following definition.\n\\begin{definition}\nWe say that an action $\\beta=(\\beta^0, \\beta^1, \\beta^2)$ of a crossed module of Lie groups $(G,H,\\rho,\\alpha)$ on $(X_1\\rightrightarrows X_0,\\eta)$ is isometric if both $\\beta^1$ and $\\beta^2$ determine isometric actions on $(X_1,\\eta^{(1)})$.\n\\end{definition}\n Thus, the following result is clear:\n\\begin{lemma}\nThere exists a one-to-one correspondence between isometric actions of Lie $2$-groups and isometric actions of crossed modules of Lie groups.\n\\end{lemma}\nAn interesting consequence that comes up from our definition of isometric Lie $2$-group action is the following. The notion of Riemannian submersions in the Lie groupoid context was introduced in \\cite{dHF2}. So, the following result is expected:\n\\begin{proposition}\\label{PQuotient}\n\tIf $\\theta$ is a free and proper isometric $2$-action of $G_1\\rightrightarrows G_0$ on $(X_1\\rightrightarrows X_0,\\eta)$ then there is a structure of Riemannian groupoid $(X_1\/G_1\\rightrightarrows X_0\/G_0, \\overline{\\eta})$ so that the canonical projection $\\pi=(\\pi_1,\\pi_0):(X_1\\rightrightarrows X_0)\\to (X_1\/G_1\\rightrightarrows X_0\/G_0)$ becomes a Riemannian groupoid submersion.\n\\end{proposition}\n\\begin{proof}\n\tWe start by exhibiting the Lie groupoid structure on $X_1\/G_1\\rightrightarrows X_0\/G_0$. This fact was stated for instance in \\cite{GZ} but without proof so that we exhibit a proof of it by seek of completeness. It is well known that $X_j\/G_j$ admits a unique manifold structure so that $\\pi_j$ (for $j=0,1$) is a surjective submersion. We define source and target maps respectively as $\\overline{s}([p])=[s_X(p)]$ and $\\overline{t}([p])=[t_X(p)]$ for all $p\\in X_1$. As consequence of Identities \\eqref{Equivariant} these maps are well defined and, moreover, both of them are surjective submersions since $\\pi_j$, $s$, and $t$ are so. We have that $([p],[q])\\in (X_1\/G_1)_2$ if and only if $\\overline{s}([p])=\\overline{t}([q])$ which in turn holds true if and only if $s_X(p)=g_0t_X(q)$ for some $g_0\\in G_0$. Thus, we define $\\overline{m}([p],[q])=[m_X(p,gq)]$ for some $g\\in G_1$ such that $t_G(g)=g_0$. It is simple to check that $\\overline{m}$ does not depend on the choice of $g$ and it is well defined because of Property \\eqref{MultAction}. This is also clearly smooth and associative since $m_X$ is associative and Identity \\eqref{MultAction} is satisfied. The unit map and the inversion are respectively defined by $\\overline{u}([x])=[u_X(x)]$ and $\\overline{i}([p])=[i_X(p)]$ for all $x\\in X_0$ and $p\\in X_1$. It is also easy to verify that this maps are well defined, smooth, and they satisfy the required groupoid conditions.\n\t\n\tConsider, for $j=0,1$, the induced Riemannian metric $\\overline{\\eta}^{(j)}=(\\pi_j)_\\ast \\eta^{(j)}$ on $X_j\/G_j$ making of $\\pi_j$ a Riemannian submersion. Moreover, note that\n\t$$\\overline{i}_\\ast \\overline{\\eta}^{(1)}=(\\overline{i}\\circ \\pi_1)_\\ast \\eta^{(1)}=(\\pi_1\\circ i_X)_\\ast\\eta^{(1)}=(\\pi_1)_\\ast \\eta^{(1)}=\\overline{\\eta}^{(1)},$$\n\tsince $i$ is an isometry, and\n\t$$\\overline{s}_\\ast \\overline{\\eta}^{(1)}=(\\overline{s}\\circ \\pi_1)_\\ast \\eta^{(1)}=(\\pi_0\\circ s_X)_\\ast\\eta^{(1)}=(\\pi_0)_\\ast \\eta^{(0)}=(\\pi_0\\circ t_X)_\\ast\\eta^{(1)}=(\\overline{t}\\circ \\pi_1)_\\ast \\eta^{(1)}=\\overline{t}_\\ast \\overline{\\eta}^{(1)},$$\n\tso that $\\overline{s}_\\ast \\overline{\\eta}^{(1)}=\\overline{t}_\\ast \\overline{\\eta}^{(1)}=\\overline{\\eta}^{(0)}$. Therefore, $(X_1\/G_1\\rightrightarrows X_0\/G_0,\\overline{\\eta})$ with $\\overline{\\eta}=(\\overline{\\eta}^{(1)},\\overline{\\eta}^{(0)})$ is a Riemannian groupoid and $\\pi=(\\pi_1,\\pi_0)$ is a Riemannian submersion of groupoids, as desired.\n\\end{proof}\n\n\\begin{remark}\nLet us denote by $\\eta^\\ast$ the dual metric associated to a Riemannian metric $\\eta$. From \\cite{dHF} we know that if $\\pi:E\\to B$ is a submersion and $\\lbrace \\eta_1,\\cdots,\\eta_k\\rbrace$ is a collection of $\\pi$-transverse metrics then its tangent average $\\frac{1}{k}\\sum_{l=1}^k\\eta_l$ fails to be $\\pi$-transverse again in general. Nevertheless, its cotangent average $\\frac{1}{k}\\left( \\sum_{l=1}^k\\eta_l^\\ast\\right)^\\ast$ is always $\\pi$-transverse which sometimes makes more advantageous to take a cotangent space point of view in the study of Riemannian submersions.\n\\end{remark}\n\nWe will deal with the classical approach used to ensure the existence of invariant Riemannian metrics on $G$-manifolds to provide a similar construction in our context. Namely:\n\n\\begin{lemma}\\label{Existence1}\nIf $G$ is a compact Lie group seen as a Lie unit 2-group $G\\rightrightarrows G$ acting on $(X_1\\rightrightarrows X_0,\\eta)$, then there exists a $1$-metric $\\overline{\\eta}$ making of such an action isometric. In particular, if $G$ is compact and $X_1\\rightrightarrows X_0$ is proper then there always exists a $1$-metric making of such a $2$-action isometric.\n\\end{lemma}\n\\begin{proof}\nLet us consider the normalized Haar measure $\\mu$ on $G$. We define $\\overline{\\eta}=(\\overline{\\eta}^{(1)},\\overline{\\eta}^{(0)})$ by averaging its dual as follows:\n$$(\\overline{\\eta}^{(1)})^\\ast:=\\int_{G}(\\theta^1)^\\ast_g (\\eta^{(1)})^\\ast d\\mu(g).$$\nThe dual metric $(\\overline{\\eta}^{(0)})^\\ast$ has a similar defining formula but using $\\eta^{(0)}$ and $\\theta^0$ instead of $\\eta^{(1)}$ and $\\theta^1$. It is simple to check that $\\overline{\\eta}=(\\overline{\\eta}^{(1)},\\overline{\\eta}^{(0)})$ is the $1$-metric we are looking for. Last part of the assertion above follows from the fact that every proper Lie groupoid admits $n$-metrics \\cite{dHF}.\n\\end{proof}\nA similar result can be obtained only by requiring that $G\\rightrightarrows G$ acts properly on $(X_1\\rightrightarrows X_0,\\eta)$. Same proof as in Lemma \\ref{Existence1} goes through by applying \\cite[Lem. 4.2]{K}. Let us assume for a moment that $G_1$ is compact so that $G_0$ is also compact. If $\\mu_1$ is the normalized Haar measure on $G_1$ then, by uniqueness, the pushforward measure $s_{G\\ast}\\mu_1$ agrees with the normalized Haar measure $\\mu_0$ on $G_0$ since $s_G$ a surjective Lie group homomorphism and\n\\begin{equation}\\label{PushHaar}\n\\int_{G_0}f d(s_{G\\ast}\\mu_1)=\\int_{G_1} f\\circ s_G d\\mu_1,\n\\end{equation}\nfor each continuous function $f:G_0\\to \\mathbb{R}$. Analogously, $t_{G\\ast}\\mu_1=\\mu_0$ and $i_{G\\ast}\\mu_1=\\mu_1$. Note also that the fact that $i_{G\\ast}\\mu_1=\\mu_1$ and $s_G\\circ i_G=t_G$ immediately implies that $t_{G\\ast}\\mu_1=s_{G\\ast}\\mu_1$. Thus, we are in conditions to state:\n\n\\begin{theorem}\\label{Existence2}\nIf $G_1\\rightrightarrows G_0$ is a Lie $2$-group, with $G_1$ compact, acting on $(X_1\\rightrightarrows X_0,\\eta)$, then there exists a $1$-metric $\\overline{\\eta}$ making of such an $2$-action isometric.\n\\end{theorem}\n\\begin{proof}\nAs it was done in Lemma \\ref{Existence1}, by dual averaging  we define:\n$$(\\overline{\\eta}^{(1)})^\\ast:=\\int_{G_1}(\\theta^1)^\\ast_g(\\eta^{(1)})^\\ast d\\mu_1(g)\\quad\\textnormal{and}\\quad (\\overline{\\eta}^{(0)})^\\ast:=\\int_{G_0}(\\theta^0)^\\ast_{g_0}(\\eta^{(0)})^\\ast d\\mu_0(g_0).$$\nThe result will follow from the fact that $\\theta$ is a $2$-action, $\\eta$ is already a $1$-metric, and Identity \\eqref{PushHaar} holds true. Indeed, on the one hand, a straightforward computation using the fact that $i$ is an isometry of $(X_1,\\eta^{(1)})$ verifying both Identity \\eqref{Equivariant} and  $i_{G\\ast}\\mu_1=\\mu_1$ implies that $i_X$ is an isometry of $(X_1,\\overline{\\eta}^{(1)})$ as well. The fact that $s_X:X_1\\to X_0$ is a Riemannian submersion tells us that $ds_X(p)^\\ast: T_{s_X(p)}^\\ast X_0 \\to \\textnormal{ker}(ds_X(p))^\\circ$ is an isometry for all $p\\in X_1$ where $\\textnormal{ker}(ds_X(p))^\\circ$ denotes the annihilator of the vectors tangent to the fiber. Given $p\\in X_1$, covectors  $\\alpha,\\beta\\in T_{s_X(p)}^\\ast X_0$, and $g\\in G_1$ such that $s_G(g)=g_0$ we get the following chain of equalities:\n\n\n\n\n\n\n \n\\begin{eqnarray*}\n(\\overline{\\eta}^{(0)}_{s_X(p)})^\\ast (\\alpha,\\beta) &=& \\int_{G_0}(\\eta^{(0)}_{g_0s_X(p)})^\\ast ((\\theta^0_{g_0})^\\ast(\\alpha ),(\\theta^0_{g_0})^\\ast(\\beta ))d\\mu_0\\\\\n\\eqref{PushHaar}&=& \\int_{G_1}(\\eta^{(0)}_{s_G(g)s_X(p)})^\\ast ((\\theta^0_{s_G(g)})^\\ast(\\alpha ),(\\theta^0_{s_G(g)})^\\ast(\\beta ))d\\mu_1\\\\\n& = & \\int_{G_1}(\\eta^{(1)}_{gp})^\\ast (ds_X(gp)^\\ast((\\theta^0_{s_G(g)})^\\ast(\\alpha )),ds_X(gp)^\\ast((\\theta^0_{s_G(g)})^\\ast(\\beta )))d\\mu_1\\\\\n& = & \\int_{G_1}(\\eta^{(1)}_{gp})^\\ast ((\\theta^1_{g})^\\ast(ds_X(p)^\\ast(\\alpha )),(\\theta^1_{g})^\\ast(ds_X(p)^\\ast(\\beta )))d\\mu_1\\\\\n&= & (\\overline{\\eta}^{(1)}_{p})^\\ast (ds_X(p)^\\ast(\\alpha ),ds_X(p)^\\ast(\\beta )),\n\\end{eqnarray*}\nfrom which we conclude that $s_X$ is also Riemannian for the averaged metrics. With analogous computations we get a similar conclusion for $t_X:X_1\\to X_0$ so that the result follows as claimed.\n\\end{proof}\n\n\\begin{remark}\\label{Rmk4}\nIt is worth noticing that if we consider an $n$-metric on $X_n$ and take into account the simplicial approach described in Remark \\ref{Rmk2} then by applying similar averaging arguments as in the proof of Theorem \\ref{Existence2} it is possible to show the existence of another $n$-metric on $X_n$ that is invariant by the action of $G_n$ on $X_n$.  \n\\end{remark}\n\nThe \\emph{tangent groupoid} $TX_1\\rightrightarrows TX_0$ is obtained by applying the tangent functor at both manifolds of arrows and objects and all of its structural maps. If $S \\subset X_0$ is a saturated manifold (only an orbit for instance) then we can restrict the groupoid structure to $X_{S}=s_X^{-1}(S)=t_X^{-1}(S)$, thus obtaining a Lie subgroupoid $X_S\\rightrightarrows S$ of $X_1\\rightrightarrows X_0$. Furthermore, the Lie groupoid structure of $TX_1\\rightrightarrows TX_0$ can be restricted to define a new Lie groupoid $\\nu(X_S)\\rightrightarrows \\nu(S)$ having the property that all of its structural maps are fiberwise isomorphisms.\n\nThe notion of \\emph{linearization} around an orbit in the Lie groupoid setting is well known and it was a unsolvable problem for a long time when considering for instance only proper groupoids. We recommend the reader to visit \\cite{CS} with the aim of getting a quick introduction and contextualization about such a notion.\n\\begin{definition}\nSuppose that we have a Lie $2$-action of $G_1\\rightrightarrows G_0$ on $X_1\\rightrightarrows X_0$ and let us pick a $G_0$-invariant saturated submanifold $S\\subset X_0$. We say that $X_1\\rightrightarrows X_0$ is \\emph{equivariant weakly linearizable} at $S$ if there exist $G$-invariant Lie groupoid neighborhoods $\\widetilde{V}\\rightrightarrows V$ of $X_{S}\\rightrightarrows S$ in $\\nu(X_S)\\rightrightarrows \\nu(S)$ (seen as the zero section) and $\\widetilde{U}\\rightrightarrows U$ of $X_S\\rightrightarrows S$ in $X_1\\rightrightarrows X_0$, and a $G$-equivariant isomorphism of Lie groupoids $\\phi: (\\widetilde{V}\\rightrightarrows V)\\xrightarrow[]{\\cong} (\\widetilde{U}\\rightrightarrows U)$ which is the identity on $X_S\\rightrightarrows S$.\n\\end{definition}\nThe $G$-invariant property of the Lie groupoid neighborhood in $\\nu(X_S)\\rightrightarrows \\nu(S)$, used in the previous definition, makes sense because of the following facts. Let us assume in this case that we are given with a $2$-metric and that $G_2$ acts on $(X_2,\\eta^{(2)})$ by isometries (see Remark \\ref{Rmk2}). It is simple to check that every $2$-action $\\theta=(\\theta^1,\\theta^0)$ of $G_1\\rightrightarrows G_0$ on $X_1\\rightrightarrows X_0$ induces a $2$-action $T\\theta=(T\\theta^1,T\\theta^0)$ of $G_1\\rightrightarrows G_0$ on $TX_1 \\rightrightarrows TX_0$ by differentiating the actions $\\theta^1$ and $\\theta^0$. Let us pick a $G_0$-invariant saturated submanifold $S$ in $X_0$. This implies that $X_{S}$ is $G_1$-invariant and that $(X_{S})_2$ is $G_2$-invariant. If we respectively use $\\eta^{(2)}$, $\\eta^{(1)}$, and $\\eta^{(0)}$ to identify $\\nu(X_{S})_{2}\\cong \\nu((X_{S})_2)$ with $(T(X_{S})_2)^\\perp$,  $\\nu(X_{S})$ with $TX_S^\\perp$, and $\\nu(S)$ with $TS^\\perp$ then it follows that the $2$-action $T\\theta$ restrict to a well defined $2$-action $\\overline{T\\theta}$ of $G_1\\rightrightarrows G_0$ on $\\nu(X_{S})\\rightrightarrows \\nu(S)$ since $\\theta$ is an isometric $2$-action. Furthermore, the latter fact also implies that the exponential maps $\\textnormal{exp}^{(2)}$, $\\textnormal{exp}^{(1)}$, and $\\textnormal{exp}^{(0)}$ are equivariant local diffeomorphisms. As isometries preserve (horizontal) geodesics then by following the classical proofs of the equivariant tubular neighborhood theorem in \\cite[VI. Thm. 2.2]{B} and \\cite[Thm. 4.4]{IK,K} with the proof of weakly linearization theorem given in \\cite[Thm. 5.11]{dHF} we easily obtain:\n\n\\begin{proposition}[Equivariant weakly groupoid linearization]\\label{Lin1}\nLet $G_1\\rightrightarrows G_0$ be a Lie $2$-group, with $G_1$ compact, acting by isometries on $(X_1\\rightrightarrows X_0,\\eta)$. Then there exists an equivariant weakly linearization of $X_1\\rightrightarrows X_0$ around any $G_0$-invariant saturated submanifold $S$ in $X_0$.\n\\end{proposition}\nObserve that we may get a similar conclusion by requiring either that $G_1\\rightrightarrows G_0$ proper (or $s$-proper) and $G_0$ compact or else that $G_1\\rightrightarrows G_0$ acts properly.\n\n\n\\begin{definition}\\label{DefSlice}\nLet $\\theta$ be a $2$-action of $G_1\\rightrightarrows G_0$ on $X_1\\rightrightarrows X_0$ with $G_0$ acting freely. A \\emph{groupoid slice} at $x_0\\in X_0$ is defined to be a Lie subgroupoid $S_{1_{x_0}}\\rightrightarrows S_{x_0}$ of $X_1\\rightrightarrows X_0$ such that $S_{x_0}$ and $S_{1_{x_0}}$ are standard slices at $x_0$ and $1_{x_0}$, respectively.\n\\end{definition}\nThe following is a straightforward result.\n\\begin{lemma}\\label{IsoOrbit}\nTake any $x_0\\in X_0$. There are natural structures of:\n\\begin{itemize}\n\\item Lie 2-subgroup between the $G$-isotropy groups $\\textnormal{Iso}_{G_1}(1_{x_0})\\rightrightarrows \\textnormal{Iso}_{G_0}(x_0)$, and\n\\item Lie subgroupoid between the $G$-orbits $G_1\\cdot 1_{x_0}\\rightrightarrows G_0\\cdot x_0$.\n\\end{itemize}\n\\end{lemma}\nIt is important to mention that we asked $G_0$ to act freely in order to have a well defined groupoid composition on the Lie groupoids defined in the previous lemma. Let us state a version of the Slice Theorem in our context. Namely:\n\n\\begin{proposition}[Groupoid slice]\\label{SliceThm}\nIf $\\theta$ is a free and proper $2$-action of a Lie $2$-group $G_1\\rightrightarrows G_0$ on a proper groupoid $X_1\\rightrightarrows X_0$ then there exists a groupoid slice $S_{1_{x_0}}\\rightrightarrows S_{x_0}$ at each $x_0\\in X_0$.\n\n\\end{proposition}\n\\begin{proof}\nConsider the induced $2$-action of $\\textnormal{Iso}_{G_1}(1_{x_0})\\rightrightarrows \\textnormal{Iso}_{G_0}(x_0)$ on $X_1\\rightrightarrows X_0$. By the properness of $X_1\\rightrightarrows X_0$ and by applying Theorem \\ref{Existence2} together with Remark \\ref{Rmk4}, we may fix a $2$-metric on $X_2$ and use it to construct another $2$-metric on $X_2$ in such a way $\\textnormal{Iso}_{G_1}(1_{x_0})\\rightrightarrows \\textnormal{Iso}_{G_0}(x_0)$ acts isometrically on $X_1\\rightrightarrows X_0$. As in the classical case \\cite[Thm. 3.49]{AB}, we define $S_{x_0}$ by setting $S_{x_0}=\\exp^{(0)}_{x_0}(B_\\epsilon(0))$ where $B_\\epsilon(0)$ is an open ball of radius $\\epsilon>0$ around the origin in the normal space $\\nu_{x_0} (G_0 \\cdot x_0)$ to the $G_0$-orbit through $x_0$ (normal domain).\n\nAs $G_1\\cdot 1_{x_0}\\rightrightarrows G_0\\cdot x_0$ is a Lie subgroupoid of $X_1\\rightrightarrows X_0$ we have a well defined Lie subgroupoid $\\nu(G_1\\cdot 1_{x_0})\\rightrightarrows \\nu(G_0\\cdot x_0)$ of $TX_1\\rightrightarrows TX_0$ so that we may also consider the Lie groupoid $V_{B_\\epsilon(0)}\\rightrightarrows B_\\epsilon(0)$ where $V_{B_\\epsilon(0)}=\\overline{ds}_{1_{x_0}}^{-1}(B_\\epsilon(0))\\cap\\overline{dt}_{1_{x_0}}^{-1}(B_\\epsilon(0))$. By shrinking $B_\\epsilon(0)$ if necessary we may assume that $V_{B_\\epsilon(0)}$ is an open ball around the origin in the normal space $\\nu_{1_{x_0}} (G_1\\cdot 1_{x_0})$ to the $G_1$-orbit through $1_{x_0}$ on which $\\exp^{(1)}_{1_{x_0}}$ is well defined. Therefore, we now set $S_{1_{x_0}}= \\exp^{(1)}_{1_{x_0}}(V_{B_\\epsilon(0)})$. Hence, by arguing with similar arguments as those used to prove the multiplicative property of the exponential maps associated to a Riemannian $2$-metric in \\cite{dHF} together with the equivariant property these exponential maps have, we conclude that $S_{1_{x_0}}\\rightrightarrows S_{x_0}$ is the Lie subgroupoid of $X_1\\rightrightarrows X_0$ we are looking for.\n\\end{proof}\nThe previous result suggests that we may somehow prove a kind of $(G_1\\rightrightarrows G_0)$-equivariant ``Tubular Neighborhood Theorem'' for the $G$-orbit groupoid $G_1\\cdot 1_{x_0}\\rightrightarrows G_0\\cdot x_0$. To do so, we need to consider the following construction which has been defined in \\cite{HOV} (a particular case of such a construction can be found in \\cite[Lem. 9.1.2]{hsz}). \n\\begin{remark}\\label{AssociatedBundle}\nLet $\\pi: (P_1\\rightrightarrows P_0)\\to (X_1\\rightrightarrows X_0)$ be a  groupoid principal $2$-bundle with structural Lie $2$-group $G_1\\rightrightarrows G_0$. Assume that there exists a left $2$-action of $G_1\\rightrightarrows G_0$ over another Lie groupoid $F_1\\rightrightarrows F_0$. Given this data we can construct two associated fiber bundles $E_j:= P_j\\times_{G_j}F_j$ over $X_j$ for $j=0,1$. These are defined as the quotient spaces $(P_j\\times F_j)\/G_j$ with respect to the actions $g_j\\cdot (p_j,f_j)=(p_jg_j^{-1},g_jf_j)$, for all $g_j\\in G_j$, $p_j\\in P_j$ and $f_j\\in F_j$, together with projections $\\overline{\\pi}_j([p_j,f_j])=\\pi_j(p_j)$ onto $X_j$. From \\cite{HOV} we know that there exists a natural Lie groupoid structure $E_1\\rightrightarrows E_0$ for which the projection $\\overline{\\pi}: (E_1\\rightrightarrows E_0)\\to (X_1\\rightrightarrows X_0)$ becomes a Lie groupoid fibration. The source and target maps are the obvious ones, namely:\n$$s_E([p,f])=[s_P(p),s_F(f)]\\qquad\\textnormal{and}\\qquad t_E([p,f])=[t_P(p),t_F(f)],$$\nand the groupoid composition is defined as follows. If $s_E([p,f])=t_E([q,l])$ then there is $g_0\\in G_0$ such that $(s_P(p)g_0^{-1},g_0s_F(f))=(t_P(q),t_F(l))$. So, we set\n$$m_E([p,f],[q,l])=[(pg^{-1})\\ast q,(gf)\\ast l],$$\nfor some $g$ inside the $s_G$-fiber at $g_0$. The latter equality does not depend on the choice of $g\\in s_G^{-1}(g_0)$.\n\\end{remark}\n\nLet $\\theta$ be a free $2$-action of $G_1\\rightrightarrows G_0$ on $X_1\\rightrightarrows X_0$ and consider the classical $G$-invariant tubular neighborhoods $\\textnormal{Tub}(G_1\\cdot 1_{x_0})=\\theta^1(G_1,S_{1_{x_0}})$ and $\\textnormal{Tub}(G_0\\cdot x_0)=\\theta^0(G_0,S_{x_0})$ of $G_1\\cdot 1_{x_0}$ and $G_0\\cdot x_0$, respectively. It is clear that as consequence of Proposition \\ref{SliceThm} we have that $\\textnormal{Tub}(G_1\\cdot 1_{x_0})\\rightrightarrows \\textnormal{Tub}(G_0\\cdot x_0)$ is a Lie subgroupoid of $X_1\\rightrightarrows X_0$ and that the $2$-action $\\theta$ restricts well over it. Furthermore, $\\textnormal{Tub}(G_1\\cdot 1_{x_0})\\rightrightarrows \\textnormal{Tub}(G_0\\cdot x_0)$ determines an open Lie groupoid neighborhood of the $G$-orbit groupoid $G_1\\cdot 1_{x_0}\\rightrightarrows G_0\\cdot x_0$. \n\nFrom Lemma \\ref{IsoOrbit}, Proposition  \\ref{SliceThm}, the quotient construction described in Proposition \\ref{PQuotient}, and Remark \\ref{AssociatedBundle}, we easily deduce that:\n\\begin{lemma}\nThere exists a principal groupoid $2$-bundle $\\pi:(G_1\\rightrightarrows G_0)\\to (G_1\/\\textnormal{Iso}_{G_1}(1_{x_0})\\rightrightarrows G_0\/\\textnormal{Iso}_{G_0}(x_0))$ with structural Lie $2$-group $\\textnormal{Iso}_{G_1}(1_{x_0})\\rightrightarrows \\textnormal{Iso}_{G_0}(x_0)$, yielding an associated groupoid fibration $$(G_1\\times_{\\textnormal{Iso}_{G_1}(1_{x_0})}S_{1_{x_0}}\\rightrightarrows G_0\\times_{\\textnormal{Iso}_{G_0}(x_0)}S_{x_0})\\to (G_1\/\\textnormal{Iso}_{G_1}(1_{x_0})\\rightrightarrows G_0\/\\textnormal{Iso}_{G_0}(x_0)),$$ \nwith groupoid fiber $S_{1_{x_0}}\\rightrightarrows S_{x_0}$.\n\\end{lemma}\nWe are now in conditions to state our version of the Equivariant Tubular Neighborhood Theorem in this context. \n\\begin{theorem}[Equivariant groupoid tubular neighborhood]\\label{ETubular}\nSuppose that $\\theta$ is a free and proper $2$-action of a Lie $2$-group $G_1\\rightrightarrows G_0$ on a proper groupoid $X_1\\rightrightarrows X_0$. For every $x_0\\in X_0$ there exists a $(G_1\\rightrightarrows G_0)$-equivariant Lie groupoid isomorphism\n$$\\Psi: (G_1\\times_{\\textnormal{Iso}_{G_1}(1_{x_0})}S_{1_{x_0}}\\rightrightarrows G_0\\times_{\\textnormal{Iso}_{G_0}(x_0)}S_{x_0}) \\xrightarrow[]{\\cong} (\\textnormal{Tub}(G_1\\cdot 1_{x_0})\\rightrightarrows \\textnormal{Tub}(G_0\\cdot x_0)).$$\n\\end{theorem}\n\\begin{proof}\nThe left $2$-action of $G_1\\rightrightarrows G_0$ on $G_1\\times_{\\textnormal{Iso}_{G_1}(1_{x_0})}S_{1_{x_0}}\\rightrightarrows G_0\\times_{\\textnormal{Iso}_{G_0}(x_0)}S_{x_0}$ we will consider here is the one given by $\\overline{g_j}\\cdot [g_j,f_j]=[\\overline{g_j}g_j,f_j]$ for all $\\overline{g_j},g_j\\in G_j$ and $f_j\\in S_j$. Here $S_1=S_{1_{x_0}}$ and $S_0=S_{x_0}$.\nThe Lie groupoid isomorphism $\\Psi$ is defined as \n$$\\Psi^j([g_j,f_j])=\\theta^j(g_j,f_j).$$\nAs consequence of \\cite[Thm. 3.57]{AB} we only have to check that $\\Psi$ defines indeed a Lie groupoid morphism. By using the structural maps defined in Remark \\ref{AssociatedBundle} we have that\n$$(\\Psi^0\\circ s_E)([g,f])=\\Psi^0([s_G(g),s_X(f)])=s_G(g)s_X(f)=s_X(gf)=(s_X\\circ \\Psi^1)([g,f]).$$\nWe can similarly obtain that $\\Psi^0\\circ t_E=t_X\\circ \\Psi^1$. Moreover, by applying Formula \\eqref{MultAction} we get\n\\begin{eqnarray*}\n\\Psi^1([g,f]\\ast [h,l]) &=& \\Psi^1([(g\\overline{g}^{-1})\\ast h,(\\overline{g}f)\\ast l])=((g\\overline{g}^{-1})\\ast h)\\cdot((\\overline{g}f)\\ast l)\\\\\n&=& (gf)\\ast (hl)=\\Psi^1([g,f])\\ast \\Psi^1([h,l]).\n\\end{eqnarray*}\nHence, the result follows as desired.\n\\end{proof}\nWe finish this section by briefly commenting that it is possible to define a groupoid $G$-orbit type ``stratification'' for $X_1\\rightrightarrows X_0$. We say that $x_0$ and $y_0$ in $X_0$ have the same $G$-\\emph{orbit type} if there exists a Lie groupoid isomorphism $\\Phi$ between $G_1\\cdot 1_{x_0}\\rightrightarrows G_0\\cdot x_0$ and $G_1\\cdot 1_{y_0}\\rightrightarrows G_0\\cdot y_0$ such that $\\Phi^1$ is $G_1$-equivariant. This automatically implies that $\\Phi^0$ is $G_0$-equivariant. It is clear that this $G$-orbit type requirement defines an equivalent relation $\\sim$  on $X_0$ for which we denote by $M_{x_0}^\\sim$ the equivalent class at $x_0\\in X_0$ associated to such a relation. We claim that $M_{x_0}^\\sim$ is saturated in $X_0$, that is, $s_X^{-1}(M_{x_0}^\\sim)=t_X^{-1}(M_{x_0}^\\sim)$. If $p\\in s_X^{-1}(M_{x_0}^\\sim)$ then $s_X(p)\\sim x_0$ so that there is a $(G_1\\rightrightarrows G_0)$-equivariant isomorphism between $G_1\\cdot 1_{s_X(p)}\\rightrightarrows G_0\\cdot s_X(p)$ and $G_1\\cdot 1_{x_0}\\rightrightarrows G_0\\cdot x_0$. It is simple to check that $\\Phi_p$ defined by $\\Phi_p^1(g1_{t_X(p)})=g1_{s_X(p)}$ and $\\Phi_p^0(g_0t_X(p))=g_0s_X(p)$ defines another $(G_1\\rightrightarrows G_0)$-equivariant isomorphism between $G_1\\cdot 1_{t_X(p)}\\rightrightarrows G_0\\cdot t_X(p)$ and $G_1\\cdot 1_{s_X(p)}\\rightrightarrows G_0\\cdot s_X(p)$ so that by taking $\\Phi\\circ \\Phi_p$ we conclude that $t_X(p)\\sim x_0$. The other inclusion may be similarly verified. Thus, by setting $M_{1_{x_{0}}}^\\sim=s_X^{-1}(M_{x_0}^\\sim)=t_X^{-1}(M_{x_0}^\\sim)$ we obtain a collection of topological groupoids $\\lbrace M_{1_{x_{0}}}^\\sim \\rightrightarrows M_{x_0}^\\sim\\rbrace_{x_0\\in X_0}$ which somehow stratifies the Lie groupoid $X_1\\rightrightarrows X_0$. The previous observation suggests that by combining classical ideas from \\cite[s. 3.5]{AB} with some of the results obtained in this section it could be possible to show that each of the $M_{1_{x_{0}}}^\\sim \\rightrightarrows M_{x_0}^\\sim$ is a honest Lie subgroupoid. Nevertheless, the local $G$-orbit type notion from the classical case does not extended directly in our case. \\emph{We conjecture that our guess is true}.\n\\begin{comment}\n\\subsection{Linearization of equivariant groupoid fibrations}\nBy a \\emph{fibration} over $X_1\\rightrightarrows X_0$ we mean a Lie groupoid morphism $\\pi:(E_1\\rightrightarrows E_0)\\to (X_1\\rightrightarrows X_0)$, from another Lie groupoid $E_1\\rightrightarrows E_0$ such that:\n\\begin{enumerate}\n\t\\item the base map $\\pi_0:E_0\\to X_0$ is a surjective submersion, and \n\t\\item the map $\\tilde{\\pi}:E_1\\to X_1{}_s\\times_{\\pi_0} E_0$,  $e_1\\mapsto (\\pi_1(e_1),s_E(e_1))$, is a surjective submersion.\n\\end{enumerate}\nIt follows from (2) that the map $\\pi_1:E_1\\to X_1$ is also a surjective submersion. The \\emph{fiber} over a point $x_0\\in X_0$ to be the Lie subgroupoid:\n\\[ \\pi^{-1}(x_0):=(\\pi^{-1}_1(1_{x_0})\\rightrightarrows \\pi^{-1}_0(x_0))\\subset E.\\] \n\nSuppose that $G_1\\rightrightarrows G_0$ is a Lie $2$-group action on both $E_1\\rightrightarrows E_0$ and $X_1\\rightrightarrows X_0$. We say that the fibration $\\pi$ is $(G_1\\rightrightarrows G_0)$-equivariant if $\\pi_1$ is $G_1$-equivariant and $\\pi_0$ is $G_0$-equivariant.\n\nA \\emph{cleavage} of $\\pi$ is a smooth map $\\sigma:X_1 {}_{s_X}\\times_{\\pi_0}E_0\\to E_1$ which is section of the surjective submersion $\\tilde{\\pi}$. Note that we have an induced left action of the Lie group $G_1 {}_{s_G}\\times_{\\textnormal{id}}G_0$ (group structure induced by the direct product of groups) on $X_1 {}_{s_X}\\times_{\\pi_0}E_0$ given by $(g,g_0)(x,e_0)=(gx,g_0e_0)$. Therefore, it follows that $\\tilde{\\pi}(ge)=(g,s_G(g))\\tilde{\\pi}(e)$ for all $g\\in G_1$. With this in mind it is reasonable to say that a cleavage $\\sigma$ is equivariant if $\\sigma(gx,g_0e_0)=g\\sigma(x,e_0)$ for all $(g,g_0)\\in G_1 {}_{s_G}\\times_{\\textnormal{id}}G_0$. The cleavage is said to be \\emph{unital} if it preserves identities, meaning that \n$\\sigma(1_{\\pi_0(e_0)},e_0) = 1_{e_0}$, and is defined to be \\emph{flat} if it is closed under multiplication, namely $\\sigma(x\\ast y,e_0) =\n\\sigma(x,t_E(\\sigma(y,e_0)))\\ast \\sigma(y,e_0)$. A fibration admitting a flat unital cleavage is called a \\emph{split fibration}.\n\nWe want to adapt the results from \\cite{dHF2} about linearization of groupoid fibrations to the equivariant setting described above. Note that if $S$ is a $G_0$-invariant saturated submanifold in $X_0$ then $\\pi_0^{-1}(S)$ is so in $E_0$.\n\n\\begin{definition}\nWe say that $\\pi$ is \\emph{equivariant linearizable} around a $G_0$-invariant saturated submanifold $S\\subset X_0$ if there are equivariant weakly linearizations $\\widetilde{\\phi}$ of $E_1\\rightrightarrows E_0$ around $\\widetilde{S}=\\pi_0^{-1}(S)$ and $\\phi_X$ of $X_1\\rightrightarrows X_0$ around $S$ forming a commutative diagram\n$$\\xymatrix{\n\t\\nu(E_{\\widetilde{S}})\\supset \\widetilde{U} \\ar[r]^{\\widetilde{\\phi}} \\ar@{>}[d]_{\\overline{d\\pi}} & \\widetilde{V}\\subset E \\ar@{>}[d]^{\\pi}\\\\\n\t\\nu(X_S)\\supset U\\ar[r]_{\\phi}  & V\\subset X\n}.$$\n\\end{definition}\n\nFrom now on let us consider a Lie $2$-group $G_1\\rightrightarrows G_0$ with $G_1$ compact. Suppose that $\\pi$ is a Riemannian submersion, that is, both $E_1\\rightrightarrows E_0$ and $X_1\\rightrightarrows X_0$ can be equipped with $2$-metrics for which $\\pi^{(2)}$ becomes a Riemannian submersion. As consequence of Proposition \\ref{Existence2} we have that both $E_1\\rightrightarrows E_0$ and $X_1\\rightrightarrows X_0$ can be equipped with $(G_1\\rightrightarrows G_0)$-invariant $2$-metrics $\\widetilde{\\eta}$ and $\\eta$ in such a way $\\pi$ is still a Riemannian submersion.\n\\begin{lemma}\\label{Lin2}\nIf $\\pi$ is a Riemannian submersion then the equivariant exponential maps of $\\widetilde{\\eta}$ and $\\eta$ define an equivariant linearization of $\\pi$ around any $G_0$-invariant saturated submanifold $S$.\n\\end{lemma}\n\\begin{proof}\nAfter applying Proposition \\ref{Lin1} it follows that the proof of \\cite[Prop. 4.2.2]{dHF2} goes through in this context since $\\pi$ is formed by equivariant maps.\n\\end{proof}\nOn the one hand, let us consider the Lie groupoid determined by the homotopy fiber product $E'=X\\widetilde{\\times}_X E$ over $\\textnormal{id}_X$ and $\\pi$ which always exists since $\\pi$ is a groupoid fibration. The manifolds of objects and arrows of $E'$ are respectively given by\n$$E'_0=\\lbrace (x_0,k,e_0)\\in X_0\\times X_1\\times E_0:\\ \\pi_0(e_0)\\xleftarrow[]{\\it k}x_0\\rbrace,$$\nand \n$$E'_1=\\lbrace (x,k,e)\\in X_1\\times X_1\\times E_1:\\ s_X(p)=s_X(k),\\ \\pi_0(s_E(e))=t_X(k)\\rbrace.$$\nIts structural maps are defined in an obvious way by using those from $E$ and $X$. On the other hand, by using the Lie $2$-group $G_1\\rightrightarrows G_0$ we may obtain another Lie $2$-group determined by the homotopy fiber product $G'=G\\widetilde{\\times}_G G$ over the identity morphism $\\textnormal{id}_G$. As above, the manifolds of objects and arrows of $G'$ are respectively given by\n$$G'_0=\\lbrace (g_0,n,h_0)\\in G_0\\times G_1\\times G_0:\\ h_0\\xleftarrow[]{\\it n}g_0\\rbrace,$$\nand \n$$G'_1=\\lbrace (g,n,h)\\in G_1\\times G_1\\times G_1:\\ s_G(g)=s_G(n),\\ s_G(h)=t_G(n)\\rbrace.$$\nThe group structure in both $G'_0$ and $G'_1$ is the induced one from the direct product of groups so that the structural maps of $G'$ are clearly Lie group homomorphisms.\n\\begin{lemma}\\label{Lin3}\nThe $2$-actions of $G_1\\rightrightarrows G_0$ on both $E_1\\rightrightarrows E_0$ and $X_1\\rightrightarrows X_0$ induce a natural $2$-action of $G'_1\\rightrightarrows G'_0$ on $E'_1\\rightrightarrows E'_0$.\n\\end{lemma}\n\\begin{proof}\nIt is straightforward to check that the expressions\n$$(g_0,n,h_0)\\cdot (x_0,k,e_0)=(g_0x_0,nk,h_0e_0)\\quad\\textnormal{and}\\quad(g,n,h)\\cdot (x,k,e)=(gx,nk,he),$$\ndefine a $2$-action of $G'_1\\rightrightarrows G'_0$ on $E'_1\\rightrightarrows E'_0$ since $\\pi$ is $(G_1\\rightrightarrows G_0)$-equivariant. \n\\end{proof}\nThe morphism $\\pi$ has a canonical factorization \n$$\\xymatrix{\n\t& E' \\ar[dr]^{\\pi'}  & \\\\\n\tE \\ar@{>}[ur]^{\\iota}\\ar[r]_{\\pi} & & X\n}$$\nwhere the maps $\\iota$ and $\\pi'$ are respectively given, on objects and arrows, by $\\iota(e_0)=(\\pi_0(e_0), u_X(\\pi_0(e_0)),e_0)$, $\\iota(e)=(\\pi_1(e), u_X(\\pi_0(s_E(e)),e)$ and $\\pi'(x_0,k,e_0)=x_0$, $\\pi'(x,k,e)=x$.\n\nIf we apply the same factorization to $\\textnormal{id}_G$ we get a morphism of Lie $2$-groups $\\iota_G:G\\to G'$ defined as $\\iota_G(g_0)=(g_0,u_G(g_0),g_0)$ on objects and $\\iota_G(g)=(g,u_G(s_G(g)),g)$ on arrows. By using the $2$-action from Lemma \\ref{Lin3} we obtain that $\\iota$ is equivariant of type $\\iota_G$, meaning that $\\iota(g_0e_0)=\\iota_G(g_0)\\iota(e_0)$ and $\\iota(ge)=\\iota_G(g)\\iota(e)$ since $\\pi$ is $(G_1\\rightrightarrows G_0)$-equivariant. Analogously, $\\pi'((g_0,n,h_0)\\cdot (x_0,k,e_0))=\\textnormal{id}'_G(g_0,n,h_0)\\pi'(x_0,k,e_0)$ and $\\pi'((g,n,h)\\cdot (x,k,e))=\\textnormal{id}'_G(g,n,h)\\pi'(x,k,e)$.\n\n\nFrom \n\n\\textcolor{red}{I got stuck. I need to review the ideas I initially had}.\n\\end{comment}\n\n\n\n\n\\subsection{Orthogonal Lie 2-groups}\nLet $G_1\\rightrightarrows G_0$ be a Lie 2-group. Consider the pairs $L=(L^1,L^0)$ and $R=(R^1,R^0)$ where $L^j$ and $R^j$ for $j=0,1$ are respectively the actions of $G_j$ on itself determined by left and right multiplications. It is simple to check that as consequence of Identity \\eqref{MultiProduct} and the fact that the structural maps of $G_1\\rightrightarrows G_0$ are Lie group homomorphisms it follows that $L$ and $R$ determine left $2$-actions of $G_1\\rightrightarrows G_0$ on itself.\n\\begin{definition}\nA Lie $2$-group $G_1\\rightrightarrows G_0$ is said to be \\emph{orthogonal} if it may be equipped with a $1$-metric for which both $L$ and $R$ are isometric $2$-actions. Such a $1$-metric will be called \\emph{bi-invariant}.\n\\end{definition}\nWe will think of $1$-metrics on a Lie $2$-algebra $\\mathfrak{g}_1\\rightrightarrows \\mathfrak{g}_0$ as pairs of inner products $\\langle \\cdot,\\cdot\\rangle=(\\langle \\cdot,\\cdot\\rangle^{(1)},\\langle \\cdot,\\cdot\\rangle^{(0)})$ verifying the required conditions of $1$-metric. Let $G_1\\rightrightarrows G_0$ be a Lie 2-group with respective Lie 2-algebra $\\mathfrak{g}_1\\rightrightarrows \\mathfrak{g}_0$. We denote by $\\textnormal{Ad}=(\\textnormal{Ad}^1,\\textnormal{Ad}^0)$ the $2$-action of $G_1\\rightrightarrows G_0$ on $\\mathfrak{g}_1\\rightrightarrows \\mathfrak{g}_0$ determined by the adjoint actions $\\textnormal{Ad}^j$ of $G_j$ on $\\mathfrak{g}_j$ for $j=0,1$. This $2$-action will be called \\emph{adjoint $2$-action} of $G_1\\rightrightarrows G_0$. \n\n\\begin{proposition}\\label{Bi-invariant1}\nA Lie $2$-group $G_1\\rightrightarrows G_0$ is orthogonal if and only if there exists a $1$-metric $\\langle \\cdot,\\cdot\\rangle$ on its Lie algebra $\\mathfrak{g}_1\\rightrightarrows \\mathfrak{g}_0$ for which the adjoint $2$-action is by linear isometries.\n\\end{proposition}\n\\begin{proof}\nIt is clear that if $\\eta$ is a bi-invariant $1$-metric then $\\eta_e=(\\eta^1_{e_1},\\eta^0_{e_0})$, where $e_j$ is the identity element in $G_j$, defines a $1$-metric on $\\mathfrak{g}_1\\rightrightarrows \\mathfrak{g}_0$ for which the adjoint $2$-action is by linear isometries. Conversely, given such a $\\langle \\cdot,\\cdot\\rangle$, we define $\\eta$ on $G_1\\rightrightarrows G_0$ by setting\n$$\\eta^{(1)}_g(v,w)=\\langle d(L^1_{g^{-1}})_g(v),d(L^1_{g^{-1}})_g(w)\\rangle^{(1)}.$$\nThe metric $\\eta^{(0)}$ has a similar defining formula but using $L^0$ and $\\langle \\cdot,\\cdot\\rangle^{(0)}$ instead of $L^1$ and $\\langle \\cdot,\\cdot\\rangle^{(1)}$. It is clear that $\\eta^{(1)}$ is bi-invariant. Therefore, it remains to prove that $\\eta$ defines indeed a $1$-metric. Firstly, for $g\\in G_1$ and $v,w\\in T_g G_1$ we have\n\\begin{eqnarray*}\ni^\\ast_g\\eta^{(1)}(v,w) & = & \\langle d(L^1_{i(g)^{-1}})_{i(g)}(di_g(v)),d(L^1_{i(g)^{-1}})_{i(g)}(di_g(w))\\rangle^{(1)}\\\\\n&=&  \\langle d(L^1_{i(g^{-1})}\\circ i)_g(v),d(L^1_{i(g^{-1})}\\circ i)_g(w)\\rangle^{(1)}\\\\\n& = & \\langle d(i\\circ L^1_{g^{-1}})_g(v),d(i\\circ L^1_{g^{-1}})_g(w)\\rangle^{(1)}\\\\\n&=&\\langle di_{e_1}(d(L^1_{g^{-1}})_g(v)), di_{e_1}(d(L^1_{g^{-1}})_g(v))\\rangle^{(1)}=\\eta^{(1)}_g(v,w).\n\\end{eqnarray*}\nSecondly, let $g_0\\in G_0$ and $v,w\\in T_{g_0}G_0$. It is clear that there are $\\tilde{v},\\tilde{w}\\in \\textnormal{ker}(ds(g))^{\\perp_{\\eta^{(1)}}}$ with $s(g)=g_0$ such that $ds_g(\\tilde{v})=v$ and $ds_g(\\tilde{w})=w$. Thus\n\\begin{eqnarray*}\n(s_\\ast \\eta^{(1)})_{g_0}(v,w) & = & \\eta^{(1)}_g(\\tilde{v},\\tilde{w}) = \\langle d(L^1_{g^{-1}})_g(\\tilde{v}),d(L^1_{g^{-1}})_g(\\tilde{w})\\rangle^{(1)}\\\\\n& = & \\langle ds_{e_1}(d(L^1_{g^{-1}})_g(\\tilde{v})), ds_{e_1}(d(L^1_{g^{-1}})_g(\\tilde{w}))\\rangle^{(0)}\\\\\n&=& \\langle d(s\\circ L^1_{g^{-1}})_g(\\tilde{v}),d(i\\circ L^1_{g^{-1}})_g(\\tilde{w})\\rangle^{(0)}\\\\\n&=& \\langle d(L^0_{g_0^{-1}}\\circ s)_g(\\tilde{v}),d(L^0_{g_0^{-1}}\\circ s)_g(\\tilde{w})\\rangle^{(0)}\\\\\n&=& \\langle d(L^0_{g_0^{-1}})_{g_0}(v),d(L^0_{g_0^{-1}})_{g_0}(v)\\rangle^{(0)}=\\eta^{(0)}_{g_0}(v,w).\n\\end{eqnarray*}\nAnalogously, $t_\\ast \\eta^{(1)}=\\eta^{(0)}$. So, the result follows.\n\\end{proof}\nFrom now on we assume that the Lie groups we are working with are connected. It is well known that bi-invariant metrics on a Lie group are in one-to-one correspondence with inner products on its Lie algebra for which the adjoint representation determines infinitesimal isometries \\cite{Me,Mi}. A Lie $2$-algebra $\\mathfrak{g}_1\\rightrightarrows \\mathfrak{g}_0$ is said to be \\emph{orthogonal} if it admits a $1$-metric $\\langle \\cdot,\\cdot\\rangle=(\\langle \\cdot,\\cdot\\rangle^{(1)},\\langle \\cdot,\\cdot\\rangle^{(0)})$ for which the adjoint representation $\\textnormal{ad}^1:\\mathfrak{g}_1\\to \\textnormal{Der}(\\mathfrak{g}_1)$ acts by infinitesimal isometries on $(\\mathfrak{g}_1,\\langle \\cdot,\\cdot\\rangle^{(1)})$. Therefore, as consequence of Proposition \\ref{Bi-invariant1} and \\cite[Lem. 7.2]{Mi} we get:\n\\begin{corollary}\\label{MilnorCharacterization}\nA Lie $2$-group is orthogonal if and only if its Lie algebra $\\mathfrak{g}_1\\rightrightarrows \\mathfrak{g}_0$ is orthogonal.\n\\end{corollary}\nFrom Theorem \\ref{Existence2} we get that:\n\\begin{corollary}\nEvery proper Lie $2$-group with either $G_1$ or $G_0$ compact can be endowed with a bi-invariant $1$-metric.\n\\end{corollary}\n\n\\begin{remark}\nSuppose that we have a crossed module of Lie algebras $(\\mathfrak{g},\\mathfrak{h},\\partial,\\mathcal{L})$ which is $\\mathcal{L}$-orthogonal in the sense of \\cite{Ba,FMM}. That is, there exist inner products $\\langle \\cdot,\\cdot\\rangle_\\mathfrak{g}$ and $\\langle \\cdot,\\cdot\\rangle_\\mathfrak{h}$ such that $\\mathcal{L}$ acts by infinitesimal isometries on $(\\mathfrak{h},\\langle \\cdot,\\cdot\\rangle_\\mathfrak{h})$ and the adjoint representation of $\\mathfrak{g}$ acts by infinitesimal isometries on $(\\mathfrak{g},\\langle \\cdot,\\cdot\\rangle_\\mathfrak{g})$. Note that this directly implies that the adjoint representation of $\\mathfrak{h}$ acts by infinitesimal isometries on $(\\mathfrak{h},\\langle \\cdot,\\cdot\\rangle_\\mathfrak{h})$. Recall that the associated Lie $2$-algebra constructed with the crossed module data has $\\mathfrak{g}_1=\\mathfrak{h}\\rtimes \\mathfrak{g}$ with Lie algebra structure provided by the semi-direct product with respect to $\\mathcal{L}$. Therefore, a straightforward computation allows us to conclude that the adjoint representation of $\\mathfrak{g}_1$ acts by infinitesimal isometries with respect to $\\langle \\cdot,\\cdot\\rangle_\\mathfrak{h}+\\langle \\cdot,\\cdot\\rangle_\\mathfrak{g}$ if and only if $\\mathcal{L}=0$. As consequence, there is no a canonical correspondence between our notion of orthogonal Lie $2$-algebras and the notion of $\\mathcal{L}$-orthogonal crossed module of Lie algebras which is known in the literature.\n\\end{remark}\nGiven an orthogonal Lie $2$-algebra $\\mathfrak{g}_1\\rightrightarrows \\mathfrak{g}_0$ we may split $\\mathfrak{g}_1$ as a direct sum of ideals $\\mathfrak{g}_1=\\mathfrak{h}\\oplus \\mathfrak{h}^{\\perp_1}$ where $\\mathfrak{h}=\\textnormal{ker}(s)$. As the unit map $u$ is a canonical bisection we would expect that $u(x)\\in \\mathfrak{h}^{\\perp_1}$ for all $x\\in \\mathfrak{g}_0$ but, however, this is not true in general unless we assume ``non-canonical\" identifications. We say that an orthogonal Lie $2$-algebra is \\emph{trivial} if the latter condition holds true. This notion comes up by the following simple result.\n\n\\begin{lemma}\\label{0-orthogonal}\nIf $\\mathfrak{g}_1\\rightrightarrows \\mathfrak{g}_0$ is a trivial orthogonal Lie $2$-algebra then $\\textnormal{ad}^1_{u(x)}|_\\mathfrak{h}=0$ for all $x\\in \\mathfrak{g}_0$. In consequence, $\\mathfrak{h}$ is abelian and $\\textnormal{im}(t|_\\mathfrak{h})\\subseteq \\mathfrak{z}(\\mathfrak{g}_0)$.\n\\end{lemma}\n\\begin{proof}\nOn the one hand, note that for any $y\\in \\mathfrak{h}$ and $z\\in \\mathfrak{g}$ one gets\n$$\\langle \\textnormal{ad}^1_{u(x)}(y),z\\rangle^{(1)}=\\langle u(x),\\textnormal{ad}^1_{y}(z)\\rangle^{(1)}=0,$$\nsince $\\mathfrak{h}$ is an ideal. Thus, as $\\langle \\cdot,\\cdot\\rangle^{(1)}$ is nondegenerate we get that $\\textnormal{ad}^1_{u(x)}|_\\mathfrak{h}=0$ for all $x\\in \\mathfrak{g}_0$. On the other hand, for all $y,y'\\in \\mathfrak{h}$ and $x\\in \\mathfrak{g}$ it follows that $[y,y']=\\textnormal{ad}^1_{u(t(y))}(y')=0$ and $0=t([u(x),y])=[x,t(y)]$ since $t$ is a Lie algebra homomorphism.\n\\end{proof}\n\nMotivated by the previous result we set our notion of $0$-orthogonal crossed module. Namely:\n\\begin{definition}\nA crossed module of Lie algebras $(\\mathfrak{g},\\mathfrak{h},\\partial,0)$ is called 0-\\emph{orthogonal} if there exist inner products $\\langle \\cdot,\\cdot\\rangle_\\mathfrak{g}$ and $\\langle \\cdot,\\cdot\\rangle_\\mathfrak{h}$ such that the adjoint representation of $\\mathfrak{g}$ acts by infinitesimal isometries on $(\\mathfrak{g},\\langle \\cdot,\\cdot\\rangle_\\mathfrak{g})$ and the fibers of the canonical projection $\\pi:\\mathfrak{g}\\to \\mathfrak{g}\/\\textnormal{im}(\\partial)$ are equidistant.\n\\end{definition}\nNote that directly from the definition we get that $\\mathfrak{h}$ is abelian and $\\textnormal{im}(\\partial)\\subseteq \\mathfrak{z}(\\mathfrak{g})$. With this in mind we have:\n\\begin{proposition}\nThere exists a one-to-one correspondence between trivial orthogonal Lie $2$-algebras and $0$-orthogonal crossed modules of Lie algebras.\n\\end{proposition}\n\\begin{proof}\nIf $\\mathfrak{g}_1\\rightrightarrows \\mathfrak{g}_0$ is a trivial orthogonal Lie $2$-algebra then the result follows from Lemma \\ref{0-orthogonal} after setting $\\langle \\cdot,\\cdot\\rangle_\\mathfrak{g}=\\langle \\cdot,\\cdot\\rangle^{(0)}$ and $\\langle \\cdot,\\cdot\\rangle_\\mathfrak{h}=\\langle \\cdot,\\cdot\\rangle^{(1)}|_{\\mathfrak{h}\\times \\mathfrak{h}}$. Conversely, let us consider a $0$-orthogonal crossed module $(\\mathfrak{g},\\mathfrak{h},\\partial,0)$ and set $\\langle \\cdot,\\cdot\\rangle^{(1)}=\\langle \\cdot,\\cdot\\rangle_\\mathfrak{h}+\\langle \\cdot,\\cdot\\rangle_\\mathfrak{g}$ and $\\langle \\cdot,\\cdot\\rangle^{(0)}=\\langle \\cdot,\\cdot\\rangle_\\mathfrak{g}$. From the construction of the Lie $2$-algebra associated to $(\\mathfrak{g},\\mathfrak{h},\\partial,0)$ we easily see that $s$ is a linear Riemannian submersion. Now, on $\\mathfrak{g}_1=\\mathfrak{h}\\rtimes \\mathfrak{g}$ we get\n$$\\langle i(x,y),i(x',y')\\rangle^{(1)} = \\langle x,x'\\rangle_\\mathfrak{h}+\\langle y+\\partial(x),y'+\\partial(x')\\rangle_\\mathfrak{g}=\\langle x,x'\\rangle_\\mathfrak{h}+\\langle y,y'\\rangle_\\mathfrak{g}=\\langle (x,y),(x',y')\\rangle^{(1)},$$\nsince the fibers of the canonical projection $\\pi:\\mathfrak{g}\\to \\mathfrak{g}\/\\textnormal{im}(\\partial)$ are equidistant. As $t=s\\circ i$ it follows that $t$ is also a linear Riemannian submersion.\n\nThe Lie bracket on $\\mathfrak{g}_1$ is explicitly given as\n$$[(x,y),(x',y')]_0=(0,[y,y']_\\mathfrak{g}).$$\nHence, the adjoint representation of $\\mathfrak{g}_1$ acts by infinitesimal isometries with respect to $\\langle \\cdot,\\cdot\\rangle_\\mathfrak{h}+\\langle \\cdot,\\cdot\\rangle_\\mathfrak{g}$ since adjoint representation of $\\mathfrak{g}$ acts by infinitesimal isometries with respect to $\\langle \\cdot,\\cdot\\rangle_\\mathfrak{g}$.\n\\end{proof}\n\n\\begin{comment}\n\\textcolor{green}{Problem:} \\textcolor{blue}{If possible, interpret \\cite[Lem. 7.5]{Mi} with the aim of giving a complete characterization of orthogonal Lie $2$-groups}.\n\n\n\n\\textcolor{green}{Problem:} \\textcolor{blue}{Try to extend the construction called double orthogonal extension introduced in \\cite{MPh} by using the cohomology for Lie $2$-algebras of either \\cite{A} or \\cite{LSZ}}.\n\\end{comment}\n\n\n\\subsection{Some examples and applications}\nIt this short subsection we exhibit some toy examples and interesting applications in which isometric Lie $2$-actions naturally appear.\n\n\\begin{example}\nClassical isometric actions of Lie groups $G$ on Riemannian manifolds $M$ are recovered from isometric $2$-actions of unit Lie $2$-groups $G \\rightrightarrows G$ acting upon unit groupoids $M \\rightrightarrows M$.\n\\end{example}\n\n\\begin{example}\nLet $(X_1\\rightrightarrows X_0,\\eta)$ be a Riemannian groupoid and $(\\xi,v)$ be a complete multiplicative Killing vector field on $X_1\\rightrightarrows X_0$. Here we consider $\\xi$ a Killing vector field on $(X_1,\\eta^{(1)})$ and $v$ a Killing vector field on $(X_0,\\eta^{(0)})$ (as consequence of Lemma \\ref{Rmk1} it is actually enough to ask that $\\xi$ is Killing). From \\cite{MX} we know that the pair of flows defined by $(\\xi,v)$ determine global automorphisms on $X_1\\rightrightarrows X_0$ so that we get a well defined isometric $2$-action of $\\mathbb{R}\\rightrightarrows \\mathbb{R}$ on $(X_1\\rightrightarrows X_0,\\eta)$. In particular, the flow of the multiplicative vector field $((1_\\xi)_{X_1},\\xi_{X_0})$ formed by the fundamental vector fields of an isometric $2$-action determines another isometric $2$-action.\n\\end{example}\n\n\\begin{example}\\label{ExampleOrthogonal}\nLet $G$ be an orthogonal Lie group and $H\\leq G$ be a normal Lie subgroup. It is clear that $H$ acts on $G$ by left multiplication, leading to the action Lie groupoid $H\\times G\\rightrightarrows G$. Note that its space of arrows has a group structure, namely the semi-direct product by the\nconjugation action $C_g(h)=ghg^{-1}$ of $G$ on $H$ so that we get a well defined Lie $2$-group. More importantly, by applying the gauge trick construction behind Proposition 4.7 and Example 4.9 in \\cite{dHF} we conclude that it is possible to cook up a $1$-metric on $H\\rtimes G\\rightrightarrows G$ made out from the initial bi-invariant metric on $G$ in such a way it becomes an orthogonal Lie $2$-group.\n\\end{example}\n\n\n\n\n\\begin{example}\nLet $(M,\\mathcal{F})$ be a regular Riemannian foliation and consider a free and proper isometric foliated action $G\\times (M,\\mathcal{F})\\to (M,\\mathcal{F})$ of a Lie group $G$. From \\cite[Thm. 3.7]{GZ} it is known that the Lie $2$-group $G\\rtimes G\\rightrightarrows G$, as defined in Example \\ref{ExampleOrthogonal}, determines a canonical $2$-action on the holonomy groupoid $\\textnormal{Hol}(M,\\mathcal{F})\\rightrightarrows M$ which  extends the given action of $G$ on $M$. The Riemannian metric on $M$ completely determines a $0$-metric on $\\textnormal{Hol}(M,\\mathcal{F})\\rightrightarrows M$ which can be extended to a $1$-metric \\cite[Ex. 3.12]{dHF}. Thus, the fact that $G$ acts on $M$ isometrically implies that the extended $2$-action of $G\\rtimes G\\rightrightarrows G$ on $\\textnormal{Hol}(M,\\mathcal{F})\\rightrightarrows M$ is by isometries.\n\\end{example}\n\n\nIn the next example we will consider the notion of multiplicative $2$-connection $\\omega=(\\omega^1,\\omega^2)$ on a groupoid principal $2$-bundle $\\pi:(P_1\\rightrightarrows P_0)\\to (X_1\\rightrightarrows X_0)$ with structural Lie $2$-group $G_1\\rightrightarrows G_0$ as defined for instance in \\cite{HOV} (see also \\cite{CCK}). \n\\begin{example}[Principal groupoid warping]\nLet us prove that if $(X_1\\rightrightarrows X_0,\\eta)$ is a Riemannian groupoid and $G_1\\rightrightarrows G_0$ is orthogonal then there exists a $1$-metric $\\overline{\\eta}$ on $P_1\\rightrightarrows P_0$ for which the $2$-action of $G_1\\rightrightarrows G_0$ is isometric and such that $\\pi=(\\pi_1,\\pi_0):(P_1\\rightrightarrows P_0)\\to (X_1\\rightrightarrows X_0)$ is a Riemannian groupoid submersion. Consider the associated $1$-metric $\\langle \\cdot,\\cdot\\rangle$ on the Lie $2$-algebra $\\mathfrak{g}_1\\rightrightarrows \\mathfrak{g}_0$ of $G_1\\rightrightarrows G_0$ and define \n\t$$\\overline{\\eta}^{(1)}(v,w)=\\eta^{(1)}(d\\pi_1(v),d\\pi_1(w))+\\langle \\omega^1(v),\\omega^1(w)\\rangle^{(1)}.$$\n\tThe metric $\\overline{\\eta}^{(0)}$ is similarly defined by using instead $\\eta^{(0)}$, $\\pi_0$, $\\langle \\cdot,\\cdot\\rangle^{(0)}$, and $\\omega^0$. It is simple to check that this expression yields a well defined right $G_j$-invariant metric on $P_j$ for which $\\pi_j:P_j\\to X_j$ becomes a Riemannian submersion ($j=0,1$). This is because $\\pi_j$ is constant along the action orbits, $\\omega^j$ is of $\\textnormal{Ad}^j$-invariant type, and $\\textnormal{ker}(d\\pi_j)^{\\perp_{\\overline{\\eta}^{(j)}}}=\\textnormal{ker}(\\omega^j)$. Let us verify that $\\overline{\\eta}=(\\overline{\\eta}^{(1)},\\overline{\\eta}^{(0)})$ determines a $1$-metric on $P_1\\rightrightarrows P_0$. Note that, by abusing a little on the notation, we may rewrite $\\overline{\\eta}^{(j)}$ in a simpler way as $\\overline{\\eta}^{(j)}=(\\pi_j)^\\ast\\eta^{(j)}+(\\omega^j)^\\ast \\langle \\cdot,\\cdot\\rangle^{(j)}$. Recall that $\\pi$ and $\\omega:(TP_1 \\rightrightarrows TP_0)\\to (\\mathfrak{g}_1\\rightrightarrows \\mathfrak{g}_0)$ are Lie groupoid morphisms. So, on the one hand we get\n\t\\begin{eqnarray*}\n\t\t(i_P)^\\ast \\overline{\\eta}^{(1)} & = & (\\pi_1\\circ i_P)^\\ast\\eta^{(1)}+(\\omega^1\\circ i_P)^\\ast \\langle \\cdot,\\cdot\\rangle^{(1)}=(i_X\\circ \\pi_1)^\\ast\\eta^{(1)}+(d(i_G)_{e_1}\\circ \\omega^1)^\\ast \\langle \\cdot,\\cdot\\rangle^{(1)}\\\\\n\t\t&=& (\\pi_1)^\\ast\\eta^{(1)}+(\\omega^1)^\\ast \\langle \\cdot,\\cdot\\rangle^{(1)}=\\overline{\\eta}^{(1)}.\n\t\\end{eqnarray*}\n\tOn the other hand, if $v\\in \\textnormal{ker}(d(s_P)_{p})^{\\perp_{\\overline{\\eta}^{(1)}}}$ then it is simple to verify that the identities $\\pi_0\\circ s_P=s_X\\circ \\pi_1$ and $\\omega^0\\circ s_P=d(s_G)_{e_1}\\circ \\omega^1$ imply that $d(\\pi_1)_p(v)\\in \\textnormal{ker}(d(s_X)_{\\pi_1(p)})^{\\perp_{\\eta^{(1)}}}$ and $\\omega^1(v)\\in \\textnormal{ker}(d(s_G)_{e_1})^{\\perp_{\\langle \\cdot,\\cdot\\rangle^{(1)}}}$. Let us pick $v_1,v_2\\in \\textnormal{ker}(d(s_P)_{p})^{\\perp_{\\overline{\\eta}^{(1)}}}$. Thus,\n\t\\begin{eqnarray*}\n\t\t\\overline{\\eta}^{(0)}_{s_P(p)}(ds_P(p)(v_1),ds_P(p)(v_2)) & = & \\eta^{(0)}_{\\pi_0(s_P(p))}(d(\\pi_0)_{s_P(p)}(ds_P(p)(v_1)),d(\\pi_0)_{s_P(p)}(ds_P(p)(v_2)))\\\\\n\t\t& + & \\langle \\omega^0(ds_P(p)(v_1)),\\omega^0(ds_P(p)(v_2))\\rangle^{(0)}\\\\\n\t\t& = & \\eta^{(0)}_{s_X(\\pi_1(p))}(d(s_X)_{\\pi_1(p)}(d\\pi_1(p)(v_1)),d(s_X)_{\\pi_1(p)}(d\\pi_1(p)(v_1)))\\\\\n\t\t& + & \\langle d(s_G)_{e_1}(\\omega^1(v_1)),d(s_G)_{e_1}(\\omega^1(v_2))\\rangle^{(0)}\\\\\n\t\t& = & \\eta^{(1)}_{\\pi_1(p)}(d\\pi_1(p)(v_1),d\\pi_1(p)(v_1)) +  \\langle \\omega^1(v_1),\\omega^1(v_2)\\rangle^{(0)}\\\\\n\t\t& = & \\overline{\\eta}^{(1)}(v_1,v_2).\n\t\\end{eqnarray*}\n\tAnalogously, it follows that $(t_P)_\\ast \\overline{\\eta}^{(1)}=\\overline{\\eta}^{(0)}$. Hence, we have shown that $(P_1\\rightrightarrows P_0,\\overline{\\eta})$ is a Riemannian groupoid for which $G_1\\rightrightarrows G_0$ acts isometrically and $\\pi$ is a Riemannian submersion of groupoids, as claimed.\n\\end{example}\nNext example comes motivated by a beautiful construction known in the literature as \\emph{Cheeger deformation}. The classical construction can be found for instance in \\cite[s. 6.1]{AB}.\n\\begin{example}[Cheeger groupoid deformation]\n\tSuppose that $(G_1\\rightrightarrows G_0, Q)$ is an orthogonal Lie 2-group, with $G_1$ compact, acting isometrically on a Riemannian groupoid $(X_1\\rightrightarrows X_0,\\eta)$. On the Lie groupoid product $X_1\\times G_1\\rightrightarrows X_0\\times G_0$ we can consider the 1-metric $\\eta\\oplus \\frac{1}{\\tau}Q$.\n\t\n\tThere is a natural free $2$-action of $G_1\\rightrightarrows G_0$ on $X_1\\times G_1\\rightrightarrows X_0\\times G_0$ where the $G_j$-actions, $j=1,0$, on $X_j\\times G_j$ are given by\n\t\n\t\\begin{equation}\\label{CheegerD}\n\t\th_j\\cdot(p_j,g_j)=(h_jp_j,h_jg_j),\\qquad p_j\\in X_j,\\ h_j,g_j\\in G_j.\n\t\\end{equation} \n\t\n\tWe claim that the quotient groupoid $\\frac{X_1\\times G_1}{G_1}\\rightrightarrows\\frac{X_0\\times G_0}{G_0}$ determined by the previous actions is isomorphic to $X_1\\rightrightarrows X_0$. Let us consider the groupoid principal $2$-bundle $G_1\\rightrightarrows G_0$ over the point groupoid $\\lbrace e_1\\rbrace\\rightrightarrows\\lbrace e_0\\rbrace$ with structural Lie $2$-group $G_1\\rightrightarrows G_0$. By applying Remark \\ref{AssociatedBundle} we know that by using the $2$-action of $G_1\\rightrightarrows G_0$ on $X_1\\rightrightarrows X_0$ we can construct the associated Lie groupoid bundle $(X_1\\times_{G_1}G_1\\rightrightarrows X_0\\times_{G_0}G_0)\\to (\\lbrace e_1\\rbrace\\rightrightarrows\\lbrace e_0\\rbrace)$. It is important to notice that $X_1\\times_{G_1}G_1\\rightrightarrows X_0\\times_{G_0}G_0$ is precisely the quotient groupoid $\\frac{X_1\\times G_1}{G_1}\\rightrightarrows\\frac{X_0\\times G_0}{G_0}$. Therefore, as the new Lie groupoid bundle has groupoid fiber $X_1\\rightrightarrows X_0$ and base groupoid $\\lbrace e_1\\rbrace\\rightrightarrows\\lbrace e_0\\rbrace$ then we have the desired isomorphism. Under this identification the canonical groupoid projection $\\pi=(X_1\\times G_1\\rightrightarrows X_0\\times G_0)\\to (X_1\\rightrightarrows X_0)$ is formed by the maps $\\pi_j(p_j,g_j)=g_j^{-1}p_j$.\n\t\n\tObserve that the $2$-action \\eqref{CheegerD} is also isometric. Thus, as consequence of Proposition \\ref{PQuotient} there is a unique 1-metric $\\eta_\\tau$ on $X_1\\rightrightarrows X_0$ making of the projection $\\pi=(X_1\\times G_1\\rightrightarrows X_0\\times G_0,\\eta\\oplus \\frac{1}{\\tau}Q)\\to (X_1\\rightrightarrows X_0,\\eta_\\tau)$ a Riemannian groupoid submersion. \n\t\n\tIt is important to point out that by construction $\\eta_\\tau \\mapsto \\eta$ when $\\tau$ goes to $\\infty$ and the $2$-action of $G_1\\rightrightarrows G_0$ on $(X_1\\rightrightarrows X_0,\\eta_\\tau)$ is still isometric when $\\tau>0$. More importantly, the 1-parameter family of $1$-metrics $\\eta_\\tau$ on $X_1\\rightrightarrows X_0$ varies smoothly with $\\tau$ and extends smoothly to $\\tau = 0$ with $\\eta_0 = \\eta$. Hence, $\\eta_\\tau$ with $\\tau\\geq  0$ is a deformation of $\\eta$ by other $(G_1\\rightrightarrows G_0)$-invariant metrics on  $X_1\\rightrightarrows X_0$ which we call \\emph{Cheeger groupoid deformation} of $\\eta$. The reader is recommended to visit \\cite[s. 6,1]{AB} for specific details about the last assertions in the classical case. \n\\end{example}\n\n\\begin{remark}\nIt is worth mentioning that the notion of isometric 2-action we have introduced in this section was used in \\cite{OV} as one of the fundamental ingredients needed to construct an equivariant double Morse-Bott complex which computes the equivariant cohomology of a $2$-action as defined in \\cite{OBT}. \n\\end{remark}\n\n\n\\section{Isometries and multiplicative Killing vector fields}\\label{S:4}\nThe aim of this section is to bring a description of the group of isometries associated to a $0$-metric on a Lie groupoid. This will give rise to a notion of geometric Killing vector field on a quotient Riemannian stack. We start by describing what would be our attempt to set the \\emph{diffeomorphism group} of a differentiable stack. Some of the references we shall be following throughout are \\cite{Ma,OW} and \\cite[App. D]{An}. A \\emph{bisection} of a Lie groupoid $X_1\\rightrightarrows X_0$ is a smooth map $\\sigma:X_0\\to X_1$ such that $s_X\\circ \\sigma=\\textnormal{id}_{X_0}$ and $\\iota_{\\sigma}:X_0\\to X_0$ defined by $\\iota_{\\sigma}(x):=t_X(\\sigma(x))$ is a diffeomorphism. The set of all bisections of $X_1\\rightrightarrows X_0$ will be denoted by $\\textnormal{Bis}(X)$. This has the structure of an infinite-dimensional Lie group where the multiplication of two bisections $\\sigma$ and $\\sigma'$ is given by $\\sigma\\bullet \\sigma'(x):=\\sigma((t_X\\circ \\sigma')(x))*\\sigma'(x)$ for all $x\\in X_0$; see for instance \\cite{SW} and \\cite[s. 1.4]{Ma}. Let us denote by $\\textnormal{Aut}(X)$ the group of Lie groupoid automorphisms of $X_1\\rightrightarrows X_0$. Given a bisection $\\sigma \\in \\textnormal{Bis}(X)$ one has an inner automorphism $I_{\\sigma}:X_1\\to X_1$ defined by $$I_{\\sigma}(p):=\\sigma(t_X(p))*p* i_X(\\sigma(s_X(p))).$$\n\nClearly, $I_{\\sigma}$ covers the map $\\iota_{\\sigma}$. This inner automorphism allows us to define what we call the \\emph{crossed module of automorphisms of a Lie groupoid} $(\\textnormal{Aut}(X),\\textnormal{Bis}(X),I,\\alpha)$ where the map $\\alpha$ is defined as $\\alpha_{\\Phi}(\\sigma):=\\Phi\\circ \\sigma \\circ \\phi^{-1}$ for all $\\Phi \\in \\textnormal{Aut}(X)$ covering $\\phi:X_0\\to X_0$ and $\\sigma \\in \\textnormal{Bis}(X)$. Accordingly, we have a 2-group $\\textnormal{Bis}(X)\\ltimes \\textnormal{Aut}(X)\\rightrightarrows \\textnormal{Aut}(X)$ called the \\emph{2-group of Lie groupoid automorphisms}. \n\\begin{proposition}\n\tThe orbit space of the 2-group of Lie groupoid automorphisms equals the set of Lie groupoid automorphisms up to smooth natural equivalences. Namely:\n$$\\textnormal{Aut}(X)\/\\textnormal{Bis}(X)=\\left\\lbrace [\\Phi]\\,|\\, \\Phi\\in \\textnormal{Aut}(X),\\quad \\Psi\\sim \\Phi \\Leftrightarrow \\exists_{\\alpha}\\left(\\Psi\\stackrel{\\alpha}{\\Rightarrow}\\Phi\\right)\\right\\rbrace.$$\n\\end{proposition}\n\\begin{proof}\n\tLet us describe the orbit of an element in $\\Phi\\in \\textnormal{Aut}(X)$ covering $\\phi$. If we pick $\\sigma\\in \\textnormal{Bis}(X)$ and $\\Psi \\in \\textnormal{Aut}(X)$ covering $\\psi$ such that $I_{\\sigma}\\Phi=\\Psi$ then it follows that \n$$\\Psi(p)=I_{\\sigma}(\\Phi(p))=\\sigma(\\phi(t_X(p)))*\\Phi(p)*i_X(\\sigma(\\phi(s_X(p)))),$$\nfor some $p \\in X_1$ so that $\t\\Psi(p)*\\sigma(\\phi(s_X(p)))=\\sigma(\\phi(t_X(p)))*\\Phi(p)$. Therefore, by setting $\\alpha:=\\sigma\\circ \\phi$ we get a smooth natural transformation $\\Phi\\stackrel{\\alpha}{\\Rightarrow}\\Psi$. Conversely, note that if $\\Phi\\stackrel{\\alpha}{\\Rightarrow}\\Psi$  a smooth natural transformation then for some $p\\in X$ it holds $\\alpha(t_X(p))*\\Phi(p)=\\Psi(p)*\\alpha(s_X(p))$, thus obtaining that $t_X(\\Phi(p))=s_X(\\alpha(t_X(p)))$ and $s_X(\\Psi(p))=t_X(\\alpha(s_X(p)))$. On the one hand, by setting $\\sigma:=\\alpha\\circ \\phi^{-1}$ it follows that $s_X\\circ\\sigma=\\textnormal{id}_{X_0}$. On the other hand, observe that\n$$ \\psi(s_X(p))=t_X(\\alpha(s_X(p)))=t_X(\\sigma(\\phi( s_X)))(p)=\\iota_{\\sigma}(\\phi(s_X(p))).$$\nHence, we have obtained that $i_{\\sigma}=\\psi\\circ \\phi^{-1}$ so that it is a diffeomorphism. \n\\end{proof}\nLet $G_1\\rightrightarrows G_0$ be a Lie $2$-group acting on $X_1\\rightrightarrows X_0$ by the left.\n\n\\begin{lemma}\\label{2ActionDecomposition}\nThe normal subgroup $H=\\textnormal{ker}(s_{G})$ acts on $X_1\\rightrightarrows X_0$ by bisections and $G_0$ acts by Lie groupoid automorphisms. Moreover, the right multiplication map defined on $s_X$-fibers for each arrow is $H$-equivariant. \n\\end{lemma}\n\\begin{proof}\n\tConsider $h\\in H$ and define the map $\\sigma_{h}:X_0\\to X_1$ as $\\sigma_h(x):=h1_x.$ It is clear that $s_X(\\sigma_h(x))=x$ and $t(\\sigma_h(x))=\\rho(h)x$ so that $\\sigma_h$ is a well defined bisection.  Let us now take $g\\in G_0$ and define the map $\\Sigma_g:X_1\\to X_1$ as $\\Sigma_g(x):=1_{g}x$. Equation \\eqref{MultAction} implies that\n\t\\[1_{g}(x*y)=(1_g*1_g)(p*q)=(1_{g}p)*(1_{g}q),\\]\n\tfor all $g\\in G_0$ and $(p,q)\\in X_2$, thus obtaining that $\\Sigma_g$ is a Lie groupoid morphism which clearly satisfies $(\\Sigma_g)^{-1}=\\Sigma_{g^{-1}}$. Note that $s_{X}(hp)=s_{X}(p)$ for all $p\\in X_1$ and $h\\in H$ so that the left action of $H$ on $X_1$ preserves the $s_X$-fibers. Therefore, for each $y\\xleftarrow[]{\\it p}x$ the right action $R_{p}:s_X^{-1}(y)\\to s_X^{-1}(x)$ satisfies that\n\t\\[R_{p}(hq)=(hq)*(1_{e}p)=(h*1_e)(q*p)=hR_{p}(q).\\]\n\tIn consequence, $R_p$ is $H$-equivariant as claimed.\n\\end{proof}\nSame result can be obtained if we consider right 2-actions instead of left ones.\n\\begin{lemma}\\label{LemmaIso1}\nThere is a natural morphism of crossed modules of Lie groups $(\\sigma,\\Sigma):(G,H,\\rho,\\alpha) \\to (\\textnormal{Aut}(X),\\textnormal{Bis}(X),I,\\alpha)$ where $\\sigma_h$ and $\\Sigma_g$ are defined as in Lemma \\ref{2ActionDecomposition}.\n\\end{lemma}\n\\begin{proof}\nLet us check that $\\Sigma \\circ \\rho=I\\circ \\sigma$ and $\\sigma_{\\alpha_{g}(h)}=\\alpha_{\\Sigma_g}(\\sigma_h)$ for all $g \\in G$ and $h\\in H$. Firstly, for $ p\\in X_1$ and $h\\in H$ we obtain\n\\begin{eqnarray*}\n\t\tI_{\\sigma_h}(p)&=&\\sigma_h(t_X(p))*p*i(\\sigma_h(s_X(p)))=(h1_{t_X(p)})*p*i(h1_{s_X(p)})\\\\\n\t\t&=&(h*e)(1_{t_X(p)}*p)*i(h1_{s_X(p)})=hp*i_{G}(h)1_{s(p)}=1_{\\rho(h)}p=\\Sigma_{\\rho(h)}(p).\n\t\\end{eqnarray*}\n\tSecondly, for $x \\in X_0$, $g\\in G_0$ and $h\\in H$ we get\n$$\n\\alpha_{\\Sigma_g}(\\sigma_h)(x) =\\Sigma_g(h1_{(g^{-1}x)})=1_gh1_{g^{-1}}1_x=\\alpha_{g}(h)1_x=\\sigma_{\\alpha_g(h)}(x).\n$$\n\\end{proof}\nWe are now in conditions to define an \\emph{infinitesimal 2-action} associated to a right Lie 2-group action. Let $A_X$ be the Lie algebroid of $X_1\\rightrightarrows X_0$ and for each $\\xi \\in \\mathfrak{g}_0$ we denote its fundamental vector fields as $\\tilde{\\xi}$. We also denote by $\\mathfrak{X}_{m}(X)$ the set of multiplicative vector fields on $X_1\\rightrightarrows X_0$ and by $(\\mathfrak{X}_{m}(X),\\Gamma(A_{X}),\\delta,D)$ its associated crossed module. Here we have that $\\delta(\\alpha)=\\alpha^r-\\alpha^l$ and $D_{(\\xi,v)}\\alpha=[\\xi,\\alpha^r]|_{X_0}$ for all $\\alpha\\in \\Gamma(A_X)$ and $(\\xi,v)\\in \\mathfrak{X}_{m}(X)$; see \\cite{OW} for specific details.\n\n\\begin{theorem}\\label{InfinitesimalAction}\nSuppose that we have a right 2-action of $G_1\\rightrightarrows G_0$ on $X_1\\rightrightarrows X_0$. Then there is a canonical homomorphism of Lie 2-algebra $j=(j_{-1},j_0):(\\mathfrak{g},\\mathfrak{h},\\partial,\\mathcal{L})\\to (\\mathfrak{X}_{m}(X),\\Gamma(A_{X}),\\delta,D)$ defined by $j_{-1}(\\xi)=\\left.\\tilde{\\xi}\\right|_{X_0}$ and  $j_0(\\zeta)=(\\tilde{1_{\\zeta}},\\tilde{\\zeta})$ for all $\\xi \\in \\mathfrak{h}$ and $\\zeta \\in \\mathfrak{g}$.\n\\end{theorem}\n\\begin{proof}\nTo see that $j$ is a well defined morphism we have to check that for any $\\xi \\in \\mathfrak{h}$ its fundamental vector field $\\tilde{\\xi}$ belongs to $\\mathfrak{X}_{inv}^{s}(X)$ and that for any $\\zeta \\in \\mathfrak{g}$ it holds that $(\\tilde{1_{\\zeta}},\\tilde{\\zeta})$ is in $\\mathfrak{X}_{m}(X)$. The latter assertion is clear since if $\\exp{t\\zeta}\\in G_0$ then Lemma \\ref{2ActionDecomposition} implies that the pair of flows $(\\varphi_t^{\\tilde{1_\\zeta}},\\varphi_t^{\\tilde{\\zeta}})$ determines a Lie groupoid morphism. Now, if $\\xi \\in \\mathfrak{h}$ then again from Lemma \\ref{2ActionDecomposition} it follows that the flow of its fundamental vector field lies inside the $s$-fibers since $s_{X}(\\varphi_t^{\\tilde{\\xi}}(p))=s_{X}(p\\exp(t\\xi))=s_{X}(p)$. Therefore, $\\tilde{\\xi}\\in \\mathfrak{X}^s(X)$. Furthermore, for $y\\xleftarrow[]{\\it p}x$ and $q\\in s_X^{-1}(y)$ one has that\n$$\nR_{p}(\\varphi^{\\tilde{\\xi}}_t(q)) =(q\\exp(t\\xi))*p=(q*p)(\\exp{t\\xi}*1_e)=\\varphi_t^{\\tilde{\\xi}}(q*p)=\\varphi_t^{\\tilde{\\xi}}(R_p(q)),\n$$\nthus obtaining that $d(R_p)_q(\\tilde{\\xi}_q)=\\tilde{\\xi}_{q*p}$ so that $\\tilde{\\xi}\\in \\mathfrak{X}_{inv}^s(X)$ and $\\left.\\tilde{\\xi}\\right|_{X_0}\\in \\Gamma(A_X)$.\n\nLet us finally verify that for $\\xi\\in \\mathfrak{h}$ and $\\zeta \\in \\mathfrak{g}$ it satisfies that $\\delta(j_{-1}(\\xi))=j_{0}(\\partial \\xi)$ and $j_{-1}(\\mathcal{L}_{\\eta}\\xi)=D_{j_0(\\eta)}(j_{-1}(\\xi))$. On the one hand, by using the flow of vector field $\\delta(j_0(\\xi))=\\left.\\tilde{\\xi}\\right|_{X_0}^r-\\left.\\tilde{\\xi}\\right|_{X_0}^l$ and Equation \\eqref{MultAction} we get\n\\begin{eqnarray*}\n\\varphi_t^{\\delta(j_{-1}(\\xi))}(p)&=&\\varphi_{t}^{\\tilde{\\xi}}(1_{t_{X}(p)})*p*\\iota_X(\\varphi_t^{\\tilde{\\xi}}(1_{s_X(p)}))=1_{t_{X}(p)}\\exp(t\\xi)*p*i_X(1_{s_{X}(p)}\\exp(t\\xi))\\\\\n&=&(1_{t_X(p)}*p)\\exp(t\\xi)*1_{s_X(p)}i_G(\\exp(t\\xi))=p(\\exp(t\\xi)*i_G(\\exp(t\\xi)))\\\\\n&=&p(1_{t_G(\\exp(t\\xi))})=p1_{\\exp(t\\partial(\\xi))}=p\\exp(t1_{\\partial \\xi})=\\varphi_t^{{j_0(\\partial \\xi)}}(p).\n\t\\end{eqnarray*}\n\tHence, $\\delta(j_{-1}(\\xi))=j_{0}(\\partial \\xi)$. On the other hand, observe that\n\t\\[    D_{j_0(\\eta)}(j_{-1}(\\xi))=\\left.\\left[\\tilde{1_{\\eta}},\\tilde{\\xi}\\right]\\right|_{X_0}=\\left.\\tilde{\\left[1_{\\eta},\\xi\\right]}\\right|_{X_0}=\\left.\\tilde{\\mathcal{L}_{\\eta}(\\xi))}\\right|_{X_0}=j_{-1}(\\mathcal{L}_{\\eta}\\xi).\\]\n\\end{proof}\n\\subsection{Isometries of a 0-metric}\nRecall that a $0$-metric on a Lie groupoid $X_1\\rightrightarrows X_0$ is a Riemannian metric $\\eta$ on $X_0$ which is transversely invariant by the canonical left action of $X_1\\rightrightarrows X_0$ on $X_0$, compare \\cite{dHF,PPT}. Note that this is the same that requiting that $\\eta$ is transversely invariant by the action of the group of bisections $\\textnormal{Bis}(X)\\times X_0\\to X_0$ which is defined by $\\sigma\\cdot x:=\\iota_{\\sigma}(x)$. It is clear that this action preserves the orbits so that it induces a well defined action on the normal space of a orbit. In consequence, $\\eta$ is a 0-metric if and only if for all $\\sigma \\in \\textnormal{Bis}(X)$ the map $\\overline{d\\iota_{\\sigma}}:(\\nu_x(\\mathcal{O}),\\overline{\\eta})\\to (\\nu_{\\iota_{\\sigma}(x)}(\\mathcal{O}),\\overline{\\eta})$ is a linear isometry.\n\nLet $(X_1\\rightrightarrows X_0,\\eta)$ be a Lie groupoid equipped with a 0-metric $\\eta$ and consider the following sets \n$$\\textnormal{Bis}_{\\eta}(X)=\\left\\lbrace \\sigma \\in \\textnormal{Bis}(X)\\,|\\, \\iota_{\\sigma}^*\\eta=\\eta\\right\\rbrace\\quad\\textnormal{and}\\quad \\mathrm{Iso}(X,\\eta)=\\left \\lbrace (\\Phi,\\phi)\\in \\mathrm{Aut}(X)\\,|\\, \\phi^*\\eta=\\eta \\right\\rbrace.$$\n\n\\begin{proposition}\\label{IsoStrong1}\nThe quadruple $(\\mathrm{Iso}(X,\\eta),\\textnormal{Bis}_{\\eta}(X),I,\\alpha)$ determines a sub-crossed module structure  of $(\\textnormal{Aut}(X),\\textnormal{Bis}(X),I,\\alpha)$.\n\\end{proposition}\n\\begin{proof}\nIt is clear that $I(\\textnormal{Bis}_{\\eta}(X))\\subseteq \\mathrm{Iso}(X,\\eta)$. As $I_{\\alpha_{\\Phi}(\\sigma)}=\\Phi I_{\\sigma}\\Phi^{-1}$ for all $\\sigma \\in \\textnormal{Bis}_{\\eta}(X)$ and $\\Phi \\in \\textnormal{Iso}(X,\\eta)$ then when restricting to unities we have that $\\iota_{\\alpha_{\\Phi}(\\sigma)}=\\phi\\circ \\iota_{\\sigma}\\circ \\phi^{-1}$, thus obtaining an isometry.\n\\end{proof}\nThe Lie 2-group associated to the crossed module $(\\mathrm{Iso}(X,\\eta),\\textnormal{Bis}_{\\eta}(X),I,\\alpha)$ will be called \\emph{Lie 2-group of strong isometries} of $(X_1\\rightrightarrows X_0,\\eta)$. Let us now consider the sets \n$$\\Gamma_{\\eta}(A_X)=\\left\\lbrace \\alpha \\in \\Gamma(A_X)\\,|\\, \\rho(\\alpha) \\in \\mathfrak{o}(X_0,\\eta)\\right\\rbrace\\quad\\textnormal{and}\\quad \\mathfrak{o}_{m}(X)=\\left\\lbrace (\\xi,v)\\in  \\mathfrak{X}_{m}(X)\\,|\\, v \\in \\mathfrak{o}(X_0,\\eta)\\right\\rbrace$$\nwhere $\\mathfrak{o}(X_0,\\eta)$ denotes the Lie algebra of Killing vector fields of $(X_0,\\eta)$. In these terms we may describe the infinitesimal version of the Lie $2$-group of strong isometries as follows. \n\\begin{proposition}\\label{Strongkilling}\nThe quadruple $(\\mathfrak{o}_{m}(X),\\Gamma_{\\eta}(A_X),\\delta,D)$ defines a sub-crossed module structure of $(\\mathfrak{X}_{m}(X),\\Gamma(A_{X}),\\delta,D)$.\n\\end{proposition}\n\\begin{proof}\nFirstly, note that $\\mathfrak{o}_{m}(X)$ is a Lie subalgebra of $\\mathfrak{X}_{m}(X)$ since $\\mathfrak{o}(X_0,\\eta)$ is a Lie algebra. Secondly, if $\\alpha, \\beta \\in \\Gamma_{\\eta}(A_X)$ then $\\rho([\\alpha,\\beta])=[\\rho(\\alpha),\\rho(\\beta)]\\in \\mathfrak{o}(X_0,\\eta)$ so that $[\\alpha,\\beta]\\in \\Gamma_{\\eta}(A)$. If $(\\xi,v)\\in \\mathfrak{o}_{m}(X)$ and $\\alpha \\in \\Gamma_{\\eta}(A)$ then we have by definition that $D_{\\xi}(\\alpha)=\\left.[\\xi,\\alpha^r]\\right|_{X_0} \\in \\Gamma(A_X)$. However, the equivariance identity implies that $\\delta(D_{\\xi}(\\alpha))=[\\xi,\\delta(\\alpha)]$. Therefore, it holds that $\\left.\\delta(D_{\\xi}\\alpha)\\right|_{X_0}=\\left.[\\xi,\\delta(\\alpha)]\\right|_{X_0}$ which is the same thing that saying\n$\\rho(D_{\\xi}\\alpha)=[v,\\rho(\\alpha)] \\in \\mathfrak{o}(X_0,\\eta)$ since $v$ is also a Killing vector field.\n\\end{proof}\nThe Lie $2$-algebra associated to the crossed module $(\\mathfrak{o}_{m}(X),\\Gamma_{\\eta}(A_X),\\delta,D)$ will be called \\emph{Lie 2-algebra of strong multiplicative Killing vector fields} of $(X_1\\rightrightarrows X_0,\\eta)$.\n\\begin{remark}\nIt is worth noticing that if we have an isometric action of a Lie $2$-group $G_1\\rightrightarrows G_0$ on $(X_1\\rightrightarrows X_0,\\eta)$ then the assertions of Lemma \\ref{LemmaIso1} and Theorem \\ref{InfinitesimalAction} can be rewritten in terms of $(\\mathrm{Iso}(X,\\eta),\\textnormal{Bis}_{\\eta}(X),I,\\alpha)$ and $(\\mathfrak{o}_{m}(X),\\Gamma_{\\eta}(A_X),\\delta,D)$, respectively.\n\\end{remark}\nA diffeomorphism $\\phi:X_0\\to X_0$ is said to be a \\emph{transversal isometry} of $(X_0,\\eta)$ if $\\overline{d\\phi}:\\nu(\\mathcal{O}_x)\\to \\nu(\\mathcal{O}_{\\phi(x)})$ is a fiberwise isometry for every groupoid orbit $\\mathcal{O}_x$ in $X_0$. Consider the set\n$$\\textnormal{Iso}_{\\textnormal{w}}(X,\\eta)=\\left \\lbrace (\\Phi,\\phi)\\in \\mathrm{Aut}(X)\\,|\\, \\phi\\ \\textnormal{transversal isometry of}\\ (X_0,\\eta)\\right\\rbrace.$$\nNote that for every $\\sigma\\in \\textnormal{Bis}(X)$ it follows that $\\iota_\\sigma$ is a transversal isometry of $(X_0,\\eta)$. Thus, by arguing as in Proposition \\ref{IsoStrong1} we easily get that:\n\\begin{lemma}\nThe quadruple $(\\mathrm{Iso}_{\\textnormal{w}}(X,\\eta),\\textnormal{Bis}(X),I,\\alpha)$ determines a sub-crossed module structure  of $(\\textnormal{Aut}(X),\\textnormal{Bis}(X),I,\\alpha)$.\n\\end{lemma}\nThe Lie $2$-group determined by the crossed module $(\\mathrm{Iso}_{\\textnormal{w}}(X,\\eta),\\textnormal{Bis}(X),I,\\alpha)$ is called \\emph{Lie 2-group of weak isometries} of $(X_1\\rightrightarrows X_0,\\eta)$. Let us denote by $\\mathfrak{o}^{\\textnormal{w}}(X_0,\\eta)$ the set of vector fields on $X_0$ whose flow determines a (local) transversal isometry of $(X_0,\\eta)$ and consider the set $\\mathfrak{o}_{m}^{\\textnormal{w}}(X)=\\left\\lbrace (\\xi,v)\\in  \\mathfrak{X}_{m}(X)\\,|\\, v \\in \\mathfrak{o}^{\\textnormal{w}}(X_0,\\eta)\\right\\rbrace$. Observe that if $v_1,v_2\\in \\mathfrak{o}^{\\textnormal{w}}(X_0,\\eta)$ then for $t$ small enough we have that the commutator flow $\\varphi_{-\\sqrt{t}}^{v_2}\\varphi_{-\\sqrt{t}}^{v_1}\\varphi_{\\sqrt{t}}^{v_2}\\varphi_{\\sqrt{t}}^{v_1}$ is a transversal isometry so that $\\mathfrak{o}^{\\textnormal{w}}(X_0,\\eta)$ is a Lie subalgebra of $\\mathfrak{X}(X_0)$. From \\cite{SW} we know that $\\textnormal{Lie}(\\textnormal{Bis}(X))$ is identified with $\\Gamma(A_X)$. Therefore, by using similar arguments as those in Proposition \\ref{Strongkilling} we obtain a description of the Lie $2$-group of weak isometries of $(X_1\\rightrightarrows X_0,\\eta)$. Namely:\n\\begin{proposition}\nThe quadruple \n$(\\mathfrak{o}_{m}^{\\textnormal{w}}(X),\\Gamma(A_X),\\delta,D)$ defines a sub-crossed module structure of $(\\mathfrak{X}_{m}(X),\\Gamma(A_{X}),\\delta,D)$.\n\\end{proposition}\nAccordingly, the Lie $2$-algebra associated to the crossed module $(\\mathfrak{o}_{m}^{\\textnormal{w}}(X),\\Gamma(A_X),\\delta,D)$ will be called \\emph{Lie 2-algebra of weak multiplicative Killing vector fields} of $(X_1\\rightrightarrows X_0,\\eta)$.\n\\begin{theorem}\nLet $(X_1\\rightrightarrows X_0,\\eta)$ be a Riemannian groupoid with $\\eta$ a 1-metric and $X_1\\rightrightarrows X_0$ proper. Then there always exists a multiplicative vector field $(\\xi,v)$ with $v \\in \\mathfrak{o}^{\\textnormal{w}}(X_0,\\eta)$.\n\\end{theorem}\n\\begin{proof}\nLet us pick $v_1 \\in \\mathfrak{o}^{\\textnormal{w}}(X_0,\\eta)$. Since $s$ is a surjective submersion there is a vector field $\\xi_1$ on $X_1$ such that $\\xi_1$ is $s_X$-related with $v_1$. The fact that $\\overline{ds}:\\nu(G_{\\mathcal{O}})\\to \\nu(\\mathcal{O})$ is a fiberwise isometry ($s_X$ is a Riemannian submersion) implies that the flow of $\\xi_1$ is a local transversal isometry of $(X_1,\\eta)$ since $\\xi_1$ and $v_1$ are $s$-related.\n\nLet us now consider a proper Haar measure system $\\lbrace \\mu^x\\rbrace$ for $G\\rightrightarrows M$. By following results proved in \\cite{CS} we can construct a multiplicative vector field $\\xi:X_1\\to TX_1$ by taking the average with respect to $\\lbrace \\mu^x\\rbrace$:\n$$\\xi(p)=\\int_{a\\in t^{-1}(s(p))}dm_{(pa,a^{-1})}(\\xi_1(ap),di_a(\\xi_1(a))\\mu(a).$$\nThis vector field is $s_X$-related with the vector field $v$ on $X_0$ defined as\n$$v(x)=\\int_{a\\in t^{-1}(x)}dt_a(\\xi_1(a))\\mu(a).$$\nTherefore, our result will follow once we prove that the flow of $v$ is a local transversal isometry of $(X_0,\\eta)$ since $\\xi$ and $v$ are $s_X$-related. However, this follows from a straightforward computation after noting that the flow of $v$ is given by $\\varphi^v_r(x)=\\int_{a\\in t^{-1}(x)}(t\\circ \\varphi^{\\xi_1}_r)(a)\\mu(a)$ and $\\overline{dt}:\\nu(G_{\\mathcal{O}})\\to \\nu(\\mathcal{O})$ is a fiberwise isometry since $t_X$ is also a Riemannian submersion.\n\\end{proof}\n\n\\subsection{Morita invariance}\nIn this subsection we apply some of the result from \\cite{OW} to our context in order to define a notion of geometric Killing field on a quotient Riemannian stack.\n\nLet us start by introducing some necessary terminology.  A \\emph{Morita map} is a groupoid morphism $\\phi:(X_1\\rightrightarrows X_0) \\to (Y_1\\rightrightarrows Y_0)$ which is \\emph{fully faithful} and \\emph{essentially surjective}, in the sense that the source\/target maps define a fibred product of manifolds $X_1 \\cong (X_0 \\times X_0) \\times_{ (Y_0\\times Y_0)} Y_1$ and that the map $Y_1 \\times _{Y_0}X_0\\to X_0$ sending $(\\phi^0(x)\\to y)\\mapsto y$ is a surjective submersion \\cite{dH,MM}. An important fact shown in \\cite{dH} is that a Lie groupoid morphism is a Morita map if and only if it yields an isomorphism between \\emph{transversal data}. That is, the morphism must induce: a homeomorphism between the orbit spaces, a Lie group isomorphism $X_x\\cong Y_{\\phi^0(x)}$ between the isotropies and isomorphisms between the normal representations $X_x\\curvearrowright \\nu_x\\to Y_{\\phi^0(x)}\\curvearrowright \\nu'_{\\phi^0(x)}$. We think of a quotient stack as a Lie groupoid up to Morita equivalence in the sense that two Lie groupoids $X$ and $Y$ define the same stack if there is a third groupoid $Z$ and Morita maps $Z\\to X$ and $Z\\to Y$ \\cite{dH}. These two Morita maps may be assumed to be Morita fibrations (surjective submersion at the level of objects) \\cite{MM}. The quotient stack associated to the Lie groupoid $X_1\\rightrightarrows X_0$ will be denoted by $[X_0\/X_1]$. \n\nTwo Riemannian metrics $\\eta_1$ and $\\eta_2$ on $X_1\\rightrightarrows X_0$ are said to be \\emph{equivalent} if they induce the same inner product on the normal vector spaces over the groupoid orbits \\cite{dHF2}. More generally, we define a \\emph{Riemannian Morita map} (\\emph{fibration}) $\\phi: (Z_1 \\rightrightarrows Z_0) \\to (X_1\\rightrightarrows X_0)$ as a Morita map between Riemannian groupoids that induces isometries on the normal vector spaces to the groupoid orbits $\\nu_z(\\mathcal{O}^Z)\\to \\nu_{\\phi(z)}(\\mathcal{O}^X)$ (Riemannian submersion at the level of objects). By using this terminology we have that  $\\eta_1$ and $\\eta_2$ are equivalent if and only if the identity $\\textnormal{id}: (X_1 \\rightrightarrows X_0,\\eta_1) \\to (X_1\\rightrightarrows X_0,\\eta_2)$ is a Riemannian Morita map.\n\nFollowing \\cite{OW}, given a Morita fibration $\\phi: (Z_1 \\rightrightarrows Z_0) \\to (X_1\\rightrightarrows X_0)$ we denote the set of projectable sections \n$$\\Gamma(A_Z)^\\phi=\\lbrace \\alpha \\in \\Gamma(A_Z):\\textnormal{there exists}\\ \\alpha'\\in \\Gamma(A_X)\\ \\textnormal{such that}\\ \\phi\\alpha=\\alpha'\\phi\\rbrace.$$\nIf $\\alpha \\in \\Gamma(A_Z)$ then the surjectivity of $\\phi$ at the level of objects implies that there exists at most one section $\\alpha'\\in \\Gamma(A_X)$ such that  $\\phi\\alpha=\\alpha'\\phi$ so that it follows that there is a natural linear map $\\phi_\\ast : \\Gamma(A_Z)^\\phi\\to \\Gamma(A_X)$. We denote by $\\Gamma(A_Z)^\\phi \\hookrightarrow  \\Gamma(A_Z)$ the inclusion map. It is clear that we can similarly define the set of projectable multiplicative sections $\\mathfrak{X}_m(Z)^\\phi$, a natural map $\\phi_\\ast: \\mathfrak{X}_m(Z)^\\phi\\to \\mathfrak{X}_m(X)$ and an inclusion $\\mathfrak{X}_m(Z)^\\phi\\hookrightarrow \\mathfrak{X}_m(Z)$. From \\cite{OW} it follows that $(\\mathfrak{X}_m(Z)^\\phi,\\Gamma(A_{Z})^\\phi,\\delta,D)$ is a sub-crossed module of $(\\mathfrak{X}_m(Z),\\Gamma(A_{Z}),\\delta,D)$ and both maps $\\phi_\\ast: (\\mathfrak{X}_m(Z)^\\phi,\\Gamma(A_{Z})^\\phi,\\delta,D)\\to (\\mathfrak{X}_m(X),\\Gamma(A_{X}),\\delta,D)$ and $(\\mathfrak{X}_m(Z)^\\phi,\\Gamma(A_{Z})^\\phi,\\delta,D)\\hookrightarrow(\\mathfrak{X}_m(Z),\\Gamma(A_{Z}),\\delta,D)$  are morphisms of crossed-modules. More importantly,\n$$(\\mathfrak{X}_m(Z),\\Gamma(A_{Z}),\\delta,D) \\hookleftarrow (\\mathfrak{X}_m(Z)^\\phi,\\Gamma(A_{Z})^\\phi,\\delta,D) \\xrightarrow[]{\\it \\phi_\\ast} (\\mathfrak{X}_m(X),\\Gamma(A_{X}),\\delta,D),$$\nare quasi-isomorphisms of crossed modules. For specific details visit \\cite{OW}.\n\\begin{lemma}\\label{MoritaLemma1}\nLet $\\phi: (Z_1 \\rightrightarrows Z_0,\\eta^Z) \\to (X_1\\rightrightarrows X_0,\\eta^X)$ be a Morita Riemannian fibration. Then\n\\begin{itemize}\n\\item $(\\mathfrak{o}_m(Z)^\\phi,\\Gamma_{\\eta}(A_{Z})^\\phi,\\delta,D)$ is a sub-crossed module of $(\\mathfrak{o}_m(Z),\\Gamma_\\eta(A_{Z}),\\delta,D)$,\n\\item the inclusion $(\\mathfrak{o}_m(Z)^\\phi,\\Gamma_{\\eta}(A_{Z})^\\phi,\\delta,D)\\hookrightarrow(\\mathfrak{o}_m(Z),\\Gamma_{\\eta}(A_{Z}),\\delta,D)$ is a morphism of crossed modules, and\n\\item the projection $\\phi_\\ast: (\\mathfrak{o}_m(Z)^\\phi,\\Gamma_\\eta(A_{Z})^\\phi,\\delta,D)\\to (\\mathfrak{o}_m(X),\\Gamma_\\eta(A_{X}),\\delta,D)$ is a morphism of crossed modules.\n\\end{itemize}\nMoreover,\n$$(\\mathfrak{o}_m(Z),\\Gamma_\\eta(A_{Z}),\\delta,D) \\hookleftarrow (\\mathfrak{o}_m(Z)^\\phi,\\Gamma_\\eta(A_{Z})^\\phi,\\delta,D) \\xrightarrow[]{\\it \\phi_\\ast} (\\mathfrak{0}_m(X),\\Gamma_\\eta(A_{X}),\\delta,D),$$\nare quasi-isomorphisms of crossed modules. Same conclusion holds true for the weak counterpart.\n\\end{lemma}\n\\begin{proof}\nBy using similar arguments as those in Lemma \\ref{Rmk1} if follows that if $v$ is a (weak) Killing vector field on $(Z_0,\\eta^Z)$ then $\\phi_\\ast(v)$ is also a (weak) Killing vector field on $(X_0,\\eta^X)$. Therefore, the result follows by applying Proposition 7.4 and Theorem 7.3 from \\cite{OW} after restricting the structure.\n\\end{proof}\nThe following result is clear.\n\\begin{lemma}\\label{MoritaLemma2}\nIf $\\eta_1$ and $\\eta_2$ are equivalent Riemannian metrics on $X_1\\rightrightarrows X_0$ then the crossed modules $(\\mathfrak{o}_{m}^{\\textnormal{w}}(X,\\eta_1),\\Gamma(A_X),\\delta,D)$ and $(\\mathfrak{o}_{m}^{\\textnormal{w}}(X,\\eta_2),\\Gamma(A_X),\\delta,D)$ agree.\n\\end{lemma}\n\nSuppose that $X$ and $Y$ are Morita equivalent Lie groupoids so that there is a third Lie groupoid $Z$ with Morita fibrations $Z\\to X$ and $Z\\to Y$. From \\cite{dHF2} we know that if $\\eta^X$ is a Riemannian metric on $X$ then there exists a Riemannian metric $\\eta^Z$ on $Z$ that makes the fibration $Z\\to X$ Riemannian. We can slightly modify $\\eta^Z$ by a cotangent averaging procedure so that we get another Riemannian metric $\\tilde{\\eta}^Z$ on $Z$ which descends to $Y$ defining a Riemannian metric $\\eta^Y$ making of the fibration $Z\\to Y$ Riemannian. It turns out that these pullback and pushforward constructions are well-defined and mutually inverse modulo equivalence of metrics. This is because $\\eta^Z$ and $\\tilde{\\eta}^Z$ turn out to be equivalent. In this case we refer to $(X,\\eta^X)$ and $(Y,\\eta^Y)$ as being \\emph{Morita equivalent Riemannian groupoids}. It suggests a definition for Riemannian metrics over differentiable stacks. Namely, a stacky metric on the orbit stack $[X_0\/X_1]$ associated to a Lie groupoid $X_1\\rightrightarrows X_0$ is defined to be an equivalence class $[\\eta]$ of a Riemannian metric $\\eta$ on $X$. For further details the reader is recommended to visit \\cite{dHF2}. \n\n\\begin{theorem}\\label{MoritaTheorem}\nIf $(X_1 \\rightrightarrows X_0,\\eta^X)$ and $(Y_1 \\rightrightarrows Y_0,\\eta^Y)$ are Morita equivalent Riemannian groupoids then the crossed modules $(\\mathfrak{o}_{m}^{\\textnormal{w}}(X),\\Gamma(A_X),\\delta,D)$ and $(\\mathfrak{o}_{m}^{\\textnormal{w}}(Y),\\Gamma(A_Y),\\delta,D)$ are isomorphic in the derived category of crossed modules. In consequence, the following quotient spaces are isomorphic:\n$$\\mathfrak{o}_{m}^{\\textnormal{w}}(X)\/\\textnormal{im}(\\delta)\\cong \\mathfrak{o}_{m}^{\\textnormal{w}}(Y)\/\\textnormal{im}(\\delta).$$\n\\end{theorem}\n\\begin{proof}\nThis result is consequence of Lemmas \\ref{MoritaLemma1} and \\ref{MoritaLemma2} together with Theorem 7.4 and Corollary 7.2 from \\cite{OW}.\n\\end{proof}\nMotivated by the previous result and \\cite[Def. 8.1]{OW} we set up the following definition.\n\\begin{definition}\nLet $(X_1 \\rightrightarrows X_0,\\eta)$ be a Riemannian groupoid. A \\emph{geometric Killing vector field} on the quotient Riemannian stack $([X_0\/X_1],[\\eta])$ is defined to be \n$$\\mathfrak{o}([X_0\/X_1],[\\eta]):=\\mathfrak{o}_{m}^{\\textnormal{w}}(X)\/\\textnormal{im}(\\delta).$$\n\\end{definition}\nOn the one hand, if we consider proper \u00e9tale Riemannian groupoids then geometric Killing vector fields recover the classical notions of Killing vector fields on both Riemannian manifolds and Riemannian orbifolds as defined for instance in \\cite{BZ}. On the other hand, if we consider the Riemannian groupoid $\\textnormal{Hol}(M,\\mathcal{F})\\rightrightarrows M$ associated to a regular Riemannian foliation $(M,\\mathcal{F})$ then geometric Killing vector fields recover the notion of transverse Killing vector fields as defined in \\cite[p. 84]{Mo}.\n\nWe finish this section by proving a dimensional result for Riemannian foliation groupoids.  A \\emph{foliation groupoid} is a Lie groupoid $X_1 \\rightrightarrows X_0$ whose space of objects $X_0$ is Hausdorff and whose isotropy groups $X_x$ are discrete for all $x\\in X_0$. For instance, every \\'etale  Lie groupoid with Hausdorff objects manifold is a foliation groupoid. The converse is not true, however every foliation groupoid is Morita equivalent to an \\'etale groupoid. As shown in \\cite{C}, being a foliation groupoid is equivalent to the associated Lie algebroid anchor map $\\rho:A_X\\to TX_0$ being injective. As a consequence, the manifold $X_0$ comes with a regular foliation $\\mathcal{F}$ tangent to the leaves of $\\mathrm{im}(\\rho)\\subseteq TX_0$. Note that if $X_1 \\rightrightarrows X_0$ is source-connected then the leaves of $\\mathrm{im}(\\rho)\\subseteq TX_0$ coincide with the groupoid orbits. \n\n\\begin{proposition}\n\tIf $(X_1 \\rightrightarrows X_0,\\eta)$ is a Riemannian foliation groupoid then the algebra of geometric Killing vector fields on $([X_0\/X_1],[\\eta])$ has finite dimension.\n\\end{proposition}\n\\begin{proof}\nLet $\\iota: T\\hookrightarrow X_0$ be a complete transversal submanifold to the orbit foliation $\\mathcal{F}$ of $X_1 \\rightrightarrows X_0$  and consider its restricted groupoid $X_T\\rightrightarrows T$. As $X_T\\rightrightarrows T$ is \u00e9tale and Morita equivalent to $X_1 \\rightrightarrows X_0$ (see \\cite[136]{MM}), from Theorem \\ref{MoritaTheorem} it follows that \n$$\n\t\t\\mathfrak{o}([X_0\/X_1],[\\eta])=  \\mathfrak{o}_m^{\\textnormal{w}}(X_{T}, \\iota^*\\eta)\/\\textnormal{im}(\\delta)\\cong \\mathfrak{o}(T)^{X}. \n$$\nHere $\\mathfrak{o}(T)^{X}$ denotes the transversal Killing vector fields that are invariant by the normal action. Therefore, $\\mathrm{dim}\\mathfrak{o}([X_0\/X_1],[\\eta])\\leq \\frac{1}{2}\\mathrm{codim}(\\mathcal{F})(\\mathrm{codim}(\\mathcal{F})+1).$\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Abstract}\n{\\footnotesize\nSupernova 1604 is the last  Galactic supernova for which historical records exist.\nJohannes Kepler's name is attached to it, as he published\na detailed account of the observations made  by himself and European colleagues.\nSupernova 1604 was very likely a Type Ia supernova, which exploded\n 350~pc to 750~pc above the Galactic plane.\n Its supernova remnant, known as Kepler's supernova remnant, \n shows clear evidence for interaction with\n nitrogen-rich material in the north\/northwest part of the remnant,\n which, given the height above the Galactic plane, must find its\n origin in mass loss from the supernova progenitor system.\n The combination of a Type Ia supernova and the presence\n of circumstellar material makes Kepler's supernova remnant\n a unique object to study the origin of Type Ia supernovae.\n The evidence suggests that\n  the progenitor binary system  of supernova 1604\n consisted of a carbon-oxygen white dwarf and an evolved\n companion star, which most likely was  in the (post) asymptotic giant branch of its evolution.\n A problem with this scenario is that the companion star must have survived\n the explosion, but no trace of its existence has yet been found, despite a deep search.\n }\n \\tableofcontents\n\n \n \n \n \\section{Introduction}\n\\label{sec:intro}\n\nSupernova (SN) 1604 \\index{SN\\,1604 (SNR G4.5+6.8)} is the last of the historical supernovae. We know of at least\ntwo other supernovae with later explosion dates, those associated\nwith the supernova remnants Cassiopeia A     \\index{Cassiopeia A (Cas A, G111.7-2.1)}  \\index{SNR G111.7-2.1 (Cas A)}\nand G1.9+0.3 \\citep{green08}, \\index{SNR G1.9+0.3}\nbut no known historical records of the associated supernova events exist.\n\nThe discovery of SN\\,1604 occurred on October 9, 1604 (Gregorian calendar),\nearly in the evening when all interested in astronomy gazed the sky\nto watch the conjunction of  Mars, Jupiter and Saturn.\nThis special circumstance probably helped the early\ndiscovery of the supernova at around \n20 days before\nmaximum brightness \\citep{stephenson02}.\nThe occurrence of a new star in conjunction with three bright planets \nlead to fierce debates among contemporary astronomers,\nmost of whom still held partial medieval world views, with many elements\nof astrology. Of all these astronomers, it is in particular\nJohannes Kepler's \\index{Kepler, Johannes} name that is now attached to this supernova, \nbecause  he wrote a book ``De Stella Nova in Pede Serpentarii'' (\"On the new star in the foot of the Snake [Ophichius]\")\nabout the new star,\nin which he\npublished his own observations of the supernova and those by various European  colleagues,\nand discusses the importance of the new\nstar, including a possible link to the \\index{star of Bethlehem} star of Bethlehem.\nKepler only started observing SN 1604 relatively late, October 17, due to unfortunate  weather \ncircumstances in Prague, his city of residence.\nKepler and his contemporaries attached a great deal of metaphysical significance to the appearance of\nthe new star: {\\em ``The star's significance is a difficult matter to establish and we can be sure of only one thing: \nthat either the star signifies nothing at all for Mankind or it signifies something of such exalted importance that it is beyond the grasp and understanding of any man''}.\nKepler himself, in fact, was a bit more cautious about the metaphysical implications of the event than some of his contemporaries \\citep{granada05}.\n\nCurrently, the  remnant of SN 1604, labeled G4.5+6.8, \\index{SNR G4.5+6.8} but\nusually called \\index{Kepler's supernova remnant (SNR G4.5+6.8)} Kepler's supernova remnant (SNR), or even \"Kepler\" for short,\nis one of the best studied SNRs.  It appears that Kepler's SNR has indeed a wide ranging significance,\nbut not for metaphysical reasons, but  for understanding the nature of  \\index{Type Ia supernova} Type Ia supernovae.\nSN 1604 was almost certainly a  Type Ia supernova, but, peculiarly, the SNR itself is interacting\nwith a nitrogen-rich \\index{CSM (circumstellar medium)} \\index{circumstellar medium} circumstellar medium (CSM) at a distance of 2-3 pc from the explosion centre, \nsuggesting that the progenitor system had significant \\index{stellar mass loss} \\index{stellar wind} mass loss. \n\nAs described in more detail in Sections 3 and 6, two basic types of supernova Type Ia progenitor models exists  \\index{single degenerate Type Ia model} \\index{double degenerate Type Ia model} \n\\citep{maoz14}: 1) the single degenerate model,\naccording to  which a CO white dwarf \\index{white dwarf} accretes  from an (evolved) stellar companion until the pressure\nin the core leads to the thermonuclear explosion \\index{thermonuclear explosion} of the white dwarf, at the moment that the progenitor\nhas  a mass approaching the \\index{Chandrasekar mass} Chandrasekar mass (1.38 {M$_\\odot$}), and 2) the \ndouble degenerate model, according to which Type Ia explosions are triggered by the merging of two white dwarfs.\nThe time scale from birth of the binary system to the merging of the two white dwarfs is generally expected to  be a few $10^9$~yr \\citep{maoz14}, \nsince the two stars are brought closer to each other due  gravitational radiation.\nThe fact that Kepler's SNR is interacting with CSM is, therefore, more consistent\nwith the single degenerate scenario. However, as will be discussed, this scenario is also not fully consistent\nwith the properties of Kepler's SNR, in particular with the lack of a surviving donor star.\n\n\\begin{figure}\n\\includegraphics[trim=0 300 0 0,clip=true,width=\\textwidth]{Fig_1_kepler_mosaic.jpg}\n\\caption{\\label{fig:4images}\nKepler's SNR at different wavelengths: a) Radio VLA (4.85 GHz) map \\citep{delaney03}; b) 24$\\mu$m dust emission as\nobserved by the Spitzer MIPS instrument \\citep{blair07}; c) optical image (H$\\alpha$, [NII] and  [OIII]) obtained\nby the Hubble Space Telescope\/ACS instrument (credit: ESA\/NASA, The Hubble Heritage Team (STScI\/AURA));\nd) Chandra X-ray image in the iron-rich 0.7-1.0 keV band \\citep{reynolds07}.\nIntensity scaling of Fig. a, b, and d are square-root scalings, bringing out fainter details.\n}\n\\end{figure}\n\n\\section{The supernova remnant, its distance and its multiwavelength properties}\n\\label{sec:properties}\n\n\\subsection{Position, distance estimates and SN 1604 as a runaway system}\n\nKepler's SNR was discovered by Walter Baade  \\index{Walter Baade}  as \"a small patch of nebulosity close to the expected\nplace\" \\citep{baade43}. \\citet{minkowski59} revealed that the spectrum of the optical nebula displayed strong [NII]\nemission as compared to H$\\alpha$ emission. \nThe H$\\alpha$\/[NII] is concentrated toward the north\/northwestern region of the SNR and a bar-shape\nregion across the center, roughly oriented in the NW-SE direction (Fig.~\\ref{fig:4images}).\nThe overall extent of the SNR is more clear from radio and X-ray images of the SNR, revealing\na roughly spherical shell with two protrusions (\"Ears\") in the NW and SE \\citep[e.g.][]{delaney03,reynolds07}.\nThe SNR's center is located at $\\alpha_\\mathrm{J2000}=17{\\rm h}30{\\rm m}41{\\rm s} ,\\ \n\\delta_\\mathrm{J2000}=-21^\\circ 29^\\prime 32^{\\prime\\prime}$, in Galactic coordinates\n$l=4.52^\\circ, b=6.82^\\circ$. The angular radius of the SNR is approximately 1.8{$^\\prime$}\\ (not taking\ninto account the \"Ears\").\n\nThe distance to the SNR is uncertain, with estimates varying from $d=3.2$~kpc \\citep{danziger80} to  \n$d=12$~kpc \\citep[][]{vandenbergh73}.\nThe long distance was based on the maximum visual magnitude of the supernova event of\n$V_\\mathrm{max}\\approx -2.5$, as estimated by \\citet{baade43} from the historical records.\nAfter correcting for absorption \\citep[$A_\\mathrm{V}=2.17-3.47$,][]{vandenbergh77,danziger80}, the peak\nmagnitude can then be compared to the  absolute magnitude of Type Ia supernovae of $M_\\mathrm{V}=-19.0 \\pm 0.5$.\nA Hubble parameter was adopted of $H_0=100$~km\/s\/Mpc.\n\nMore recent distance estimates, not based on the historical peak brightness \\index{peak brightness} of the\nsupernova,  vary widely, ranging from $d=3.9\\pm 1.4$~kpc \\citep{sankrit05}\nbased on combining proper motion and Doppler broadening of H$\\alpha$ emission, \n up to, or even beyond, 6-7~kpc based on the angular size of\nthe SNR, combined with the energetics of the supernova \\citep[][see {Section}~\\ref{sec:dynamics} for details]{aharonian08,chiotellis12,patnaude12}.\nRadio absorption measurements\ntoward the SNR are consistent with a distance in the range 4.8-6.4~kpc \\citep{reynoso99}.\n\nThe Galactic latitude of Kepler's SNR ($b=6.8${$^{\\circ}$}) implies a large height above the Galactic plane\nof $z=594 d_5$~pc (with $d_5$ the distance in units of 5~kpc). The large separation from  active star\nforming regions suggests that SN 1604 was Type Ia supernova, in agreement with the reconstructed\nsupernova light-curve, as already pointed out by \\citet{baade43}. However, the large amounts of\nnitrogen present in the optical nebula can only have come from material processed by the CNO nucleosynthesis\ncycle, suggesting a  stellar wind origin for the optical nebula. This lead \\citet{bandiera87} to suggest  that\nthe progenitor was a massive star, which escaped the Galactic plane with a high velocity. The high\nproper motion of the progenitor is in agreement with observations of the kinematics of the optical nebula\n({Section}~\\ref{sec:dynamics}). Given the historical light curve, \na Type II origin of SN 1604 was excluded, but a Type Ib \\index{Type Ib supernova} supernova\nwas consistent with both the light curve and the idea of a massive runaway star as the progenitor of the supernova \\citet{bandiera87}.\n\n\n\n\n\\begin{figure}\n\\centerline{\n\\includegraphics[width=0.85\\textwidth]{Fig_2_kepler_chandra_oxygen_fel_si.png}\n}\n\\caption{\nChandra X-ray image, with the three image channels  (red, green, blue) \ncorresponding\nto oxygen emission (0.5-0.7 keV), Fe-L emission \\index{Fe-L complex} (0.7-1 keV) and Si-K \nemission (1.7-1.9 keV), respectively. It can be seen that the Si-K emission (blue)\npeaks more outward than the the Fe-L shell emission (green). Peaks in the oxygen emission\n(red\/yellowish) occur at the outskirts in the northeast and along the bar, suggesting\nthat the emission comes from shock-heated, circumstellar material.\n\\label{fig:Xray}}\n\\end{figure}\n\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=0.8\\textwidth]{Fig_3_kepler_chandra_spec_b.pdf}}\n\\caption{\nThe Chandra ACIS-S X-ray spectrum of Kepler's SNR, observed in 2006.\nThe \"bump\" between 0.7- 1.2 keV consists mainly of Fe-L emission \\index{Fe-L complex}, consisting\nof complex of lines from Fe XVII to Fe XXIV. The prominence of this Fe-L complex\nin the spectrum of Kepler's SNR is indicative of a Type Ia origin.\\label{fig:xspec}\n}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\centerline{\n\\includegraphics[width=0.85\\textwidth]{Fig_4_kepler_24mu_sik_cont.png}\n}\n\\caption{\nMultiwavelength image of Kepler's SNR. The red channel represents the 24$~\\mu$m dust emission as\nobserved by the Spitzer MIPS instrument \\citep[][]{blair07}. The green channel is the Si-K band (1.7-2.0 keV), representing\nthe bulk of the thermal X-ray emission, associated with Si-rich ejecta. The Si-K emission comes  on average from a smaller radius than the dust emission,\nindicating that the dust originates from the swept up circumstellar medium, rather than the ejecta.\nThe blue channel (enhanced with contours) shows the X-ray continuum emission in\nthe 4-6 keV band. In the SE the emission is dominated by synchrotron emission from close to the shock front. \\index{X-ray synchrotron radiation}\nIn the NW it is probably a mixture of synchrotron emission and thermal bremsstrahlung.\nThe X-ray bands are taken from the 2006 Chandra observations of Kepler's SNR \\citep{reynolds07}.\n\\label{fig:morphology}}\n\\end{figure}\n\n\\index{X-ray imaging spectroscopy}\n\\subsection{X-ray imaging spectroscopy and SN 1604 as a Type Ia supernova }\n\\label{sec:xrays}\nThe idea of a Type Ib supernova origin for SN 1604 was discredited with the advent of X-ray imaging\nspectroscopy with CCDs. These type of detectors were first used by\n\\index{ASCA X-ray satellite} ASCA, and it provides now still the most often used instrumentation for X-ray observatories\nsuch as \\index{Chandra X-ray observatory} \\index{XMM-Newton X-ray space observatory} \n\\index{Suzaku X-ray satellite} \nChandra, XMM-Newton, and Suzaku. \nThe X-ray spectrum of Kepler's SNR (Fig.~\\ref{fig:xspec}) shows that the\ndominant line emission is caused by  \\index{Fe-L complex} Fe-L transitions \\citep{kinugasa99,cassam04,reynolds07}, which is very indicative of\nType Ia SNRs, since they produce 0.3-1.3 {M$_\\odot$}\\ of Fe and Fe-group elements.\nIn addition line emission from intermediate mass elements \\index{intermediate mass elements}\nlike silicon, sulphur, argon and calcium, are prominent.\nUnlike Kepler's SNR, young core collapse SNRs on the other hand, show dominant emission from oxygen, neon and magnesium.\nRecently, \\citet{katsuda15} confirmed the high nitrogen \\index{nitrogen} abundance in Kepler's SNR using high resolution X-ray spectroscopy with the XMM-Newton reflective\ngrating spectrometer.\n\n\nThe spatial distribution of the  X-ray line emission indicates that the ejecta are layered according to element mass, with the intermediate mass elements \nlocated, on average, at a larger radius than iron \\citep[][illustrated in Fig.~\\ref{fig:Xray}]{cassam04}. This distribution is similar to that of other Type Ia SNRs like Tycho's SNR \\citep[SN\\,1572,][]{hwang97} \\index{SNR G120.1+1.4} \\index{Tycho's supernova remnant (SNR G120.1+1.4)} \n\\index{SN\\,1572 (SNR G120.1+1.4)}\nand\nSNR  0519-69.0 \\citep{kosenko10}, but Fe-L emission is more prominent in Kepler's than in Tycho's SNR, whereas SNR 0519-69.0 and\nSN\\,1006  display more \\index{SNR\\,0519-69.0} \\index{SN\\,1006 (SNR G327.6+14.6)} \\index{SNR G327.6+14.6} \noxygen line emission from the outer layers\nThe relatively high iron abundance in Kepler's SNR,\nas compared to other Type Ia SNRs, suggests that SN 1604 was an iron\/nickel-rich supernova, similar to the peculiar Type Ia supernova SN1991T \\citep[][but see Section~\\ref{sec:lightcurve}]{patnaude12}.\n\n\nRecently it has become possible to detect line emission from the uneven elements manganese (Mn) and chromium (Cr).\nThe Mn yields for supernovae are a result of the presence of neutron-rich nuclei in the progenitors, which ultimately is linked to the nitrogen abundance of\nthe progenitor at the main sequence. Hence, it can be used to infer the metallicity of the progenitor. \\citet{park13} used the Mn\/Cr ratio to show\nthat SN 1604 must originate from a star with super-solar metallicity, ruling out a halo star progenitor\n\nThe continuum X-ray emission of Kepler's SNR, best distinguished in the 4-6 keV band (blue in Fig.~\\ref{fig:morphology}), shows narrow filaments tracing the outer region of\nthe SNR. \\index{X-ray synchrotron filaments} Most of this emission, especially the narrow filaments in the SE,\n are thought to be caused by X-ray synchrotron emission from\n10-100 TeV shock accelerated electrons \\citep[][for a review]{helder12b}. The widths of the filaments can be used to estimate the post-shock \nmagnetic fields, indicating for Kepler's SNR a magnetic field strength of $B\\approx 120$~$\\mu$G. Since the 10-100~TeV lose their energy very fast in such  a magnetic field ($\\sim 10-20$~yr), the X-ray synchrotron radiation is an excellent tracer of  the shock front. Some of the X-ray synchrotron filaments\nare situated along the edges of the central bar (Fig.~\\ref{fig:morphology}), suggesting that we observe the shock front  under a\n(nearly) edge-on viewing angle. Given the position of the bar in the center of the SNR, this  viewing angle  indicates  \na peculiar, perhaps halter-like, morphology of the SNR in three dimensions \\citep{burkey13}.\n\n\\subsection{The circumstellar medium as studied in the optical and infrared}\n\\label{sec:opt_ir}\nThe shocked interstellar medium in Kepler's SNR  is best studied in the optical and infrared. The optical emission, dominated by\nhydrogen line emission, comes from northwestern part of the SNR and from the central bar. \\index{radiative shocks} \\index{non-radiative shocks}\nThe optical emission can be divided in emission caused by either radiative or non-radiative shocks \\citep{dennefeld82,fesen87,blair91,sankrit08}. \nRadiative shock emission is characterised by a mix of hydrogen line emission (Balmer and Lyman lines) and\nforbidden line emission from NII, OIII, and SII. The H$\\alpha$ line emission is in those cases  double peaked, since NII has among\nothers line emission at $\\lambda=654.9$~nm, close to H$\\alpha$ ($\\lambda=656.4$~nm).\nForbidden line emission comes from regions heated by slow shocks ($\\lesssim 200$~{km\\,s$^{-1}$}), \nwhich heat plasma to temperatures below $\\sim 10^6$~K. For this temperature regime radiative losses by optical\/UV line emission causes rapid cooling. \nFor Kepler's SNR the overall shock speed is several thousand {km\\,s$^{-1}$}. \nSo the radiative emission must come from dense knots \\citep[$n \\gtrsim 10^3$~{cm$^{-3}$}][]{leibowitz83}, in which locally the shock wave\nhas decelerated considerably. The [NII]\/hydrogen line ratios measured for radiative shocks in Kepler's SNR \nsuggests an overabundance in nitrogen of a factor 2-3.5  \\citep{leibowitz83,blair91}.\n\nThe non-radiative emission is caused by neutral hydrogen atoms entering the shock, after which they may not be immediately ionised, but first\nget into an excited state, or undergo will charge exchange  with shock-heated protons behind the shock front. H$\\alpha$ line spectra of \nnon-radiative shocks typically consist of  narrow-line emission caused by direct excitation of the neutrals, and broad-line emission caused by\nhot protons that just picked up an electron through charge exchange\nThe narrow line width and line center can be used to measure the plasma speed\nand temperature of the un-shocked gas, whereas the broad-line width can be used to measure the temperature immediately ($\\lesssim 10^{15}\/n_\\mathrm{p}$~cm)\nbehind the shock (see {Section}~\\ref{sec:dynamics}). The H$\\alpha$ flux of the non-radiative filaments can be used to infer densities as well,\nwhich appears to be of the order of $\\sim 10$~{cm$^{-3}$} \\citep{blair91}. This is surprisingly large, given the height above the Galactic plane\nof $594 d_5$~pc. \nThe optical emission from Kepler's SNR, therefore indicates that the SNR is interacting\nwith  dense, CNO processed material, which, given the isolated nature of the SNR high above the Galactic plane, \ncan only be understood by assuming that the circumstellar material originates from \nthe progenitor system of SN\\,1604.\n\nIn the infrared Kepler's SNR has been observed in the wavelength range from 12-100$\\mu$m by all major\ninfrared observatories \\index{Infrared Astronomical Satellite (IRAS)} \\index{Spitzer Space Telescope} \\index{Herschel Space observatory} \\index{Infrared Space Observatory (ISO)}\n\\citep[IRAS, ISO, Spitzer, Herschel, see][]{braun87,douvion01,blair07,gomez12,williams12}.\nAdditional emission for wavelength larger than 100~$\\mu$m, even up to 850~$\\mu$m may be present \\citep{morgan03,gomez12}, but\nat long wavelengths background subtraction is difficult, and even at 100~$\\mu$m there is already a factor three discrepancy\nbetween IRAS and Herschel based flux measurements \\citep{gomez09}.\n\n\\index{dust emission}\nInfrared emission from SNRs is caused by warm dust, heated by collisions with hot electrons and ions. The dust temperature\n is established by the equilibrium between collisional heating and thermal emission \\citep[][]{draine81,dwek87}. \n\n Both heating\nand emission depend on the surface area of the dust grains, and for large dust grains the temperature is independent of the size\nof the dust particles. However, since the wavelength of the radiation and the dust particle size can be comparable, the dust emission\ncannot be adequately described by Stefan-Boltzmann's law. Moreover, the emission spectrum also depends on the composition of the dust.\n\nInfrared photometry of Kepler's SNR indicates dust temperatures of 80--120~K \\citep{braun87,douvion01,blair07,gomez12,williams12},\nwhereas more detailed spectra obtained with the ISO-SWS and Spitzer-IRS spectrometers  \\citep[respectively][]{douvion01,williams12} \nindicate that the dust particles themselves are  silicates (silicon oxides). Interestingly, the modelling of the spectra gives rise\nto different interpretations. \\citet{douvion01} conclude that the dust comes from regions with densities of $n_\\mathrm{e}=500-7500$~cm$^{-3}$\nwith plasma temperatures of $(0.4-6)\\times 10^5$~K. On the other hand, \\citet{williams12} estimates the  densities to be  $n_\\mathrm{e}\\approx 5$~cm$^{-3}$ in the South\nto $n_\\mathrm{e}\\approx 50$~cm$^{-3}$ in the North, but assumes that the plasma temperatures are consistent with the X-ray measured\ntemperatures ($T_\\mathrm{e}\\approx 10^7$~K, and  ion temperatures a factor ten higher). Both conditions may arise\nin Kepler's SNRs as the radiative shocks are going through very dense regions, whereas the non-radiative shocks are responsible\nfor the X-ray emitting plasma. Indeed, several of the regions analysed by \\citet{williams12} contain non-radiative shock emission. The infrared spectrum\nof the region in which radiative shock emission is dominant is characterised by bright forbidden line emission in the infrared (among others  [Si II], [Fe II]). \nIt is  important to note  that the location of the dust emission indicates that\nall the dust is associated with shocked CSM,\nand not with supernova ejecta. This appears to be a characteristic of Type Ia supernovae, which do not seem to produce dust in their ejecta \\citep{williams12}.\nThis is in contrast to a core collapse SNR like Cas A \\citep{lagage96} or SN 1987A, where dust clearly is formed from\nsupernova ejecta. \\index{Cassiopeia A (Cas A, G111.7-2.1)}  \\index{SNR G111.7-2.1 (Cas A)}\n\\index{SN 1987A}\n\\index{dust emission}\n\\index{dust production}\n\n\n\\section{The dynamics of Kepler's SNR}\n\\label{sec:dynamics}\n\n\\subsection{Velocity measurements}\nThe earliest measurements of  velocities in Kepler SNR's were  radial velocity measurements of the bright optical\nknots, indicating line emission that is blue shifted by about 230 {km\\,s$^{-1}$} \\citep{minkowski59}. Optical proper motion studies\n\\citep{vandenbergh77,bandiera91}\nalso indicated velocity offsets,  indicating that the bright optical knots have a bulk velocity of 212 $d_5${km\\,s$^{-1}$}\\ in northwestern direction,\naway from the Galactic plane,\nwhereas the expansion velocity, based on spherical expansion directed away from the centre of the SNR, is only $(74\\pm 29)d_5$~{km\\,s$^{-1}$}\n\\citep{bandiera91}. The slow expansion velocity, corresponding to an expansion time scale of 32000~yr, indicates that the velocity\nof the optical knots reflects mostly the velocity of these knots prior to interaction with the SNR blast wave. \n\nThe more diffuse non-radiative H$\\alpha$ emission \\citep{fesen87} provides information on the local shock wave velocities through\nthe width of the broad H$\\alpha$- component; and the narrow line component\nprovides information on the velocity of the unshocked, partial neutral gas,\n immediately ahead of the shock.\nThe latter informs us that the unshocked gas is blue shifted with about $180$~{km\\,s$^{-1}$} \\citep{blair91,sollerman03}.\n\nThe width of the broad line H$\\alpha$ component is caused by thermal Doppler broadening \\index{Doppler broadening} of shock-heated protons.\nThe reported H$\\alpha$ line FWHM of $\\sim 1800$~{km\\,s$^{-1}$}\\ translates into  shock velocities of $1500-2000$~{km\\,s$^{-1}$} \\citep{blair91}.\nPart of the uncertainty is caused by the uncertainty in electron-ion temperature ratio. \\cite{sankrit05} combined this shock velocity measurement\nwith a measure proper motion of diffuse filament in the NW of $1.45\\pm 0.03$ {$^{\\prime\\prime}$}\/16.33 yr to infer a distance to Kepler's SNR\nof 3.0-5.0~kpc. However, note that some of the Doppler broadening widths measured by  \\cite{blair91} indicate \nshock velocities larger than 2000~{km\\,s$^{-1}$}. Notably, the filament labeled \"NW diffuse emission\" has a width of $3409 \\pm 454$~{km\\,s$^{-1}$}, which corresponds\nto a shock velocity range of 2300-4000~{km\\,s$^{-1}$} \\citep{vanadelsberg08}, or even larger if cosmic-ray acceleration \\index{cosmic rays} absorbs part of the shock\nenergy-flux into the shock, which results in lower than expected post-shock temperatures \\citep[e.g.][]{vink10a}. \n\n\n\\begin{figure}\n\\centerline{\\includegraphics[trim=0 -30 0 0 0,width=0.4\\textwidth]{Fig_5_left_expansion3.pdf}\n{\\includegraphics[width=0.6\\textwidth]{Fig_5_right_expansion2.pdf}}\n}\n\\caption{\nLeft: Difference image between images in the 1-1.5 keV band in 2000 and 2006, as observed by Chandra.\nThe shadow-like features are caused  by proper motions.\nRight: X-ray expansion measurements based on these two Chandra images in various energy bands \\citep[adapted from figure in][]{vink08b}.\n\\label{fig:expansion}\n}\n\\end{figure}\n\n\n\nThe optical velocity measurements pertain to the densest regions of Kepler's SNR. The overall\nexpansion of Kepler's SNR has been measured in the radio \\citep[][with VLA]{dickel88} and X-rays \\citep[][see Fig.~\\ref{fig:expansion}]{vink08b,katsuda08} using proper\nmotion measurements. These measurements show that the northwestern part is expanding at a slower rate than the southwestern part.\nA useful way to characterise the expansion rate is in terms of the expansion parameter \n\\begin{equation}\nm \\equiv \\frac{R\/t}{V},\n\\end{equation}\nwith $R$ the radius, $t$ the age of the SNR and $V$ the velocity. For proper motions both $R$ and $V$ depend in a similar way on the distance,\nso one can replace $R$ and $V$ with angular radius and proper motion. \nOne can also calculate the \\index{expansion age} expansion age $\\tau_\\mathrm{exp}\\equiv R\/V$, in which case $m=t\/\\tau_\\mathrm{exp}$, \\index{expansion parameter}\nwith $t$ the true age\nof the SNR. For SNRs in the Sedov-Taylor phase of the evolution we\nexpect $R \\propto t^{2\/5}$ and hence $m=2\/5$, whereas in the ejecta dominated phase one expects $m=0.7-0.9$ \\citep{chevalier82}.\nIt turns out that for Kepler's SNR in the Northwest $m=0.3-0.35$, whereas in the Southwest the expansion is faster, with $m=0.6-0.7$.\nThe expansion parameter in the northwest is lower than $m=2\/5$, which indicates that the shock  must have encountered a density enhancement.\n\\citet{vink08b} estimates that the  total excess mass toward the northwestern region is about 1~{M$_\\odot$}.\n\nThe proper measured proper motions translate in shock velocities of\n\\begin{equation*}\nV_\\mathrm{sh}= m 9920(\\theta\/2.79) d_5~\\mathrm{km\\ s^{-1}}\n\\end{equation*} \nfor an age of 400~yr, with $\\theta=2.79${$^\\prime$}\\ the angular radius of Kepler's SNR.\nGiven the measured values of $m$ the shock velocities around Kepler's SNR are, therefore, in the range of 2900-7000 $d_5$~{km\\,s$^{-1}$}.\n\n\\subsection{Hydrodynamical simulations of Kepler's SNR}\n\nAs discussed above the density enhancements in the north-northwestern regions, as well as the evidence for high density,\nnitrogen-rich material suggests that the progenitor system of SN\\,1604 suffered significant mass loss. \n\\citet{borkowski94} simulated this situation in the context of a Type Ib origin of SN\\,1604 and \\citet{chiotellis12} adapted it\nfor the case of a Type Ia scenario, which I will briefly describe here.\n\nThe basic idea is that a dense wind has emanated  from the secondary star with  a mass-loss rate of $\\dot{M} $~{M$_\\odot$}\\,yr$^{-1}$,\nand a wind velocity $v_\\mathrm{w}$. The wind must have lasted long enough for the wind material to reach the current\nradius of Kepler's SNR ($2.6 d_5$~pc) implying that the wind  must have prolonged for\nat least $t_\\mathrm{w} \\gtrsim 2.5 \\times 10^5 d_5 (v_\\mathrm{w}\/10~\\mathrm{km\\,s}^{-1}$)~yr.\nThe total mass lost by the system  is given by \n\\begin{equation}\nM_\\mathrm{wind} \\gtrsim \\dot{M}t_\\mathrm{w}= \\dot{M} \\frac{R}{v_\\mathrm{w}}=\n2.5\\left( \n\\frac{\\dot{M}}{10^{-5}\\mathrm{M_\\odot\\,yr^{-1}}}\\right)\n\\left(\\frac{v_\\mathrm{w}}{10\\ {\\rm km\\,s}^{-1}}\\right)^{-1}\nd_5 \\ \\mathrm{M_\\odot}.\n\\end{equation}\nThis means that the mass loss rate cannot have been much higher than $10^{-5}$~{M$_\\odot$}\\,yr$^{-1}$, as the total\nmass loss of the donor could not exceed the mass of a viable donor star for a Type Ia progenitor,\nwhich must lie in the range of 3-6~{M$_\\odot$}.\n\nThe optical proper motions of the knots indicate a space of around 250-280~{km\\,s$^{-1}$}\\ (212 $d_5$~{km\\,s$^{-1}$} in NW direction and 180 km\/s\ntoward us). This likely corresponds to the space velocity of the progenitor system.  The combination of a spherical wind\nand a space velocity gives rise to a bow shock around the system in the direction of the motion of the system.\nThe density of the wind as a function of radius $r$ is given by\n\\begin{equation}\n\\rho_\\mathrm{w}(r)=\\frac{\\dot{M}}{4\\pi r^2 v_\\mathrm{w}} = 7.8\\times 10^{-25}\n\\left(\\frac{\\dot{M}}{10^{-5}\\,{\\rm M_\\odot}}\\right) \\left(\\frac{v_\\mathrm{w}}{10\\ {\\rm km\\,s}^{-1}}\\right)^{-1} \\left(\\frac{R}{2.6\\,{\\rm pc}}\\right)^{-2}\\, {\\rm g\\,cm}^{-3}.\n\\end{equation}\nThe densities in the Northwest appear to be  around 50~cm$^{-3}$ ({Section}~\\ref{sec:opt_ir}). The density in the wind\nmay be lower, as the gas encountered by the SNR shock may have been shocked by the wind termination shock,\nand later by the SNR shock. However, we see that the mass loss rate cannot have been substantially lower\n$10^{-5}${M$_\\odot$}\\,yr$^{-1}$, as otherwise the wind densities would fall below the densities of the CSM component in Kepler's SNR.\n\nThe termination radius of the wind is now given by assuming  equilibrium between the ram\npressure of the wind $P_\\mathrm{w}=\\rho_\\mathrm{w}v_\\mathrm{w}^2$ and the ram\npressure of the interstellar medium, $P_\\mathrm{ISM}=\\rho_\\mathrm{ISM}v_\\mathrm{prog}^2$, with\n$v_\\mathrm{prog}$ the velocity of the progenitor system. Hence,\n\\begin{equation}\nR_\\mathrm{ts}=  1.9\\ \n\\left(\\frac{\\dot{M}}{10^{-5}\\,{\\rm M_\\odot}}\\right)^{1\/2}\n\\left(\n\\frac{v_\\mathrm{w}}{10\\,\\mathrm{km\\,s^{-1}}}\n\\right)^{1\/2} \n\\left(\\frac{v_\\mathrm{prog}}{250\\ \\mathrm{km\\,s^{-1}}}\\right)^{-1}\\left(\\frac{n_\\mathrm{ISM}}{10^{-3}\\, \\mathrm{cm^{-3}}}\\right)^{-1\/2}~\\mathrm{pc},\n\\end{equation}\nwith $n_\\mathrm{ISM}=10^{-3}$  an estimate for the ISM at 500~pc above the Galactic plane.\nThis estimate for the termination shock radius is in reasonable agreement with the radius of Kepler's SNR.\n\nThe simulations of \\citet{chiotellis12} show that  both the bow-shock like shape of Kepler's SNR as well as the measured expansion parameters\ncan be reasonably well reproduced if SN\\,1604 indeed went off inside a stellar wind bubble with a systematic velocity of 250~{km\\,s$^{-1}$}. \nIt was assumed that the ejecta mass was 1.4~{M$_\\odot$}, typical for an exploding white dwarf. \nThe simulations showed that for a distance of 4~kpc the radius of the termination shock must be around 2~pc. However,\na SNR with an \\index{explosion energy} explosion energy of $10^{51}$~erg  going off inside such a smaller bubble, would have moved through the wind bubble\nand the shock would have penetrated into the ISM, which is in disagreement with the observations. \nOne can reproduce the characteristic of Kepler's SNR at this distance, but only if the explosion \nenergy is reduced to $2\\times 10^{50}$~erg. This is in disagreement with the typical  Type Ia explosion energies of $1.2 \\times 10^{51}$~erg\n\\citep{woosley07}. Moreover, the high Fe content of Kepler's SNR rather suggest that SN\\,1604 was a relatively energetic type Ia event\n\\citep{park13}. This problem does not occur if Kepler's SNR has a distance of $\\gtrsim 6$~kpc.\n\nA related hydrodynamical study by \\citet{patnaude12}\nconcentrates  on the impact of hydrodynamics on the resulting X-ray spectra. The wind bubble evolution itself is not modelled, but  an \n$1\/r^2$ density\ndistribution is assumed. They confirm that normal Type Ia explosion energies are not consistent with a distance of Kepler's SNR\nof 4~kpc, but that instead a distance of  7~kpc should be considered. \\citet{toledo14} follow more closely the scenario of \\citet{chiotellis12},\nbut the simulations are in 3D and the wind loss is assumed to be anisotropic. As a result more structure is found in the simulated X-ray maps,\nsome of which may explain for example the central bar in Kepler's SNR (e.g. their Fig.~10). In general, the anisotropy of the wind\ncan result in a peculiar structure. \\citet{burkey13} simulated the evolution of the SNR inside a wind strongly varying\nas a function of the polar axis. However, the simulation does not take into account the effects of the velocity of the system as a whole,\nand the expansion parameter of the plasma.\n\n\\section{The progenitor system of SN\\,1604}\n\n\\subsection{Elevated circumstellar nitrogen abundances, silicates  and a single degenerate scenario for SN\\,1604}\n\\citet{chiotellis12} argued that the mass loss properties inferred for the progenitor system of SN\\,1604 suggest that the progenitor\nsystem consisted of CO white dwarf accreting wind material from a Asymptotic Giant Branch (AGB) \\index{Asymptotic Giant Branch (AGB) star} star. \nIndeed, the high mass\nloss rates and the wind expansion time scale are in good agreement with what is generally inferred for AGB stars.\nGiven the relatively high total\nmass loss ($\\sim 2.5${M$_\\odot$}, plus matter that has accreted onto the white dwarf), \nit does mean that the donor star must have been a relatively massive star at the main sequence. \n\nIn addition, the nitrogen-rich abundance of the wind,  and the silicate dust  (Sect.~\\ref{sec:opt_ir}) also gives clues\nabout the donor star. The nitrogen-richness can be explained by \\index{hot bottom burning} hot bottom burning \\citep[e.g.][]{karakas10}, a process in which\nhydrogen is burned through CNO cycle at the base of the outer convective envelope. This brings nitrogen to the surface.\nHot bottom burning occurs in stars with initial masses larger than 4-5~{M$_\\odot$}.\nSilicate dust \\index{dust emission} \\index{sillicates} predominantly forms in winds in which the carbon\/oxygen ratio is smaller than 1. Otherwise the oxygen \nbinds to carbon, making CO, leaving no oxygen for the build up of silicates. In this light it is interesting that \\citet{mcsaveney07}\nfound two AGB stars in the Large Magellanic Clouds, which have elevated nitrogen abundance and a $C\/O<1$. The masses\nof these two stars are inferred to be 4~{M$_\\odot$}\\ and 6~{M$_\\odot$}.\n\nTaken all the evidence together we seem to have a rather consistent scenario for the progenitor system of SN\\,1604 and the evolution\nof the SNR: a single degenerate white dwarf, accreting wind material from a rather massive AGB star. The mass of the donor\nthen implies that the white dwarf was also rather massive at the main sequence (5-6~{M$_\\odot$}). {Kepler's SNR provides therefore the\nbest case that at least some Type Ia supernovae are caused by single degenerate white dwarf systems.}\n\n\\subsection{Problems with a single degenerate Type Ia scenario for SN\\,1604}\n\nThere two problems that need to be solved for the single degenerate scenario for SN\\,1604, or even for any Type Ia binary scenario:\n\n1) The donor star must be relatively massive (4-6{M$_\\odot$}), and therefore a luminous star. A recent search by \\citet{kerzendorf14} did not reveal\nany bright enough star near the centre of Kepler's SNR. The former donor star \\index{donor star} is likely to still be bright, although, its outer envelope\nmay have been removed by the supernova blast. Its space velocity should be close to that of the progenitor system.\nThe lack of a donor star argues instead for a double degenerate explosion for SN\\,1604. A similar conclusion was\n drawn for other Type Ia SNRs, SNR 0509-675 and SN\\,1006 \\citep{schaefer12,gonzalez12}, \\index{SNR G327.6+14.6}\nfor which also  no bright donor stars have been found. For Tycho's SNR a donor star has been identified \\citep{ruiz04}, but this identification\nis disputed \\citep{kerzendorf13}. \\index{SNR G120.1+1.4}\n\n2) The velocity of the progenitor system must have been $\\sim 250$~{km\\,s$^{-1}$}. This implies that the system must have left\nthe Galactic plane around 3~million year ago. \nSuch a time scale is too short for  any Type Ia scenario, which should have evolutionary time scales of $>50$ Myr \\citep[e.g.][]{claeys14}.\n One can of course speculate that Kepler's SNR\nis bound to the Galactic plane, and has been oscillating far up and below the Galactic plane for  considerable longer times.\nThis still creates, however, the problem of how to slingshot away a binary system with 250~{km\\,s$^{-1}$}. Runaway stars \\index{runaway stars}\nare usually created by\nclose interactions of triple systems, but in 90\\% of the cases single stars are being ejected,\nand the highest velocities are not expected to be obtained by the ejected binaries \\citep{leonhard90}. Moreover, in the AGB scenario\nfor SN\\,1604 the binary system must have been relatively wide \\citep{chiotellis12}. Nevertheless, observationally some high velocity binaries are known.\nA well known example is the Mira system, which has a space velocity of about 100~{km\\,s$^{-1}$}, and which consists also of an AGB star and\na probable white dwarf \\citep{martin07}.\n\nProblem 1) is a problem for only the single degenerate scenario, but problem 2) is a problem for any Type Ia  scenario involving\na binary system. \n\n\\subsection{Was SN\\,1604 a core-degenerate Type Ia explosion?}\n\\label{sec:core}\nA possible solution to problem 1 is so-called core degenerate scenario \\index{core degenerate scenario for Type Ia supernovae} \n\\citep{ilkov12,tsebrenko13} according to which a single degenerate system\nmay form a common envelope at the end of the evolution of the secondary star (an AGB star). The white dwarf\nand the core of the AGB star merge at the very end of the AGB star evolution, or shortly thereafter. The resulting white dwarf may\nbe super-Chandrasekhar, but gravitational collapse followed by an explosion can be delayed due to a rapid rotation of the massive white dwarf.\n Indeed, this scenario is somewhat\nintermediate between the single degenerate and the double degenerate scenario, as the supernova is clearly caused by a double\ndegenerate merger, but the time scales involved can be  similar to the single degenerate scenario. A consequence of this scenario\nis that the supernova mass does not have to be 1.38 {M$_\\odot$}, but could be more. Another possible consequence for SN\\,1604 may be that\nthere was high density material surrounding the progenitor, consisting of the envelope removed by the common envelope event. \nNo light echo has yet been detected of SN\\,1604 \\citep{rest08}, but if it will be detected and an optical spectrum can be obtained,\nperhaps we will learn more about the event itself and the presence of material in the immediate vicinity of the supernova.\n\n\\subsection{What can we learn from the historical light curve of SN\\,1604?}\n\\label{sec:lightcurve}\n\nUp to the 1990ies the hope was that the historical light curves \\index{supernova light curve} \\index{supernova light curve (SN 1604)} \nof SN\\,1572 and SN\\,1604 could shed  light on the maximum\nbrightness of Type Ia supernovae, and hence be used to constrain the Hubble parameter \\citep[e.g.][]{danziger80,schaefer96}. \nNowadays the Hubble parameter is known\nto sufficient detail, but the historical light curves remain of interest, as it may lead to sub-typing of the historical supernovae and\nprovide estimates on the distance of  SNRs ({Section}.~\\ref{sec:properties}). \\citet{baade43} was the first one to determine\nthe historical light curve of SN\\,1604, and this was later augmented with magnitude estimates based on Korean observations\nby \\citet{clark77}. These data have been compared to supernovae light curve by among others \\citet{schaefer96}, and most\nrecently \\citet{katsuda15}.\n\nThe historical light curve of SN\\,1604 is shown in Fig.~\\ref{fig:lc} with a typical magnitude error of 0.25-0.5. The historical light curve\nis compared to a number of recently observed Type Ia supernovae (Table~\\ref{tab:vmax} for an overview). \nThese lightcurves have been scaled to the light curve of SN\\,1604\nby taking into account the distance moduli ($\\mu$) and visual extinction parameters ($A_V$) of the supernovae, and assuming\na distance of 5~kpc to SN\\,1604 (somewhat in the middle of current estimates, Sect.~\\ref{sec:properties}) and $A_V=2.8$ for the historical supernova.\nThe latter is based on the estimate of \\citet{blair91} for the non-radiative shock \\index{non-radiative shocks} \nemission.\n\nAccording to Type supernova light curve models, the peak magnitude of Type Ia supernovae is determined by\nthe amount of radioactive $^{56}$Ni produced in the explosion (typically $\\sim 0.7${M$_\\odot$}), whereas the steepness\nof the decline after maximum is determined by the diffusion of heat generated by radioactivity to the photosphere of\nthe supernova; the more mass the slower the decline \\citep{arnett79,cappellaro97,dado15}. Hence,\nthe low peak magnitude and fast decline of SN\\,1991bg has been attributed to a sub-Chandrasekhar Type Ia explosion \\index{sub-Chandrasekhar Type Ia supernova}\n\\citep{turatto96,drout13}, whereas the very bright and slow declining SN\\,2009dc is thought to indicate \na super-Chandrasekhar \\index{super-Chandrasekhar Type Ia supernova} Type Ia \\citep{silverman11}. \nThe latter has a light curve that is even broader\nand brighter than SN\\,1991T, and its light curve has to be scaled by $\\Delta V=0.5$ in order to provide even an approximate\nfit to SN\\,1604.\n\n\nRecently \\citet{patnaude12} suggested that SN\\,1604 may have been similar to the bright Type Ia SN\\,1991T.\nFor a distance of 5~kpc the peak absolute peak magnitude of SN\\,1991T and SN\\,1604 match reasonably\nwell, but SN\\,1991T has a much brighter late time light curve, and a slower post maximum decline. \nIn order to make SN\\,1604 as luminous as\nSN\\,1991T at late times,  the distance of SN\\,1604 has to be $\\sim$7~kpc.\nSo the light curve of SN\\,1991T \\index{SN 1991T} does not seem to be an overall good template for SN\\,1604. \nThe situation is even worse for the probable super-Chandrasekhar Type Ia SN,2009dc, which, \neven when scaled down in brightness,\nshows a profile that is too broad to fit the historical light curve.\n\nNormal Type Ia supernova light curves (SN\\,1994D, SN\\,1996X, SN\\,2004eo) are in reasonable agreement with\nthe historical light curve of SN\\,1604. In particular, they fit reasonably well around the peak and they agree\nreasonably well at very late times (300--400 days), but they are brighter than SN\\,1604 for days 100--200.\n\nAt the faint extreme, there are peculiar  Type Ia supernovae such as SN\\,2009dc, whose light curve has to shifted up by\n$\\Delta V=1.5$ in Fig.~\\ref{fig:lc} in order to have the peak brightness coincide with the peak of SN\\,1604.\nSuch a shift would mean that SN\\,1604 would be at a distance of 2.5~kpc,\nwhich is lower than most independent distance estimates (Sect.~\\ref{sec:properties}).\nAs \\citet{katsuda15} showed, when properly scaled to match the peak magnitude of SN\\,1604, \nthe light curve of SN\\,2009dc is rather similar to the light curve of SN\\,1604,\nas both are fastly declining in brightness after peak brightness. \\index{SN 1991bg} \\index{SN 2005ek}\nA similarly fast decline Type Ia was SN\\,2005ek, whose light curve was used to estimate the ejecta mass of \n0.3-0.7~{M$_\\odot$}, and a kinetic energy of $(2-5)\\times 10^{50}$~erg. However, SN\\,2005ek was poor in $^{56}$Ni,\nat odds with what we know of SN\\,1604 (Sect.~\\ref{sec:xrays}). \nPerhaps one can still fit SN\\,1604 with a faint Type Ia model, if one allows for a small\n(sub-Chandrasekhar) progenitor mass, but with a large fraction of $^{56}$Ni.\nThis would make a fainter supernova, with a rapid decline, but it could\nstill allow for the high iron fraction in Kepler's SNR.\nThe reason this may work is that in X-rays the fraction of iron can be much better\ndetermined than the absolute mass of iron.\nA consequence is then that the distance of Kepler's SNR is  less than 5~kpc,\nas the supernova is fainter, but \nthis can be  accommodated\nwith hydrodynamical models since the supernova is then probably\nalso a sub-energetic Type Ia \\citep[$<10^{51}$~erg,][]{chiotellis12}.\n \nSo what can we conclude from the historical light curve, with some reserve as the magnitudes derived from the historical \nobservations have to be treated with some caution, and the extinction is uncertain?\nFirst of all, the light curves of normal Type Ia seem overall in better agreement with the light curve of SN\\,1604 than\nthe light curve of SN\\,1991T.\nSecondly, the light curve of SN\\,1604 may have declined rather rapidly after peak brightness. The evidence\nfor this is uncertain, but could indicate a sub-Chandrasekhar explosion. However,\nin order to account for the high iron fraction of Kepler's SNR, SN\\,1991bg and SN\\,2005ek are probably\nnot the right models. Moreover, a sub-Chandrasekhar model, would result in a shorter distance of Kepler's SNR.\n\nA possibility that needs to be explored better is\nwhether it is possible that the fast decline of SN\\,1604 could perhaps be the result of an overall faster expansion, and\/or\na distribution of nickel closer to the photosphere. This helps the heat generated by $^{56}$Ni to diffuse  faster to the photosphere,\nresulting in a more rapid post-maximum decline.\nIn any case, the fast decline is inconsistent with a super-Chandrasekhar explosion. This is of interest given the suggestion that\nSN\\,1604 could provide a evidence for the core-degenerate model  \\citep[Sect.~\\ref{sec:core}][]{tsebrenko13},\nwhich can result in super-Chandrasekhar Type Ia explosions.\n\n\\begin{table}\n  \n  \\caption{A comparison of SN\\,1604 to the Type Ia SNe shown in Fig.~\\ref{fig:lc}. \\label{tab:vmax}\n  }\n  \\scriptsize\n  \\begin{tabular}{l rcrcll}\\hline\\hline\\noalign{\\smallskip}\n    &\t$V_\\mathrm{max}$\t&\t$A_V$\t&\t$m-M$\t&\t$M_V$\t&\tRemark\t&\tReferences\t\\\\\n    \\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n    SN\\,1604:&\\\\\n    @ 3 kpc\t&\t-2.75\t&\t2.8\t&\t12.39\t&\t-17.94\t&\t\t\t\t&\t{\\scriptsize\t\\citet{blair91}\t}\t\\\\\n&\t-2.75\t&\t3.5\t&\t12.39\t&\t-18.64\t&\t\t\t\t&\t{\\scriptsize\t\\citet{danziger80}\t}\t\\\\\n @ 6 kpc\t&\t-2.75\t&\t2.8\t&\t13.89\t&\t-19.44\t&\t\t\t\t&\t{\\scriptsize\t\\citet{blair91}\t}\t\\\\\n \t&\t-2.75\t&\t3.5\t&\t13.89\t&\t-20.14\t&\t\t\t\t&\t{\\scriptsize\t\\citet{danziger80}\t}\t\\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\nSN\\,1991T\t&\\ \\ 11.50\t& \\ \\ 0.37\t& \\ \\ \t30.57\t&\\ \\\t-19.44 \\ \\ &\t{\\scriptsize\tBright Type Ia\t}\t&\t{\\scriptsize\t\\citet{sasdelli14}\t}\t\\\\\nSN\\,1991bg \\index{SN 1991b}\t&\t13.96\t& \\ \\ \t0.12\t&\t31.61\t&\t-17.77\t&{\\scriptsize\tSub-Chandrasekhar\t}\t&\t{\\scriptsize\t\\citet{turatto96,gomez04}\t}\t\\\\\nSN\\,1994D\t&\t11.90\t& \\ \\ \t0.08\t&\t30.82\t&\t-19.00\t&\t{\\scriptsize\tNormal Type Ia\t}\t&\t{\\scriptsize\t\\citet{cappellaro97,feldmeier07}\t}\t\\\\\nSN\\,1996X\t&\t13.21\t& \\ \\ \t0.22\t&\t31.15\t&\t-18.16\t&\t{\\scriptsize\tNormal Type Ia\t}\t&\t{\\scriptsize\t\\citet{salvo01}\t}\t\\\\\nSN\\,2004eo\t&\t15.35\t&\t0.34\t&\t34.12\t&\t-19.11\t&\t{\\scriptsize\tSub-Chandrasekhar?\t}\t&\t{\\scriptsize\t\\citet{pastorello07}\t}\t\\\\\nSN\\,2009dc\t&\t15.35\t& \\ \\ \t0.31\t&\t34.93\t&\t-19.89\t&\t{\\scriptsize\tSuper-Chandrasekhar\t}\t&\t{\\scriptsize\t\\citet{silverman11}\t}\t\\\\\n\\noalign{\\smallskip}\\hline\n  \\end{tabular}\n\\end{table}\n\n\n\\begin{figure}\n\\centerline{\n\\includegraphics[trim=50 50 50 50,clip=true,width=1.\\textwidth]{Fig_6_sn1604_lc.pdf}}\n\\caption{The light curve of SN\\,1604 based on the European observations as interpreted by \\citet[][black data points]{baade43} and\nthe Korean observations collected by \\citet[][red data points]{stephenson02}.\nFor comparison the light curves of several other Type Ia supernovae have been over-plotted.\nThese  curves have been scaled to that of SN\\,1604, assuming a distance for SN\\,1604 of 5~kpc and $A_V=2.8$.\nMore or less normal Type Ia supernovae light curves have been assigned bluish colours.\nSee Table~\\ref{tab:vmax} for details.\n\\label{fig:lc}\n}\n\\end{figure}\n\n\n\\section{Conclusions}\n\nThe remnant of SN\\,1604, Kepler's SNR, is one of the most remarkable SNRs, with its high bulk velocity of ~250~{km\\,s$^{-1}$}.\nSN\\,1604 is now generally regarded to  have been a Type Ia supernova, whose\nremnant is interacting with wind material from its progenitor system. One can even infer that the donor star must have been a 4-6 {M$_\\odot$}\\ star that\nhad evolved to the AGB phase. But this single degenerate Type Ia scenario for SN\\,1604 has only one problem: a surviving donor has not been detected.\nA core-degenerate supernova scenario, may offer a viable alternative theory, but it has its own problems.\nWhatever the explosion scenario, the high space velocity of the progenitor system remains mysterious, as it requires the ejection\nof a binary system out of the Galactic plane with $\\sim 250$~{km\\,s$^{-1}$}.\n\nIn Johannes Kepler's age SN\\,1604 was recognised as  a unique, mysterious event that was thought to have profound implications for mankind. \nWe may no longer think that SN\\,1604 is of prime importance for mankind, but SN\\,1604 is of prime\nimportance for understanding Type Ia supernovae.  \nSN\\,1604 has, even more than 400 yr after discovery, still profound implications for understanding cosmic phenomena, and\nit has still not shared all its mysteries with us.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\chapter{\\MakeUppercase{R-FFT: Functional Split at IFFT\/FFT in Unified LTE and Cable Access \n\t\tNetwork}}\n\n\\section{\\MakeUppercase{Introduction}} \n\\subsection{Motivation}\nToday the user Internet connectivity \nhas been dominated by the cellular wireless and \ncable\/DSL access network technologies \nFor a long time, wireless technologies, such as LTE \nhave enjoyed large growth opportunities due to the\nproliferation of wireless smart phones and hand-held devices \nin contrast to the cable technologies~\\cite{kuzlu2013assessment}.\nWith the recent development of DOCSIS~3.1 specifications for cable\ntechnologies, Multi-System Operators (MSOs)\nare able to offer the services which are equivalent \nto the advance wireless technologies supporting gigabit \nconnectivity to the users in both upstream and downstream~\\cite{Hamzeh2015}. \nThe next generation 5G technology~\\cite{agiwal2016next} \nfocuses on the unification of heterogeneous platforms \nat the access, backhaul and core networks especially through the \nsoftwarization and virtualization of the network infrastructures.\nTowards this end, integration of cable and wireless \ntechnologies could be the first step\nin the process of unifying the heterogeneous access platform.\n5G is also envisioned to support a wide range of applications \nranging from lower data rate Internet of Things (IoT) to\nultra-reliable health monitoring, and \nlarger data rates for 4K HD video streaming~\\cite{chen2015virtual}.\nChallenges in the 5G technologies~\\cite{chin2014emerging} include backhaul complexity\nresulting from densification of radio nodes, \ninterference with the neighboring cells \nand network deployment costs~\\cite{Chih-Lin2015}. \nPromising technologies, such as Software Defined \nOptical Networking (SDN)~\\cite{thyagaturu2016software} and\nNetwork Function Virtualization (NFV)~\\cite{blenk2016survey} \nare not only addressing the challenges posed by 5G but also \nfacilitating the faster integration of \nheterogeneous-access networks while supporting the larger date rate, \nlower latencies and high reliability.\nMoreover, current network infrastructures for both wireless and cable\ntechnologies can rely on common optical fiber technologies, such as \noptical digital Ethernet~\\cite{gomes2016new}. \nTherefore, MSOs are showing large interests in\nconverging to a common access platform through generic \nprotocols and standardization,  \nsuch as CPRI~\\cite{de2016overview} for integration of heterogeneous \ninfrastructures.\n\n\\subsection{Overview of Softwarization}\n\\subsubsection{Cloud-Radio Access Networks (CRAN)}\nSoftwarization of network functions can fundamentally \nreduce implementation complexity while increasing the \nflexibility~\\cite{galis2013softwarization}. Software implementation of \nthe RAN functions\nin a cloud environment is referred to as Cloud-RAN \nor CRAN~\\cite{makhanbet2016overview}.\nIn the CRAN, the functional implementation of the \nRadio Access Network (RAN) is split \nbetween Remote Radio Units (RRUs) which corresponds to\nantenna units with RF passband processing capabilities\nfor the physical transmission of the signal, \nand Base Band Units (BBUs) which \nimplements the complementary baseband signal processing.\nCommunication between CRAN and RRU is typically enabled by\noptical technologies, such as Passive Optical Networks (PON) and\nOptical Wavelength Division Multiplexing.\nTraditionally, the connection between RRU and BBU is established \nby a cable RF link. Cable RF contributes towards considerable signal degrade \nfor the larger distances between RRU and BBU.\nIn contrast to downlink signals, \nsignal degrade in the uplink direction has more negative consequences \ndue to the lower signal levels.\nThe signal degrade between BBU and RRU can be avoided by digitizing the \nRF signal and transmitting the digitized symbols over an optical fiber.\n\nAs noted earlier, important aspect of the RRUs in the context of\nCRAN is the reduced complexity in radio equipment hardware \nwhich results in lower CAPEX\/OPEX for\nthe large scale deployment of small cells.\nRRUs in a building environment are also\nreferred to as Distributed Antenna Systems (DAS)~\\cite{heath2013current}. \nDAS play an important role in realizing the large number of \nsmall cell deployments in meeting the coverage and capacity demands for\nindoor applications. \nCentralization of BBU can \nsupport multiple RRUs which can provide a common platform for \nthe centralized management of resources. \nBBUs are typically implemented on a generic computing \nhardware, where by a specific virtual machine (VM) \nimplements the base band processing operations \non a virtualized hardware~\\cite{dai2014uplink}.\n\n\\subsubsection{Distributed Converged Cable Access Platform (DCCAP) Architectures}\nConnectivity to the Cable Modems (CM) in the Hybrid Fiber Coax (HFC) network \nis delivered in part by optical and in part by cable segment.  \nThe Cable Modem Termination System (CMTS)\nconnects the CMs to Central Office (CO)\/headend\nthrough the protocol defined by Data Over Cable Service \nInterface Specifications (DOCSIS).\nIn the traditional HFC network deployments, analog signals from the\nCMTS are carried over an optical link to a remote analog fiber node where\nthe signal is converted to analog RF signal which can be \ntransmitted over the cable link.\nBut, an analog signal over an optical fiber gradually degrades\nas the optical propagation distance is increased \nlimiting the effective HFC operational distance (i.e., from the headend to CM). \nModular Headend Architecture version 2 (MHAv2) allows \nCMTS functions to be implemented in a modular fashion allowing \nsome of the modular entities to be implemented at the newly \ndesigned digital remote nodes. \nIn order to improve the signal quality over the cable segment,\nthe remote nodes are connected to headend through the digital \noptical link where the remote node can be deployed in close\nproximity to the users reducing the effective cable length.\nDistributed Converged Cable Access Platform (DCCAP) \narchitecture implements the modular CMTS functions at the\nremote nodes.\nRemote node is similar to a RRU in the CRAN whereby the \nRAN implementation at the cloud is similar to \nCMTS implementation at the headend.\nThe Remote-PHY (R-PHY) and Remote-MACPHY (R-MACPHY) technology \nof the DCCAP \nextends the modular CMTS and \nmodular headend architectures with a \ngoal to reduce the complexity and increasing \nserviceability of the cable \ninfrastructure. In particular, R-PHY implements\nthe DOCSIS PHY functions on a \nremote node, Remote-PHY Device (RPD), \nwhile centralizing the MAC and higher layer network functions at the headend. \nWhereas, the R-MAC implements the DOCSIS PHY and MAC \nfunctions on a remote node, Remote-MACPHY Device (PMD),\nwhile centralizing the IP forwarding\nand higher layer network\nfunctions at the headend. \nCentralized DOCSIS functions for RPD and RMD \ncan be implemented as a software on a \nvirtualized entity either at the headend or cloud.\n\nThe critical aspect of CRAN and DCCAP in the time domain I\/Q\ntransport are the lower latency and higher data rate requirements\non the optical fiber~\\cite{Wubben2014}.\nA constant high bit rate and low latency \nconnection must be maintained constantly between \nbaseband processing and radio node regardless of the user traffic.\nThe analog RF signals must be transmitted and received at all times even \nwhen there is no user activity. For e.g., the passband signal with the \ncell broadcast information, reference or pilot tones must be transmitted always. \nThus, I\/Q samples of the RF passband must always be transported at the \nconstant rate at all the times. \nMoreover, the requirements of an optical fiber\nincreases linearly with increase in the \nnumber of nodes. Therefore, numerous  \ntechniques such as~\\cite{nieman2013time, nanba2013new, guo2013lte, guo2012cpri} \nhave been proposed to enable the dynamic compression of RF I\/Q samples \nfor the effective transmissions over the optical fiber.\nHowever the compression techniques are lossy because of the quantization of \nRF signal reducing the sensitivity of the receiver in the upstream.\nNevertheless, the data rate requirements between cloud and radio unit\nleads to the dedicated deployments of\noptical fiber connections and static allocations of resources \nfor the CRAN and DCCAP. Functional split architectures~\\cite{Wubben2014,maeder2014towards} \ntransport the upper layer\ndata to the radio unit such that the constraints on\nthe optical connection are relaxed. However, the additional processing at the \nradio unit increases the complexity at the node as well as the flexibility\nof the infrastructure is reduced as compared to the traditional \nCRAN is due to the distributed implementation of split RAN functions. \n\n\\subsection{Contributions and Organization}\nIn this article, we primarily focus on the trade offs and benefits of \nthe functional-split in both LTE and cable networks. In particular, \nwe focus on the split-PHY architecture which implements the IFFT\/FFT at\nthe radio units of both LTE and cable networks. \nTowards this end, the main contributions of this article are as follows:\n\\begin{itemize}\n\t\\item[i)] Overview and trade off discussions on the functional split \n\tand split-PHY architectures for both LTE and cable networks. \n\tPresented in Sec.~\\ref{sec:backgroud}.\n\t\\item[ii)] A novel cross-split interaction \n\tmechanism for remote caching and prefetching of QAM symbols\n\tto reduce the data rate required for the \n\tI\/Q transmissions between cloud and \n\tradio unit, especially in the \\textit{downstream} direction \n\twhile implementing the IFFT\/FFT at the remote nodes. \n\tPresented in Sec.~\\ref{sec:split:inter}.\n\t\\item[iii)] A novel mechanism to commonly share the \n\tinfrastructure resources among LTE and cable networks \n\tThe discussion also includes the general timing analysis \n\tfor interleaving multiple IFFT\/FFT computations \n\tof LTE and cable on a single IFFT\/FFT module.\n\tPresented in Sec.~\\ref{sec:common:infra}.\n\t\\item[v)] In the end, we present the evaluation of \n\tFFT sharing mechanism and mean packet \n\tdelay performance evaluation of \n\tthe radio unit which implements IFFT\/ FFT for the \n\tsimultaneous LTE and cable operation in comparison\n\tto existing R-PHY technology.\n\tPresented in Sec.~\\ref{sec:perf:eval}.  \n\\end{itemize} \n\n\\subsection{Related work}\nChecko et al.~\\cite{checko2015cloud} has conducted the\nCRAN technology review which includes advantages, challenges, \nSDN\/NFV applications, as well as the compression techniques \nfor I\/Q transport over the fronthaul network. \nA more detailed discussions on the internals of the \nBBU and RRU units has been discussed by Wu et al. in~\\cite{wu2015cloud}.\nThey show that cooperative signal processing in the CRAN \nachieves better spectral efficiency and improves utilization \nas compared to traditional networks. \nCRAN also provides a platform to provide computing services \nto the users through the access network. \nA comprehensive survey on Mobile Cloud Computing (MCC) in the area\nof CRAN has been conducted by \nFernando et al.~\\cite{fernando2013mobile}. \nFundamentals of the functional and protocol split uplink as well as the \ndownlink and uplink compression techniques has been \npresented in~\\cite{peng2016recent}. The impact of overhead due to \npackatization on the data rate and latency \nof the fronthaul link while supporting \nvarious functional split in the CRAN\nhas been presented in~\\cite{chang2016impact}. \n\nMiyamoto et al.in their \nwireless performance evaluation~\\cite{miyamoto2016analysis}  \nclaim that their proposed split-PHY architecture reduces the \nfronthaul bandwidth by up to 97\\% compared to traditional\nCRAN with the penalty of 2~dB SNR. However, their proposed\nsplit-PHY architecture requires specialized \ntransceiver designs and optical transmission mechanisms\nwhich may increase the cost of CRAN deployments.\nIn a typical RRH deployment scenario, \nseveral RRHs experience different\nload due to the independent channel qualities of the UEs reducing\nthe overall load RRH. \nTherefore, especially when the higher functional splits are considered,\nall the RRHs connected to the CRAN would likely\nnot result in the peak load.\nUser specific traffic variation can provide the  \nmultiplexing opportunity and the impact of such multiplexing \nfor the RRH deployments has been studied by \nChang et al.~\\cite{chang2016impact}.\nFurther to split-PHY, Nishihara et al.~\\cite{nishihara2016study} \nhave evaluated the performance of \nMACPHY-split where both MAC and PHY are implemented at the RRH. \nTheir simulation results show\nthat MACPHY-split approach reduces the bandwidth from\n10~Gbps to 600~Mbps while supporting the multiplexing.\n\nCRAN involve large number of RRH deployments \nwhich can potentially\nbe the source of large power consumptions. \nEfforts to reduce the carbon foot print over the\ncellular networks supporting both CRAN as well as traditional\ndeployments has been presented in~\\cite{hasan2011green, oh2013dynamic,wu2015energy,de2014enabling}. \nWhereas the energy saving mechanism in the wireless\nnetworks has been presented in~\\cite{suarez2012overview}.\nStephen et al.~\\cite{stephen2016green} have designed an \ncache enabled OFDM resource allocation mechanism which saves\nthe transmissions over the fronthaul link. \nHowever, the content\nhas to be saved in parts marked with identifiers in the remote node\nIn contrast to all the previous CRAN studies, we have\nconsidered the caching of\nresource elements at the RRH on the split-PHY architecture \nto reduce the data rate over the fronthaul link potentially \nsaving the energy consumption when \nuser data is not present.\n\nComplementary to the wireless cellular networks\nHFC network provides the broadband access to the residential \nusers. Bisdikian et al.~\\cite{bisdikian1996cable}\nhas presented the overview of initial designs and protocol \nmechanisms for the cable networks. \nAs compared to the earlier designs, current and \nfuture broadband access networks extends the\ncapabilities of connectivity to Gigabit \nspeeds~\\cite{effenberger2016future}. \nEven further, present cable networks can be designed to support cellular \nnetworks services as described in Gambini et al.~\\cite{Gambini2012, gambini2010lte}. \nThe economic benefits of infrastructure sharing between residential wired and \ncellular wireless networks has been identified in~\\cite{pereira2012infrastructure}.\nAdditionally, economic benefits form the\nintegration of LTE and DOCSIS has also \nbeen discussed in~\\cite{gruber2014broadband}.\n\nArticles \\cite{liu2015bandwidth,zeng2016demonstration,liu2016cpri} have\ndemonstrated efficient transmissions of CPRI equivalent \n fronthaul data up  to 256 Gbps using the optical bandwidth of 10 GHz,\nespecially for supporting MIMO LTE applications. \nHowever, the proposed mechanism in the articles use specialized \noptical transmission with constant traffic over the optical link \nwithout allowing the possibility of multiplexing. In addition, \nwe propose the novel remote caching and prefetching of QAM symbols, \nwhich saves the fiber transmissions. Especially, when there are no \nusers, optical fiber connectivity can be effectively turned off to \nsave the power or share the resources with other nodes. \nIn comparison to most of the access networks, such as, \nmacro base stations, DAS, femto cells, and HFC,\nthe traditional macro base \nstation backhauled through a microwave link at low traffic\nresults in the most \nenergy economical option~\\cite{fiorani2016joint}.\nAs cable networks are one of the sources for growing\ncarbon footprint in the Internet connectivity,\nthere has been several studies proposed to address the issue\nsuch as~\\cite{zhu2011novel,Zhu2012,Lu2013}.\n\n\\section{\\MakeUppercase{Background on Functional Split in LTE and Cable Networks}} \n\\label{sec:backgroud}\n\\subsubsection{Downstream v\/s Upstream}\n\\paragraph{Upstream} In the upstream direction the RRU\nreceives the RF signal transmitted from the users. This analog\npassband signal is down converted to baseband and digitized \nfor the transmission to BBU for baseband processing. Unlike the \ncable link in the traditional cellar network and antenna infrastructures,\nthe CRAN connects the BBU and RRU with an digital optical fiber. \nThe cable link adds significant attenuation to the upstream signal especially\ndue to the low level of received signal from the devices. Whereas, the\ndigital fiber does not constitute towards the attenuation loss as it \ncarries the signal in the digital form. To note, an extreme care \nis needed at the RRUs for digitizing the uplink signal from the users \nas an additional loss cannot be afforded due to the low signal \nlevel of uplink RF signal at the RRU. For example, if the cable link \naccounts for $2$~dB of loss and noise floor is $-120$dB, then the received\nsignal at the RRU connected a BBU over cable link must be $\\leq-118$~dB \nfor the successful detection, whereas the received signal can be $\\leq \n-120$~dB if the RRU is connected to a BBU \nthrough the digital fronthaul link, thus increasing the dynamic range of the system by 2~dB. \nAlthough Single Carrier Orthogonal\nFrequency Multiplexing (SC-OFDM) \nuplink modulation format is used in the typical deployments, \nthe technology is advancing towards the uplink OFDM systems especially\nfor the MIMO applications~\\cite{studer2016quantized, pitarokoilis2016performance}.\nTherefore, in the scope of this article we focus on the symmetrical OFDM systems\nin both upstream and downstream directions. In addition to the increased dynamic \nrange, the processing of upstream \nfor the detection and extraction of information from the RF \nuplink signals can be centrally processed at the cloud on the generic hardware, \nsuch as the general purpose processors.\n\n\\paragraph{Downstream} In the downstream direction BBU sends the \ninformation to RRUs for the generation of passband signal to\ntransmit over the physical antennas. The RRUs can easily set the \ntransmit power level gain states for RF signals. As in the upstream direction,\nthere is no significant difference in terms of \nsignal generation or the dynamic range of the systems \nbetween cable and digital fronthaul links.\nSimilar to the centralized processing of upstream at the cloud, \nthe information is centrally processed on a generic hardware, such as the general \npurpose processors to generate the baseband downlink signals.\n\n\\subsection{Functional Split in LTE}\n\\begin{figure}[t!] \\centering\n\t\\includegraphics[width=5in]{fs\/fig_CRAN.pdf}\n\t\\caption{The Cloud-RAN (CRAN).} \n   \\label{fig_CRAN}\n\\end{figure}\nFigure~\\ref{fig_CRAN} shows the\ntraditional CRAN deployment in comparison conventional cellular deployments.\nA radio base station eNB of LTE protocol stack towards the UE \ncan be functionally split and implemented flexibly over radio node and cloud\nseprated by a fronthaul link which is typically an optical fiber connectivity\nThe traditional CRAN transports the baseband time domain I\/Q samples\nover the optical fiber to the RRUs. The number of RRUs that is\nsupported over a fiber deployment depends on the amount of traffic\nthat is required to support the operations of each RRU. Suppose if \n$C$ is the capacity of the fronthaul optical connectivity and $R_i$ \nis the data rate required by $i^th$ RRU,\nthen maximum number of RRUs $N$ that can be supported over that specific\nfronthaul link is $\\max_{N} \\{\\sum_{i = 1}^N R_{i} \\}$ such at \n$\\sum_{i = 1}^N R_{i} \\leq C$. In the present deployments of CRAN\nthe resources of the fronthaul link are statically allocated with dedicated\nconnectivity. Therefore in symmetrical and homogeneous deployment, \n$R_1 = R_2 \\ldots R_N = R$ resulting in the total number of RRUs \nthat can be supported over the fronthaul link as $N = C\/R$. The main\nbottleneck for the CRAN deployments is delay and capacity of the fronthaul link $C$.\nHeterogeneous fronthaul links where multiple RRUs requirements\nof delay and data rate changing over time independent of each other \nwould require an advance resource management mechanisms to avoid \nthe capacity wastages as compared to the \nworst case tolerance deployment practices.  \n\\begin{table}[t]\n\t\\caption{Typical LTE CRAN Parameters}\n    \\label{tab:CRANparams}\n\t\\centering\n\t\\begin{tabular}{|p{1cm}|p{5cm}|p{2cm}|} \n\t\t\\hline \n\t\t\\textbf{Param.} & \\textbf{Description} &  \\textbf{Value} \\\\ [.5ex]\n\t\t$N$ & Number of RRU per CRAN &  1 \\\\\n\t\t$B$ & LTE cell bandwidth &  20 MHz  \\\\\n\t\t$K$ & Bits per I\/Q & 10 bits   \\\\\n\t\t$W$ & Number of Tx\/Rx antennas & 2 \\\\\n\t\t$T_s$ &  OFDM symbol duration & 66.6 $\\mu$s \\\\\n\t\t$f_s$ &  Sampling frequency & 30.72 MHz \\\\\n\t\t$f_c$ &  Carrier frequency & 2 GHz \\\\\t\t\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\nTo understand the fronthaul requirements of an RRU, \nwe estimate the data rate $R$ required by the traditional CRAN where \nthe baseband I\/Q is transported from the cloud BBU to RRU \nfor the most common LTE deployment scenarios. \nTable~\\ref{tab:CRANparams} summarizes the important list of parameters\nused for evaluation in the context of LTE for the \nfronthaul optical link connecting RRU to CRAN. \nThe data rate comparisons of various functional \nsplit in the LTE protocol stack has \nbeen conducted in~\\cite{maeder2014towards, Wubben2014}. In contrast, \nwe take a closer look on the data rate requirements based on the implementation \nspecifics of the protocol stack. That is, we track the flow of information \nacross multiple protocol stack layers of the LTE and \nidentify the key characteristics that dictate the \nrequirements of fronthaul link. \n\nBased on the computationally intensive operation of the FFT The data flow \nbetween BBU and RRU can be categorically divided into \ntwo types i) time domain samples and ii) frequency domain samples.  \n\n\\subsubsection{Time Domain I\/Q Forwarding}\n\\label{sec:time:iq}\nThe time domain I\/Q samples\nrepresents the RF signal in the digital form either in the passband \nor baseband representation. Typically, the digital representation of the \npassband signal requires a very large data rate depending on the \nphysical transmission frequency band. Thus, passband time domain I\/Q forwarding\nis non economical. For example, in an LTE system, the passband signal\nis sampled at twice the carrier frequency $f_c$ with each sample requiring\n10 bits for digital representation, the passband \nI\/Q data rate $R_i^P$ required over the fronthaul link is\n\\begin{equation}\n\\begin{split}\nR^P_i = N \\times W  \\times 2f_c \\times 10 \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\\\\n= 1 \\times 2 \\times 2(2 \\times 10^9) \\times 10 = 80~\\text{Gbps}.\n\\end{split}\n\\end{equation}\n\nThe baseband signal for an OFDM symbol in the \ntime-domain consists time samples equal to the number of \nOFDM subcarriers because of the symmetric input and output samples \\\nfrom the IFFT\/FFT structure.\nThe cyclic prefix is added to the OFDM signal to avoid the \ninter symbol interference. In order to reduce the constraints on the \nRF signal generation at the RRU, the baseband signal is sampled\nat the frequency of 30.72 MHz with each sample requiring\n10 bits for digital representation and an oversampling factor of 2, \nthe baseband I\/Q data rate $R_i^B$ \nrequired over the fronthaul link is\n\\begin{equation}\n\\begin{split}\nR^B_i = N \\times W  \\times (2 \\times f_s) \\times (2 \\times K) \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\\\\n= 1 \\times 2 \\times (2\\times 30.72 \\times 10^6) \\times (2 \\times 10)\\\\\n = 2.46~\\text{Gbps}.\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\n \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\n\\end{split}\n\\end{equation}\n\nAlthough the data rate is significantly reduced as compared to the passband \nI\/Q forwarding, the baseband I\/Q data rate scales linearly with the\nnumber of antennas and bandwidth. Thus, for large number of \nantennas and larger aggregated bandwidth the data rate $R^B_i$ can be very large.\n  \n\\subsubsection{Frequency Domain I\/Q Forwarding}\n\\label{sec:freq:iq}\nIn the 20 MHz LTE system One OFDM symbol duration \nincluding the cyclic prefix is $71.3~\\mu$s which \ncorresponds to 2192 samples. The useful symbol duration in OFDM \nis $66.7~\\mu$s of 2048 samples while the cyclic prefix duration \nis $4.7~\\mu$s of 144 samples. \nThus, each of 2048 samples in the OFDM symbol (excluding the cyclic prefix) \ncorresponds to 2048 subcarriers ($B_{sub}$) when transformed by the FFT. \nHowever, only 1200 of these subcarriers are used for \nsignal transmission which corresponds to 100 resource blocks (RBs) \nof 12 subcarriers each the rest is zero-padded \nand serves as guard carriers. This \nleads to $(2048-1200)\/2048=0.41=41~\\%$ of unused guard carriers.\nEach subcarrier in the OFDM is modulated by a complex value mapped \nby a QAM alphabet. LTE QAM alphabet size is based on QAM bits \nsuch as, 64 QAM and 256 QAM. If only the subcarrier information which consists of\nthe complex values taken from the QAM alphabet i.e., only the frequency domain data\nis used to transport between BBU to RRU. \nThe resulting data rate $R^F_i$ is only dependent \non the number of subcarriers $B_{sub}$. A vector of \ncomplex valued QAM alphabet symbols of size $B_{sub}$ needs to be sent \nonce every OFDM symbol duration $T_s$ i.e., the data rate \n\\begin{equation}\n\\begin{split}\nR^F_i = N \\times W \\times B_{sub} \\times T_s^{-1} \\times (2 \\times K)\n\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\\\\n= 1 \\times 2 \\times 1200 \\times (66.7\\times 10^{-6})^{-1} \\times (2 \\times 10)\\\\\n= 720~\\text{Mbps}.\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\n\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\n\\end{split}\n\\end{equation}\nIn comparison to time domain baseband I\/Q data rate $R_i^B$, the $R_i^F$ is\napproximately reduced by $(R_i^B - R_i^F)\/R_i^B = (2460-720)\/2460 = 70~\\%$. \nFurther attempts to reduce the data rate $R_i$ \nrequire moving complex functions, such as MAC to the RRU.\nThe MAC HARQ (Hybrid ARQ) \nprocesses requires complex operations and \nlarge buffers to be implemented \nat the RRU increasing the complexity of the RRU.\nThus, with the savings of 70~\\%, keeping the simplicity at the RRU\nis worthwhile to explore the transport mechanism of \nfrequency domain I\/Q symbols over the fronthaul link. \n\n\\subsection{Functional Split in Cable Distributed Converged Cable Access Platform (DCCAP) Architectures}\n\\begin{figure}[t!] \\centering\n\t\\includegraphics[width=4.5in]{fs\/fig_remote_node.pdf}\n\t\\caption{Distributed Converged Cable Access Platform (DCCAP).} \n\t\\label{fig_remote_node}\n\\end{figure}\nTraditional CCAP architecture for the HFC network implements \nthe CMTS at the headend and \ntransports analog optical signal to an Optical Remote \nNode (ORN) over the optical fiber. \nORN then converts the optical\nanalog signal to electrical RF signal to transmit over the cable \nsegment. However, such architectures which rely on\nanalog optics suffer from attenuation of the analog signal in both \noptical fiber segment as well as cable segment of the HFC network because\nof the placement of the ORN. If the ORN is deployed far from the headend,\nthe attenuation in optical signal will be dominant, similarly\nif the ORN is deployed far from the CM (users) then the attenuation in \nthe RF signal of the cable will be dominant. Modular Headend\nArchitecture (MHA) overcomes the downside of CCAP architectures by\nenabling the functional splitting of CMTS allowing the modular \nimplementation of the CMTS functions. Implementation of \nmodular CMTS functions distributively across multiple nodes\nare referred to as Distributed Converged Cable \nAccess Platform (DCCAP) Architecture. As shown in the Fig.~\\ref{fig_remote_node}\nDCCAP architecture defines a new\nRemote Node (RN) which is connected to the headend through digital\nEthernet fiber. Digital connection between the RN and headend eliminates the \nattenuation of the analog optical signal allowing the RN to be \ndeployed more deeper into the users network \nreducing the cable segment length which in turn reduces \nthe analog RF attenuation improving the overall Signal to Noise Ratio (SNR)\nat the CM. The network which connects the RN to headend is referred \nto as Converged Interconnect Network (CIN). \nMHA version 2 (MHAv2)~\\cite{MHAv2} architecture\ndefines two DCCAP architectures Remote-PHY and Remote-MACPHY.  \n \n\\subsubsection{Remote-PHY (R-PHY)}\n\\paragraph{Overview}\nIn the R-PHY architecture, the DOCSIS PHY function  \nin the CMTS protocol stack is implemented at the \nRN referred to as Remote-PHY Device (RPD), while all the\nhigher layers in the CMTS protocol stack including \nthe MAC as well as the upstream scheduler are implemented at the headend.\nA virtual-MAC (vMAC) entity virtualizes the DOCSIS MAC on a \ngeneric hardware which can be flexibly implemented either at the headend \nor cloud\/remote data center. Since the upstream scheduler is implemented\nat the headend, the request from the CM is sent to the headend for processing\nthe grants to coordinate the upstream transmissions over the cable link. \nThe extensive performance evaluation of R-PHY and R-MACPHY have been conducted\nin~\\cite{ziyad2017,thyagaturu2017r}.\nTherefore, the CM request-to-grant delay depends on the distance of \nRN from the headend (i.e., the CIN distance)~\\cite{Chapman2014, Chapman2015}.\n\\paragraph{Advantages}\nR-PHY node is simple to implement and hence the CAPEX and OPEX of the\nRPD can be reduced. At the headend, more computational resources \ncan be dedicated to support the complex schedulers for \nmode advance coordination mechanism of CM transmissions over the \nbroadcast cable medium.\n\\paragraph{Disadvantages}\nAs the distance of RPD from headend increase, the CM request-to-grant\ndelay increases due to increases RTT between RPD and headend.\nApplications such as, virtual and augmented reality requires\nUltra Low Latency (ULL) over the cable link can be impacted\nby the larger CM request-to-grant delays.\n\n\\subsubsection{Remote-MACPHY (R-MACPHY)}\n\\paragraph{Overview}\nIn the R-MACPHY architecture, the DOCSIS PHY and MAC functions along with\nthe upstream scheduler in the CMTS protocol stack are implemented at the \nRN referred to as Remote-MAC Device (RMD), while all the\nhigher layers in the CMTS protocol stack are implemented at the headend.\nSince the upstream scheduler is implemented\nat the RMD, the requests from the CMs are processed for the \nthe grants at the RMD to coordinate the upstream transmissions \nover the cable link reducing the request-to-grant delay.\nTherefore, the CM request-to-grant delay is independent of the distance of \nRN from the headend.\n\\paragraph{Advantages}\nR-MACPHY produces significantly lower delay when compared to R-PHY especially\nthe CIN distance is very large. ULL applications can be readily supported as\nthe upstream scheduler is implemented very close the CM reducing the request-to-grant\ndelay. \n\\paragraph{Disadvantages}\nRMD device has to implement the scheduler on the remote node increasing the \nCAPEX and OPEX of the remote node. As remote node implements more \nfunctions of the CMTS protocol stack in a distributed fashion,\nthe flexibility of the network reduces as\ncompared to centralized architectures, such as R-PHY.\n\n\\subsection{Functional PHY-Split}\n\\begin{figure}[t!] \\centering\n\t\\includegraphics[width=5in]{fs\/fig_split_phy.pdf}\n\t\\caption{PHY-split architecture.} \n\t\\label{fig_split_phy}\n\\end{figure} \n\nThe maximization of the \"cloudification\" of CMTS DOCSIS and \nLTE functions results in the\nincreased flexibility and reduced CAPEX \nand OPEX for the Multiple System Operators (MSOs) and \ncellular operators. The cloud implementation of DOCSIS and LTE functions \ncan be easily achieved at the same physical location, remote data center or cloud.\nBoth LTE and DOCSIS 3.1 also share similar transceiver PHY characteristics for \nthe OFDM implementation that can be exploited to design \nthe simultaneous support for LTE and DOCSIS over the HFC network which is critical to\nthe 5G type of applications, such as health monitoring and security \\cite{veeranna2016hardware}. \nThe general overview of the physical layer for LTE and \nDOCSIS is shown in the Fig.~\\ref{fig_split_phy}. In the downstream direction,\nthe data from MAC layer is\nprocessed to PHY frames and mapped to OFDM resource locations which is then \nconverted to frequency domain QAM I\/Q symbols (see~\\ref{sec:freq:iq}) based on the \nmodulation and coding schemes. The\nQAM I\/Q symbols are then IFFT transformed to \nget the complex time domain samples. \nThis time domain samples (see~\\ref{sec:time:iq}) \nare then converted \nto analog RF signal for the transmission over the cable link. To note, the processing of I\/Q \ninformation before the IFFT\/FFT module is specific to DOCSIS and LTE \nprotocol whereas the processing of \nI\/Q after the IFFT\/FFT module is relatively same for both DOCSIS and LTE.\nThus, we can separate the functions based on the IFFT\/FFT module such \nthat IFFT\/FFT is implemented at the remote node to support the\nthe architectures and mechanisms to simultaneously \noperate LTE and DOCSIS over the HFC network. \nTo the best of our knowledge, there exists no prior research \nfor simultaneously supporting LTE and cable so as to efficiently \nutilize the optical fiber (fronthaul) \nresources of the already installed HFC plant. \n\n\n\\section{\\MakeUppercase{Proposed Cross-Functional Split Interaction}} \n\\label{sec:split:inter}\n\\begin{figure}[t!] \\centering\n\t\\includegraphics[width=4.5in]{fs\/fig_LTE_cable.pdf}\n\t\\caption{Deployment of the LTE eNB RRU at the Remote Node (RN).}\n\t\\label{fig_LTE_cable}\n\\end{figure} \nThe digital optical remote node (ORN) in the DCCAP architecture \nis deployed closer to the CM (users). Close proximity of the \nORN to the residential \nsubscribers would help in establishing the wireless LTE \nconnectivity by deploying\nan LTE eNB RRU at the ORN site. As show in Fig.~\\ref{fig_LTE_cable} \ninstalling an RRU at the ORN site \nis advantageous to the operators due to the close proximity of the users. \nWith the establishment of LTE connectivity by the MSOs users can be \nwirelessly connected to the MSO's core network \nfor the Internet connectivity increasing the service capabilities of the MSOs.\nIn addition, support of LTE eNB RRU at the ORN also reuses the \nexisting HFC infrastructure reducing the CAPEX and OPEX for the MSOs in \nproviding the additional services of LTE.\n\nTraditionally, the CRAN and DCCAP architectures split the \nfunctions linearly based on the protocol stack \n(i.e., MAC, PHY etc.), \nand implements the split parts at the ORN and cloud.\nAdditional to protocol stack split, our proposed mechanism \nenables cross-split interaction through signalling\nto facilitate the caching and prefetching (see Sec~\\ref{sec:caching}) \nof redundant information between the functional splits. More\nspecifically, cross-functional split interaction involves \nthe communication of control information between\nbaseband unit and radio unit.\n\\subsection{Proposed Shared Remote-FFT (R-FFT) Node}\n\\begin{figure}[t!] \\centering\n\t\\includegraphics[width=5in]{fs\/fig_rfft.pdf}\n\t\\caption{Remote FFT (R-FFT) implements the FFT module at the remote node.}\n\t\\label{fig_rfft}\n\\end{figure} \n\nIn the uplink direction, the proposed remote node R-FFT converts the \nincoming DOCSIS RF signal from the CM to an encapsulated data bits format \nthat can be transported over digital fiber link for additional processing and\nonward forwarding at the headend. In a similar way, for the downlink, \nRF signals are generated from the incoming formatted data bits and \nsent out on the RF link to the CMs. For LTE, an eNB can use a wide \nrange of licensed spectrum with a single largest carrier component \nof 20 MHz, however the bandwidth can be extended further by carrier \naggregation technique resulting in larger effective bandwidth. \nThe R-FFT node effectively converts the upstream LTE RF signal \nfrom the wireless users and transports the signal digitally to the \nBBU\/CRAN via the digital fiber link. In the downlink direction, \nthe R-FFT node converts the digital information to an LTE RF \nsignal that can be transmitted wirelessly to the users.\n\nA Remote-DAC\/ADC (R-DAC\/ADC) would require peak rate \ntransmissions \\cite{rdac}. Considering a system with a downlink \nbandwidth of 100 MHz and a 1024 QAM modulation format, \nwhich requires 12 bits per sample, results in a total bit rate over the fiber of (100 MHz) x (12) x (Frequency sampling rate). \nFor a frequency sampling rate of 2.5 times (slightly greater than the Nyquist rate),\n the total required bit rate is 3 Gbps. In contrast, \nan equivalent R-PHY system requires a data rate of (100 MHz) x (10 Bits\/Hz) x\n(90~\\%) =  900~Mbps; whereby, 100 MHz is for the resource allocation \nacross the entire bandwidth, 10 Bits\/Hz is for the 1024 QAM, and 90~\\% \nis for the effective channel with a 10~\\% guard band.\nThis simple example comparison indicates a threefold increase of the \ndata rate in the R-DAC\/ADC system compared to the R-PHY system. \nNote that the R-DAC\/ADC system requires a constant peak \ndata rate to transport the I\/Q samples, regardless of the \namount of user data present in the RF signal. \nIn contrast, similar to R-PHY\/MAC, in R-FFT system, \nthe data rate depends on the \namount of user data. In the above example data rate calculation, \nwe have considered the maximum allocation \nof the user data in the R-PHY system. \n\nOur motivation is to address the increased fiber data rate \nthrough a balanced split among the functions within the PHY node, \nwhile keeping the simplicity of the R-DAC\/ADC and the CRAN systems. \nThe existing R-DAC\/ADC node requires some digital circuitry, \nsuch as a CPU, for the DAC and ADC control. Therefore, we believe \nthat the FFT\/IFFT implementation is not a significant additional \nburden for the remote node~\\cite{jang2016study}. \nFigure~\\ref{fig_rfft} shows the implementation of vMAC \nfor both LTE and DOCSIS at the Headend and \na common FFT module at the remote node. \nThe advantages of proposed FFT implementation at the remote node include: \n\\begin{itemize}\n\t\\item[i)] flexible deployment support for LTE and DOCSIS\n\t\\item[ii)] requires lower data rate $(R_i^F)$ to transport frequency \n\tdomain I\/Q as compared to time-domain I\/Q $(R_i^B)$.\n\t\\item[iii)] data tones carrying no information are zero valued \n\tin the frequency I\/Q samples effectively resulting a \n\tlower date-rate over the fiber channel in both LTE and DOCSIS with the\n\tchance for statistical multiplexing, and\n\t\\item[iv)] possible caching of repetitive frequency QAM I\/Q samples, \n\tsuch as Reference Signals (RS) and pilot tones. \n\\end{itemize}\n\nTo emphasize the important data rate aspect, we note that in the \nproposed shared R-FFT, the data rate required over the fiber channel is \ndirectly proportional to the user traffic. We believe this is an \nimportant characteristic of the FFT functional-split whereby we \ncan achieve multiplexing gains by combining multiple R-FFT nodes or \nby the enabling the concurrent support of LTE as illustrated in \nFig.~\\ref{fig_LTE_cable}. With the similar characteristics of DOCSIS,\nthe LTE user traffic also translates to proportional data rate \nover the fiber channel. In addition, the proposed mechanism enables \nthe complex signal processing of the PHY layer to be implemented \nat the headend. Examples of the signal processing operations include channel\nestimation, equalization, and signal recovery, which can be implemented with\ngeneral-purpose hardware and software. In addition, the processing of \ndigital bits, such as the LDPC, which is necessary for the forward error\ncorrection, can also be implemented at the headend. Thus, the proposed approach\nreduces the cost of the remote nodes and increases the flexibility of changing \nthe operational technologies. The software implementations at the headend can be\neasily upgraded while retaining the R-FFT node hardware since the node hardware\nconsists only of common platform hardware, such as elementary DAC\/ADC and \nFFT\/IFFT components. Thus, the proposed approach eases the change\/upgrade of\ntechnologies. That is, the R-FFT node has minimal impact on technology \nadvancements because the blocks within a remote node are elementary \nor independent of most technology advancements. \n\n\\subsection{Proposed Remote Caching and Prefetching}\n\\label{sec:caching}\n\\begin{figure}[t!] \\centering\n\t\\includegraphics[width=5in]{fs\/fig_caching.pdf}\n\t\\caption{Caching at the remote node.}\t\n\t\\label{fig_caching}\n\\end{figure}\nIn order to further reduce the bandwidth in addition to functional split process, \nseveral techniques, such as I\/Q compression~\\cite{nanba2013new, nieman2013time,guo2012cpri, joung2013base} \ncan be employed. In contrast, we propose resource element (time and frequency slot) \nallocation based remote caching. If some part of the information is regularly and \nrepeatedly sent over the interface, a higher (orchestration, in case of SDN) \nlevel of the signaling process can coordinate caching mechanisms. For example, \nthere is no need to transmit the downlink I\/Q samples of the pilot tones as\nthey would remain constant in the DOCSIS. Figure~\\ref{fig_caching} illustrates the overview \nof repetitive QAM symbols in the LTE and DOCSIS. There is a process of \nup-sampling and zero padding before and after the FFT\/IFFT block, \ndepending on the FFT size and the effective subcarriers, which can \npotentially be controlled and cached through the headend (depending on \nthe implementation). Zero padded QAM symbols as well as certain reserved resource elements \ncan be excluded from the \ntransmissions while signaling the changes \nto the cache management at the R-FFT node.\nAs compared to downstream, upstream information must be entirely \ntransported to the headend to process all the signal component \nreceived by the R-FFT receiver. \n\\subsubsection{LTE Networks}\n\\paragraph{Reference Signal (RS) Tones Caching}\nRS tones are the pilot subcarriers which are embedded throughout the operational \nbandwidth of the wireless system for the channel estimation to equalize the \nimpairments of received wireless signal. Disturbances to the \nwireless signal are more prominent compared to propagation of signal \nin the wired channel and therefore\nthe RS tones are added in close proximity with \neach other to accurately estimate the channel based \non the channel characteristics, such as coherence-time and coherence-bandwidth. \nFor a single antenna, the RS tones are typically spaced 6 subcarriers apart \nin frequency such that 8 RS tones exists in a single subframe, i.e, 14 OFDM symbols\nand single Resource Block (RB), i.e., 12 subcarriers of LTE. Thus, the overhead\ndue to RS tones in the LTE resource grid is\n\\begin{equation}\n\\text{RS Overhead} = \\frac{8}{12 \\times 14 } = 4.7~\\%.\n\\label{eq:rs}\n\\end{equation}\nTherefore, approximately 4.7~\\% of I\/Q transmissions over the \ndigital fiber can be saved from the RS tones caching at the \nremote node regardless of the system bandwidth.  \n\n\\paragraph{PHY Broadcast Channel (PBCH) Caching}\nPHY Broadcast Channel (PBCH) carries the Master Information Block (MIB)\nwhich is broadcasted continuously by the\neNB regardless of the user connectivity.\nMIB includes the basic information about the LTE system \nsuch as the system bandwidth and control information specific to LTE channel.\nPBCH\/MIB always uses central\n6~RBs (i.e, 72~subcarriers) for the duration of \n4~OFDM symbols to broadcast the MIB data.\nPBCH space in the resource grid is inclusive of the \nRS tones used in the calculation of Eq.~\\ref{eq:rs} and therefore\nneeds to subtracting while calculating the overhead for MIB transmission.\nPBCH\/MIB occurs once every 40~ms and there exists 4~redundant versions of \nMIB which will be broadcasted with the offset of 10~ms. Thus, \nessentially PBCH\/MIB occurs once in every 10~ms (radio frame). The \nPBCH\/MIB overhead for the entire LTE system is\n\\begin{equation}\n\\text{PBCH Over.} = \\frac{6 \\times 12 \\times 4 - (8 \\times 6)}{1200 \\times 14 \\times 10} = 0.142~\\%.\n\\label{eq:mib}\n\\end{equation}\nAlthough 0.142~\\% is less, relative overhead of PBCH\/MIB data \nincreases significantly for lower bandwidth LTE system. Moreover, our evaluation of the\nPBCH overhead considers the full RBs i.e., 1200 subcarriers, 14 OFDM symbols and 10 subframes\nin the LTE. In a real system the allocation broadly varies user density i.e, \ngenerally only part of the RBs will be used increasing the overhead effects. \nFor example, for a 1.4~MHz system (used in an IoT type of applications) \nwith full RBs, the overhead jumps to \n\\begin{equation}\n\\text{PBCH Over.}_{1.4\\text{MHz}} = \\frac{6 \\times 12 \\times 4 - (8 \\times 6)}{72 \\times 14 \\times 10} = 2.3~\\%.\n\\label{eq:mib:14}\n\\end{equation}\nTherefore, up to 2.3~\\% I\/Q transmissions over the \ndigital fiber can be saved from the PBCH\/MIB caching at the remote node.\n\n\\paragraph{Synchronization Channel Caching}\nSynchronization Channel (SCH) consists of Primary Synchronization Sequence (PSS)\nand Secondary Synchronization Sequence (SSS) which is broadcasted continuously\nby the eNB regardless of the user connectivity. PSS and SSS are the special\nsequence which helps in the cell synchronization of wireless users by identifying\nthe physical cell ID and the frame boundaries of the LTE resource grid.\nPSS\/SSS occurs every 5~ms (twice per radio frame) and uses central 6~RBs over 2 \nOFDM symbols. Similar to Eq.~\\ref{eq:mib}~and~\\ref{eq:mib:14}, the overhead due to\nPSS\/SSS in 20~MHz and 1.4~MHz system is \n\\begin{equation}\n\\begin{split}\n\\text{SCH Over.} = \\frac{6 \\times 12 \\times 4}{1200 \\times 14 \\times 10} = 0.171~\\%. \\;\\;\\;\\:\\\\\n\\text{SCH Over.}_{1.4\\text{MHz}} \n= \\frac{6 \\times 12 \\times 4}{72 \\times 14 \\times 10} = 2.8~\\%. \\;\\;\\;\\;\\;\\;\\;\n\\end{split}\n\\label{eq:sch}\n\\end{equation}\n\n\\paragraph{System Information Block (SIB) Caching}\nIn a similar way, the caching mechanism can also be extended \nto the System Information Blocks (SIBs) broadcast messages \nwhich is through the PHY Downlink Shared Channel (PDSCH) of the LTE.\nThere are 13 different types of SIBs, from SIB1 to SIB13. \nSIB1 and SIB2 are mandatory broadcast messages while the transmission of other SIBs \ndepends on the relation between the \nserving and neighbor cell configurations. On a typical deployment, \nSIB3 to SIB9 are configured and can be combined in a single message block\nfor the resource block allocation. Typical configuration of the \nRB allocation type schedules the SIB over 8 RBs across\n14~OFDM symbols in time (1~subframe) are used for the transmission of SIB1 and SIB2, \nand with the effective periodicity (with redundancy version transmission) \nof 2 radio frames (20~ms). The overhead and caching gain from the \nof SIB1 and SIB2 transmissions while subtracting the corresponding \nRS tones overhead $8 \\times 8$, i.e., 8 tones per RB for 8 RBs is\n\\begin{equation}\n\\begin{split}\n\\text{SIB Over.} = \\frac{8 \\times 12 \\times 14 - (8 \\times 8)}{1200 \\times 14 \\times 20} = 0.381~\\%. \\;\\:\\\\\n\\text{SIB Over.}_{1.4\\text{MHz}} \n= \\frac{8 \\times 12 \\times 14 - (8 \\times 8)}{72 \\times 14 \\times 20} = 6.3~\\%. \\;\\;\\;\\;\\;\n\\end{split}\n\\label{eq:sib}\n\\end{equation}\nCaching of higher order SIBs i.e., from SIB3 to SIB 9 \ncan achieve further savings, however, the resource allocation \nand periodicity can widely vary to accurately estimate the overhead.\n\nThe stationary resource elements across the time domain, such as \nSIB information is allowed to change over a larger time scale in terms of \nhours and days, at such times the cached elements are refreshed or \nre-cached through cache management and signalling procedures, see Sec.~\\ref{sec:cache:mgt}.\n \n\\subsubsection{Cable Networks}\nIn the DOCSIS~3.1 downstream pilot subcarriers are modulated \nby the CMTS with a predefined modulation pattern which is known to all the\nCMs to allow the interoperability.\nTwo types of pilot patterns are defined in the DOCSIS~3.1 for OFDM time\nfrequency grid allocations, i) continuous, and ii) scattered. \nIn the continuous pilot pattern, the pilot tones with a predefined modulation \noccur at fixed frequencies in every symbol across time. \nWhereas, in the scattered pilot pattern, the pilot tones are sweeped to \nto occur at each frequency locations but at different symbols across time.\nThe scattered pilot pattern has a periodicity of 128 OFDM symbols along the time dimension such that the pattern repeat for the next cycle. \nScattered pilots assist in the channel estimation for the\nequalization process of demodulation. In a typical deployment~\\cite{cabledeploy}, \noperational bandwidth is 192~MHz corresponding to the FFT size of 8K\nwith the 25~kHz subcarrier spacing. The total number\nof subcarriers in a 192~MHz system is 7680 out of which \nthere are 80 guard band subcarriers, 88 continuous pilot subcarriers and \n60 scattered pilot subcarriers. Therefore, the overhead due to redundant \nsubcarriers in the DOCSIS system which can be cached at the remote node is\n\\begin{equation}\n\\text{Cable Over.} = \\frac{80 + 88 + 60}{7680} = 2.9~\\%.\n\\label{eq:cable}\n\\end{equation}  \n\n\\subsection{Memory Requirements for Caching}\nCaching of frequency domain I\/Q symbols in the OFDM \ncomes at the cost of caching memory implementation at the remote node.\nEach I\/Q symbol that needs to be cached is a complex number\nwith real and imaginary part. For the purpose of evaluation, we consider \neach part of the complex number represented by 10 bits resulting\nin 20 bits in total for each frequency domain QAM symbol. The caching of\nRS tones in the LTE results in 4.7~\\% of the savings in the \nfronthaul transmissions as show in the Eq.~\\ref{eq:rs}. In the duration of \neach OFDM symbol, 2 RS tones exists for every 12~subcarriers. For the \n20~MHz system, there would be 200 RS tones. Therefore,\ntotal memory required to cache the data\nof QAM symbols corresponding\nto the RS tones is \n\\begin{equation}\n\\centering\n\\text{RS Tones Mem.} = (2 \\times 100) \\times 2 \\times 10 = 4000~\\text{bits}.\n\\label{eq:rs:mem}\n\\end{equation}  \nSimilarly, caching of PBCH, SCH and SIB data requires \n\\begin{equation}\n\\begin{split}\n\\text{PBCH Mem.} = (4\\times12\\times6 - (8 \\times 6)) \\times 2 \\times 10 \\;\\;\\;\\;\\;\\;\\; \\\\\n= 4800~\\text{bits}, \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\n\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\: \\\\\n\\text{SCH Mem.} = (6 \\times 12 \\times 4) \\times 2 \\times 10 = 5760~\\text{bits},\\;\\;\\: \\\\\n\\text{SIB Mem.} = (8 \\times 12 \\times 14-(8 \\times 8)) \n\\times 2 \\times 10 \\;\\;\\;\\;\\; \\\\ =5760~\\text{bits}.\n\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\n\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\n\\end{split}\n\\label{eq:pbch:mem}\n\\end{equation}\nWhereas, for the DOCSIS, the the memory requirements for the\ncaching of continuous and scattered pilots is  \n\\begin{equation}\n\\centering\n\\text{Pilot Tones Mem.} = (80+88+60) \\times 2 \\times 10 = 4560~\\text{bits}.\n\\label{eq:pilot:mem}\n\\end{equation}\nThus, based on Eqs.~\\ref{eq:rs}-\\ref{eq:cable}, total  \nsavings of approx. 7~\\% to 18~\\% can be achieved in the\nfronthaul transmissions when the full resource allocation over the\nentire bandwidth is considered in both LTE and DOCSIS.\nFor lower allocations i.e, when there \nexists lesser user data, the caching process can achieve much \nlarger relative benefits. \nIn an extreme case, when the user data is not present, all the cell \nspecific broadcast data information can be cached at the \nremote node such that the fronthaul transmissions \ncan be seized reducing the power consumption.\nThe total memory for the caching required at the remote node based \non the Eqs.~\\ref{eq:pbch:mem}-\\ref{eq:pilot:mem} is approximately 19120~bits.\nThe implementation of cache at the remote node \nwhich is lesser than the size of 25~KB is relatively very simple \nand of no practical burden to the existing remote node. Therefore, \nwe believe the savings of more than 7~\\% with almost negligible implementation\nburden is a significant benefit in the CRAN functional PHY-split. \n \n\\subsubsection{Signalling and Cache Management}\n\\label{sec:cache:mgt}\nSignalling mechanism facilitates the cache management processes. \nCache management operation involves, \ni) transporting the caching information to the remote nodes,\nii) updating the cached information at the remote nodes with new \ninformation, and\niii) establishing the rules for prefetching of cached resource \nelements at the remote node.\nSignalling agents at the headend\/cloud\nand remote node coordinate with each other \nthrough a separate (i.e., non-I\/Q transport)\nlogical connection between headend\/cloud and remote node.\nSome of the information which has been cached may change over time but \nat a larger timescale compared to the transmission of \nI\/Q from headend to the cloud. The signalling overhead which arises \nfrom the cache management is negligible because of the timescale of\nsignalling operations. Headend\/cloud can notify the remote node \nwith the changes \nthrough signalling to keep the information up to date at the remote node.\nPrefetching of the cached content has to be precisely executed \nwith accurate placement of the subcarriers information in the particular \ntime and frequency locations to an IFFT\/FFT computational module. \n \n\\subsection{Transport Networks and Protocols}\n\\subsubsection{Transport Networks}\nAn R-FFT node can be connected to headend through variety of transport \nnetwork solutions over the optical fiber connectivity as well as dedicated \nmmWave links. The fundamental requirement is to support the CRAN and CCAP \ndemands while virtualizing the QAM functions at the headend. \nSome of the technologies that can be considered are:\n\\paragraph{Dark Fiber} A fiber network that is deployed in excess of \nthe existing requirement is referred to as a dark fiber network. \nTypically, dark fiber resources are abundant but with limited accessibility. \nIf a headend can be connected to an R-FFT node using dark fiber, a large data \nrate with low latency connections can be established, supporting safe \nand reliable transport of required I\/Q samples.\n\\paragraph{Optical Wavelength Division Multiplexing (WDM)} WDM establishes \nend to end connectivity with the dedicated wavelength aggregated to meet \nthe user demands. WDM with the potential to reach large data rates and low \nlatencies can easily support the connectivity of the R-FFT with the headend.\n\\paragraph{Passive Optical Network (PON)} A PON network typically provides \nhigh data rates in the Point to Multipoint Ethernet services for residential \nand Fiber to The Premises (FTTP) type of applications. If PON technology is \nused to establish the connectivity, careful latency limitations must be evaluated. \nSeveral interfaces, such as CPRI, exists to support the \nconnections over the PON transport network.\n\\paragraph{Wireless} A standard and dedicated microwave or millimeter-wave \nwireless links can support large data rate and low latency connection between \ntwo nodes. However, wireless links are limited by the range and \nsometimes require Line of Sight (LoS) operation.\n\n\\subsubsection{Protocols}\nProtocol is required to coordinate the transmissions of I\/Q data over the\ntransport network. A strict latency requirement for the CRAN and DCCAP \narchitectures limits the choice of generic protocols over \nEthernet. Some of the fronthaul protocol that exists for the \ntransport of information between headend\/cloud and radio node \nare:\n\n\\paragraph{Radio over Fiber (RoF)}\nRadio over fiber (RoF) transports the radio frequency signal \nover an optical fiber link by converting the electrically modulated signals\nto optical signal. RoF signals are not converted in frequency but superimposed\nover optical signals to achieve the benefits of optical transmissions,\nsuch as reduced sensitivity to noise and interference\nRoF signals are optically distributed to end nodes where the\noptical signals are directly converted to electrical signal with\nminimal processing reducing the cost of the remote node. \nThe downside of the RoF is the analog transmission of the optical \nsignal which attenuate over distance over the fiber as compared to \nthe digital data over the fiber.\n\n\\paragraph{Common Public Radio Interface (CPRI)}\nCommon Public Radio Interface (CPRI) \\cite{CPRI} defines the protocol\nto transport the digitized I\/Q data through the encapsulated CPRI frames. \nAs compared to RoF, CPRI provides more reliable end-to-end connection between\nheadend\/cloud and the remote node. Dedicated TDM channels can be established \nto supports multiple logical connections supporting different air interfaces. \nThe downsides of the CPRI are the strict timing and synchronization \nrequirement as well as support for only the fixed functional split to \ntransport time domain I\/Q samples. To overcome the fixed functional split\nin the eCPRI is being currently developed to support a new functional \nsplit with ten folds of decrease in the data rate requirement over the \nfronthaul link. In comparison to eCPRI and CPRI protocols, \nour proposal is an enhancement to \nthe base protocols where the repeated I\/Q can be cached when frequency \nI\/Q samples are transported over the fronthaul link.\n\n\\paragraph{Open Base Station Architecture Initiative (OBSAI)}\nOpen Base Station Architecture Initiative (OBSAI)~\\cite{obsai} is \nsimilar to CPRI in which the digitized time domain I\/Q samples are \ntransported over fronthaul interface.\nIn contrast to CPRI, OBSAI interface is an IP based connection.\nA digital interface which is based on an IP logical connection \ncan be implemented over any generic Ethernet link providing a flexible\nconnectivity between headend\/cloud to remote node.\n\n\\paragraph{External PHY Interfaces}\nDownstream External PHY Interface (DEPI)~\\cite{depi} and Upstream External PHY Interface\n(UEPI)~\\cite{uepi} enable the common transport mechanisms between \nR-PHY and CCAP core. DEPI\nand UEPI are based on the Layer 2 Tunneling Protocol version 3 (L2TPv3).\nThe L2TPv3 transparently transports the Layer 2 protocols \nover a Layer 3 network creating the psedudowires (logical connections).\nFor the R-MACPHY, since only the MAC payload is required to \nbe transported to the headend for processing of upper layer\ndata, strict requirements for the latency as in the case of R-PHY \ncan be relaxed, allowing\nany generic tunneling protocol to be used between headend and R-MACPHY.\n\n\\section{\\MakeUppercase{Proposed Common Platform for LTE and Cable Networks}}\n\\label{sec:common:infra}\n\\begin{figure*}[t!] \\centering\n\t\\includegraphics[width=\\textwidth]{fs\/fig_common_fft.pdf}\n\t\\caption{Reuse of infrastructure.}\n\t\\label{fig_common_fft}\n\\end{figure*}\nThe last hop connectivity to the user devices is typically \nenabled by wireless technologies. Although, any wireless interface, \nsuch as Bluetooth, ZigBee, or WiMax, can provide the wireless \nlast hop connectivity, WiFi is a very promising technology. \nWiFi has proven to be simple and efficient for managing large \nnumbers of connections as well as to serve large\ndata rates to connected nodes. Also, importantly, WiFi has the capability\nof delivering high data rates (up to Gbps) in the unlicensed bands\nover a reasonable coverage area (up to 100 m), and in indoor \nenvironments. In contrast, the LTE wireless interface, which is widely used \nfor commercial cellular communications in licensed bands, is capable of \ndelivering up to several hundreds of Mbps over larger geographical areas \n(up to several tens of miles). Several efforts are underway to integrate \nthese two wireless technologies so as to enable even higher data rates \non the order of several Gbps. LTE-U and LTE-LAA [6] are newly developed wireless \ninterfaces where a primary LTE link opportunistically utilizes the \nunlicensed WiFi bands. The next generation personal computers \nare expected to support LTE connectivity in conjunction with \nEthernet and WLAN. \nHowever, to meet the future demands \nto support concurrent connectivity, deployment and management of\nindependent LTE and WiFi technologies by the same operator \nincurs large expenses to the operators. We envision a solution\nwhere LTE base station can be established at the \nremote node in the DCCAP architecture.\nTo our knowledge, the\nimplementation of FFT at the remote node to simultaneously support\ncable and LTE infrastructure is novel to our proposal. \n\n\\subsection{Common IFFT\/FFT for LTE and DOCSIS}\nThe protocol of LTE and DOCSIS share the same physical properties as\nthey depend on OFDM for the physical layer modulation technique.\nImplementation of the OFDM is dependent on the design of \nFFT computations~\\cite{he1998designing}.\nThe property by which both LTE and DOCSIS require \nsame IFFT\/FFT operations to be \nperformed for each OFDM modulation and demodulation can be exploited\nto reuse the existing infrastructure provided such mechanism \ncan sustained on the computing hardware.\nThus, the main motivation of the FFT implementation at the remote node \nis to bring out  the common platform at the remote \nnodes while the transmission formats are \nflexibly realized at the headend for heterogeneous \nprotocols based on OFDM.\nFigure~\\ref{fig_common_fft} describe the internals of the\nR-FFT architecture simultaneously supporting cable and LTE. \nGenerally, in the downstream direction, an IFFT operation is performed \nonce every OFDM symbol duration. The LTE OFDM symbol duration is \napproximately 72 \u00b5s, and for DOCSIS, the OFDM symbol duration \nis 84.13 \u00b5s. However, the actual time to compute IFFT can span \nfrom few microseconds to several tens of microsecond. \nConsidering that there exists a large portion of idle \ntime durations in the IFFT module during the FFT computation, \nwe can interleave the I\/Q input in time such that same \nIFFT\/FFT module can be used for multiple technologies. \nBy reusing the IFFT\/FFT computing structures we can reduce \nthe complexity of the hardware, be more power efficient, \nand reduce the cost of R-FFT node. \n\n\\subsection{FFT Computations Interleaving Timing Discussion}\n\\begin{figure}[t!] \\centering\n\t\\includegraphics[width=5in]{fs\/fig_timing.pdf}\n\t\\caption{The IFFT\/FFT interleaving.}\n\t\\label{fig_timing}\n\\end{figure}\nIn this section we preset the timing schedule analysis \nfor the interleaving of IFFT\/FFT computations on \na single computing resource. Figure~\\ref{fig_timing} shows the\nbasic timing diagram to schedule the FFT computations on the\ncomputing resource. $T_D$ and $T_L$ are the OFDM symbol durations of \nDOCSIS and LTE respectively. Similarly, $\\tau_D$ and $\\tau_L$\nare the durations of FFT computations for DOCSIS and LTE.\nWhile $p$ and $q$ are the timing indexes for the frame of reference\nfor DOCSIS and LTE independent periodic events. \nWE also assume that $T_L$ and $T_D$ start at the time index p\nand q without any offset as show seen in the Fig.~\\ref{fig_timing}\n\nFor illustration, in this article we assume the OFDM\nsymbol duration of LTE is greater then the symbol duration of \nDOCSIS, i.e., $T_L > T_D$, Consequently, the duration\nof FFT computations depends on the FFT size where we assume \n$\\tau_D > \\tau_L$, as DOCSIS has the larger FFT size compared to LTE. \nAlso, it is most likely that heterogenous technologies operate with \ndifferent OFDM symbol times. Thus, \nwe define the difference in the OFDM symbol duration as \n$\\delta= T_L - T_D$. \n\n\\paragraph{Feasibility Discussion}\nSuppose if there is no offset between the cycles \nas shown in the start of timing schedule in Fig.~\\ref{fig_timing}\nand $k\/l$ denotes the fraction of OFDM symbol durations, \nwhere, $k$ and $l$ if exists are the minimum positive integers,\nsuch that $T_L\/T_D=k\/l$,\nfor every $k$ cycles of $T_D$ there would be $l$ cycles of $T_L$. \nThat is, when two periodic signal are overlapped, \nthe difference of periodicities $T_L$ and $T_D$, $\\delta$, \nat each cycle turns out to be integer multiple of the \nprevious cycle. \nFor example, in the Fig.~\\ref{fig_timing} at the\nfirst cycle we define $\\delta = T_L - T_D$, for the second cycle\nthe difference is $2T_L-2T_D = 2\\delta$ and similarly for the third\ncycle the difference is $3\\delta$. Since $T_L > T_D$ From \nthe fundamental principles we can evaluate that after  \n$k=\\lceil T_L\/\\delta \\rceil = \\lceil T_L\/(T_L-T_D) \\rceil $ \ncycles of $T_D$, the overlapping behavior repeats. Also,\nthe  For example,\nin the Fig.~\\ref{fig_timing}, for every $k=4$, which corresponds to\n4 cycles of $T_D$ and 3 cycles of $T_L$, the behavior repeats.\nTherefore, if the stability is ensured for $k$ cycles of $T_D$ or\n$l$ cycles of $T_L$, the system is stable and feasible.\n\nThus, considering the larger OFDM symbol duration $T_L$\nas the reference, where we have $l$ cycles of $T_L$ for the secondary\nperiodicity of combined $k$ and $l$ cycles. For the system \nto be stable, the total \nFFT computation durations from combined LTE and DOCSIS \ni.e., $l\\tau_L + k\\tau_D$ should not exceed \nthe duration of $l$ cycle of $T_L$ i.e.,\n\\begin{equation}\n\\centering\nl\\tau_L + k\\tau_D \\leq lT_L.\n\\label{eq:stability}\n\\end{equation}  \nWe know that $k$ cycles of $T_D$ is equal to the $l$ cycles of $T_L$,\ni.e., $kT_D = lT_L$ and therefore, $k=\\frac{lT_L}{T_D}$. Substituting $k$\nin the Eq.~\\ref{eq:stability} we get,\n\\begin{equation}\n\\centering\n\\begin{split}\n\\left(\\frac{lT_L}{T_D}\\right) \\tau_D+l\\tau_L\\leq lT_L \\\\\n\\frac{T_L \\tau_D}{T_D} + \\tau_L \\leq T_L\\\\\n\\tau_D T_L + \\tau_L T_D \\leq T_L T_D.\n\\label{eq:stability2}\n\\end{split}\n\\end{equation}\nIn addition to Eq.~\\ref{eq:stability2}, as shown in the Fig.~\\ref{fig_timing}\nwe need to ensure that\nFFT computation  durations are sufficiently small to support the \ninterleaving. That is, for interleaving $\\tau_{\\max(T_L, T_D)}$ on\nthe periodic cycles of $\\min(T_L, T_D)$, we need at least the space of  \n\\begin{equation}\n\\centering\n\\tau_{\\max(T_L, T_D)} \\leq 2\\min(T_L, T_D)-2(\\tau_{\\min(T_L, T_D)}).\n\\label{eq:stability3}\n\\end{equation} \n\n\\paragraph{Constraints}\nThe end of each OFDM symbol duration marks the\ntrigger point for the FFT computing scheduling request. \nOur fundamental analysis is based on\nthe fact that IFFT\/FFT computing takes much less time \ncompared to the actual OFDM symbol duration, especially\nas the computing hardwares are becoming more \nadvanced~\\cite{yeh2003high, arun2016design}.\nAs this factor is dependent on the hardware we impose \nfollowing constraints to design the interleaving scheduling\nprocedure of two heterogeneous technologies on the\nsame hardware, i) the FFT computing\ndurations should not exceed their OFDM symbol duration, and\nii) the FFT computations must be finished before the start of\nnext OFDM symbol, i.e.,\n\\begin{equation}\n\\centering\n\\begin{split}\n\\tau_L < T_L, \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\:\\\\\n\\tau_D < T_D, \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\:\\\\\np\\tau_L < (p+1)T_L,\\\\\nq\\tau_D < (q+1)T_D.\n\\end{split}\n\\label{eq:comp.constr}\n\\end{equation}\nThus, in addition to Eq.~\\ref{eq:stability2}, the constraints presented in\nthe Eq.~\\ref{eq:comp.constr} must be satisfied in the process interleaving\nof computations to have no adverse effects on the operational technologies.\n\n\\paragraph{Effect of Guard Time}\nGuard time separates two consecutive computations to compensate for the \nload and read time of the I\/Q to the FFT structure while supporting the \ncoexistence of two technologies on the same computation module.\nWe define $\\theta$ as the constant guard time which is applied \nto each scheduling set of the FFT computation. Based on Eq.~\\ref{eq:stability2},\nthe impact of guard time can be evaluated as\n\\begin{equation}\n\\centering\n(\\tau_D+\\theta)T_L + (\\tau_L+\\theta)T_D \\leq T_DT_L,\n\\label{eq:guardtime}\n\\end{equation}\nwhich can be written as,\n\\begin{equation}\n\\centering\n\\tau_DT_L + \\tau_LT_D \\leq T_DT_L - \\theta(T_L+T_D).\n\\label{eq:guardtime2}\n\\end{equation}\n\n\\begin{algorithm}[t!]\n\t\\caption{Caching and FFT Computation Procedure}\n\t\\label{algo:interleaving}\n\t\\SetKwInOut{Input}{input}\n\t\\SetKwInOut{Output}{output}\n\t\\nonl {\\bfseries{1. CRAN\/Headend}} \\newline\n\t\\nl (a) Identify cachable I\/Q samples. \\newline\n\t    (b) Create Caching rules. \\newline\n\t    (c) Signal the rules and data for caching.\t  \n\t\\newline\n\t\\nonl \\If{Cached I\/Q samples requires updating}{\n\t\t\\nonl  Signal remote node for cache renew or flush.\n\t\t\\nonl }\n \\nonl {\\bfseries{2. Remote Node}}   \\newline\n\\nonl\\ForEach{OFDM Symbol in $T_D$ and $T_L$}{\t\t\n  \\nonl          \\If{Caching is enabled}{\n  \\nonl\t\t\t\t  Read cache and I\/Q mapping\\; \n  \\nonl                    Insert I\/Q cache-read to recieved I\/Q\\;}     \n  \\nonl\t    \t\\If{FFT module is free}{\n  \\nonl\t\t\t\t  Schedule I\/Q for FFT based on Eqs.~\\ref{eq:stability}-\\ref{eq:comp.constr}\\;} \n  \\nonl          \\Else{\n  \\nonl\tSchedule at completion of current execution\\; \n  \\nonl }  \\nonl}\t   \t    \n\\end{algorithm}\n\n\\paragraph{Implementation}\n\\label{sec:tau}\nSharing of remote node infrastructure for multiple technologies can \nbe in both upstream and downstream, as the computations are \nperformed independent of each other \neven for wireless full-duplex communications. \nThe FFT computation duration \n$\\tau$ can be aggregate of multiple instances of OFDM symbol, for example, in\nthe case of carrier aggregation in LTE (or channel bonding in DOCSIS), \nthere would be an OFDM symbol \nfor each carrier component resulting in $\\tau_L=\\tau_1+\\tau_2\\dots\\tau_C$, \nwhere $C$ is the number of carrier components. Similarly, computations resulting \nfrom multiple LTE eNBs at a single node can be aggregated and abstracted to \na single $\\tau_L$ resulting in a lager $\\tau_L$ can be very \nclose the symbol duration $T_L$. Proposed approach can also be \neasily extended to more than two technologies which depend \non FFT computations by sharing the the remote node.\n \n\\section{\\MakeUppercase{Performance Evaluation}}\n\\label{sec:perf:eval}\nIn this section we present the performance evaluation of the R-FFT node where\nthe infrastructure of a remote node is \nshared for two independent LTE and DOCSIS technologies.\n\\begin{figure*}[t]\n\t\\begin{tabular}{cc}\n\t\n\t\t\\includegraphics[width=2.8in]{fs\/fig_tauD.pdf} \\vspace{-0.1cm}\n\t\t&\\includegraphics[width=2.8in]{fs\/fig_tauD_U.pdf} \\vspace{-0.1cm}\\\\\n\t\t\\vspace{+0cm}\n\t\t\\scriptsize (a) $T_L=71.4~\\mu\\textit{s, } \\tau_L=14.28~\\mu s\\text{, and } \\theta=0~\\mu\\textit{s}$\n\t\t&\\scriptsize (b) $T_L=71.4~\\mu\\textit{s, } \\tau_L=14.28~\\mu s\\text{, and } \\theta=0~\\mu\\textit{s}$\\\\\n\t\n\t\t\\includegraphics[width=2.8in]{fs\/fig_theta.pdf} \\vspace{-0.1cm}\n\t\t&\\includegraphics[width=2.8in]{fs\/fig_theta_U.pdf} \\vspace{-0.1cm}\\\\\n\t\t\\scriptsize (c) $T_L=71.4~\\mu\\textit{s, } \\tau_L=14.28~\\mu s\\text{, and } T_D=80~\\mu\\textit{s}$ \n\t\t&\\scriptsize (d) $T_L=71.4~\\mu\\textit{s, } \\tau_L=14.28~\\mu s\\text{, and } T_D=80~\\mu\\textit{s}$ \\\\\n\t\\end{tabular}\n\t\\caption{Simulation results for \n\t\tFFT interleaving procedure.} \n\t\\label{fig_tau_theta}\n\\end{figure*}\n\n\\subsection{FFT Module Sharing}\nIn the initial evaluation, we verify the proposed FFT interleaving procedure. \nwe have implemented the FFT interleaving mechanism \nas a discreet event simulation framework in OMNET++.\nA LTE and DOCSIS OFDM symbols are generated every \n$T_L$ and $T_D$ durations respectively. For LTE with normal cyclic prefix\nthe OFDM symbol duration $T_L$ is 71.4~$\\mu$s, while the DOCSIS \nsymbol duration can be either 40 or 80~$\\mu$s. \nFor each OFDM symbol arrival at the radio node, \nFFT computation is queued by the scheduler. Typically, the FFT size\nfor the LTE is 2K, whereas the FFT size for the DOCSIS can be 4K or 8K\nbased on the symbol duration (i.e., 40 or 80~$\\mu$s).\nComputation duration $\\tau$ for the FFT computation \ncan vary based on the number of carrier\ncomponents and number of radio nodes supported by the \nremote node as described\nin Sec.~\\ref{sec:tau}. \nInterleaving procedure should be aware of the computation duration for\neach technology to ensure the stable operation of infrastructure sharing. \nTo evaluate the system for wide range of FFT computation duration,\nwe define a computation duration factor\n\\begin{equation}\n\\centering\n\\beta=\\frac{\\tau_L}{\\tau_D}.  \n\\end{equation}\nWe choose the FFT computation duration $\\tau_L$ to be 20~\\% of \nthe $T_L$, which is $0.20 \\times 71.432~\\mu\\textit{s} = 14.2832~\\mu\\textit{s}$ \nand the computation factor $\\beta$ is varied \nfrom $0$ to $1$ with an effective sweep of \nthe FFT computation duration $\\tau_D$ (i.e., $\\min(T_L,T_D)$) \nfrom $0$ to $T_L$, which is effectively $0 \\leq \\tau_D \\leq T_L$. \nWe primarily evaluate the FFT sharing mechanism with \ntwo performance metrics, average wait time and utilization.\nAverage wait time is the average time required to\nschedule the FFT computation after the arrival of an OFDM symbol.\nWhereas, the utilization parameter is defined as the ratio of \ntotal computation time to total elapsed time.\n\nFigure.~\\ref{fig_tau_theta} presents the performance evaluation \nof proposed FFT sharing mechanism. \nThroughout the evaluation, the OFDM symbol duration \nof LTE is kept constant $T_L=71.4~\\mu$s.\nFigure.~\\ref{fig_tau_theta}(a) \nshows the average wait time of LTE and DOCSIS as a function of \ncomputation duration factor $\\beta$ for different values \nof DOCSIS OFDM symbol duration, $T_D=40\\mu$s and $T_D=80\\mu$s.\nWe can observe from the Fig.~\\ref{fig_tau_theta}(a), the \naverage wait time of LTE increases linearly as the computation factor \n$\\beta$ is increased which corresponds to increase\nin the FFT computation duration of DOCSIS $\\tau_D$. This shows that \nthe computation duration of one technology has the direct impact on \nthe average wait time of another as they compete with each other\nfor the computational resource. We also observe that the average\nwait time of both LTE and DOCSIS \nfor $T_D = 40~\\mu$s linearly increases up to $\\beta=0.45$ and \novershoots to a very large indicating the instability of the system.\nThis is because, for the FFT computation sharing mechanism to be \nstable, the Eq.~\\ref{eq:stability2} must be satisfied, i.e, \n$\\tau_D \\leq (T_L T_D - \\tau_L T_D)\/T_L$, which is, \n$\\tau_D \\leq (40 \\times 71.4 - 14.28 \\times 40)\/71.4 = 32~\\mu\\textit{s}$. \nWhen $\\beta=0.45$, the corresponding \n$\\tau_D = 0.45 \\times T_L = 0.45 \\times 71.4~\\mu\\textit{s} = 32.13~\\mu\\textit{s}$,\nsurpassing the stability limit based on Eq.~\\ref{eq:stability2}, which is $>32~\\mu\\textit{s}$. \nThis behavior can also be observed in Fig.~\\ref{fig_tau_theta}(b) \nwhich plots the utilization as a function of\ncomputation factor $\\beta$. While the utilization of FFT computation module \nincreases linear with $\\beta$, the system \napproaches nearly $100\\%$ utilization when $\\beta>0.45$ for $T_D=40~\\mu\\textit{s}$ \nreaching the stability limit of the system. \nSimilarly, when the DOCSIS OFDM symbol duration $T_D$ is\nchanged from $40~\\mu\\textit{s}$ to $80~\\mu\\textit{s}$, the \nbehavior of average wait time is similar to $T_D = 40~\\mu\\textit{s}$, but the\nsystem becomes unstable when $\\beta > 0.9$. When $\\beta$ is $0.9$, the value \nof $\\tau_D$ is $0.9 \\times T_L = 0.9 \\times 71.4~\\mu\\textit{s} = 64.26~\\mu\\textit{s}$,\nwhich is slightly  greater than stability limit of \n$\\tau_D \\leq (80 \\times 71.4 - 14.28 \\times 80)\/71.4$ or $\\tau_D \\leq 64~\\mu\\textit{s}$. \nConsequently, we can observe from Fig.~\\ref{fig_tau_theta}(b), the utilization\nof system approaches to $100\\%$ for $\\beta\\geq0.9$ for $T_D=80~\\mu\\textit{s}$.\n   \nFig.~\\ref{fig_tau_theta} (a) and (b) corresponds to the evaluation \nwhen guard time $\\theta=0~\\mu\\textit{s}$ for different values of\nDOCSIS OFDM symbol duration $T_D=40$ and $80~\\mu\\textit{s}$. In contrast, \nFig.~\\ref{fig_tau_theta} (c) and (d) evaluates the system for different \nguard times $\\theta=2$ and $10~\\mu\\textit{s}$ for $T_D=80~\\mu\\textit{s}$. \nFig.~\\ref{fig_tau_theta} (c) plots the average wait time as a function of\ncomputation factor $\\beta$, whereas, Fig.~\\ref{fig_tau_theta} (c) plots the\nsystem utilization as a function of $\\beta$. Fig.~\\ref{fig_tau_theta} (c) \nshows the same behavior as in Fig.~\\ref{fig_tau_theta} (a), where the average \nwait time for LTE increases linearly as the $\\beta$ is increased. \nThe scheduling of $\\tau_D$ experiences \na constant delay of $2.3~\\mu\\textit{s}$ until the stability limit as \nthe computation duration of LTE is retained constant in the simulation\n$\\tau_L=14.28~\\mu\\textit{s}$. \nIn contrast to Fig.~\\ref{fig_tau_theta} (a), \nsystem reaches stability limit for lower values of $\\beta$ in (c) \nand this behavior pronounced as the guard time is increased. \nMore specifically,\nwhen $\\theta$ is increased from 0 (see Fig.~\\ref{fig_tau_theta} (a) for $T_D = 80 ~\\mu\\textit{s}$) to $2~\\mu\\textit{s}$, the system becomes \nunstable for $\\beta=0.85$ as compared to $\\beta=0.9$. \nBased on Eq.~\\ref{eq:guardtime2}, for a system with guard time to be \nstable, $\\tau_DT_L + \\tau_LT_D \\leq T_DT_L - \\theta(T_L+T_D)$ or\n$\\tau_L \\leq (T_DT_L - \\theta(T_L+T_D)-\\tau_LT_D) \/ T_L$.\nWhen $\\beta=0.85$ which corresponds to $\\tau_D=60.69~\\mu\\textit{s}$, with $\\theta=2~\\mu\\textit{s}$, the stability\ncondition evaluates to be $tau_D \\leq 59.75$, therefore, the system is\nunstable for $\\beta=0.85$ indicated by the very \nlarge in the average wait time for $\\beta>0.85$.\nSimilarly, for $\\beta=0.6$ which corresponds to $\\tau_D=42.84~\\mu\\textit{s}$, with $\\theta=10~\\mu\\textit{s}$, the stability\ncondition evaluates to be $\\tau_D \\leq 42.79$, therefore, the system is\nunstable for $\\beta=0.6$.\nAdditionally, the effect of guard time has the direct impact on the\nutilization of the system. We see from Fig.~\\ref{fig_tau_theta} (d), the maximum\nachievable utilization of the system is only up to \n94~\\% when guard time $\\theta=2~\\mu\\textit{s}$ for $\\beta>0.8369$. \nAs value of $theta$ is increased\nto $10~\\mu\\textit{s}$, the maximum achievable utilization is\nreduced to 73.5\\% for $\\beta>0.59$. \nThe saturation in the utilization also\nindicates the instability of the system, where the average wait time\ngrows to a very large value for $\\beta>0.8369$ and $\\beta>0.59$,\nwhen guard time $\\theta=2$ and $2~\\mu\\textit{s}$ respectively. \n\n\\subsection{End-to-End Delay Evaluation}\n\\begin{figure*}[t]\n\t\\begin{tabular}{cc}\n\t\n\t\t\\includegraphics[width=2.9in]{fs\/fig_cable02.pdf} \\vspace{-0.1cm}\n\t\t&\\includegraphics[width=2.9in]{fs\/fig_LTE02.pdf} \\vspace{-0.1cm}\\\\\n\t\t\\vspace{+0cm}\n\t\t\\scriptsize (a) DOCSIS Delay (end-to-end), $\\rho_c=0.2$\n\t\t&\\scriptsize (b) LTE Delay (fronthaul), $\\rho_c=0.2$, $H=0.5$\\\\\n\t\n\t\t\\includegraphics[width=2.9in]{fs\/fig_cable06.pdf} \\vspace{-0.1cm}\n\t\t&\\includegraphics[width=2.9in]{fs\/fig_LTE06.pdf} \\vspace{-0.1cm}\\\\\n\t\t\\scriptsize (c) DOCSIS Delay (end-to-end), $\\rho_c=0.6$\n\t\t&\\scriptsize (d) LTE Delay (fronthaul), $\\rho_c=0.6$, $d=50$~km\\\\\n\t\\end{tabular}\n\t\\caption{Simulation results for RFFT and RPHY.}\n\t\\label{fig_rfft_perf}\n\\end{figure*}\n\\begin{figure}[t!]\n\t\\centering\n\t\\includegraphics[width=3.5in]{fs\/fig_tput06.pdf} \n\t\\vspace{-0.3cm}\n\t\\caption{Average DOCSIS (cable) and LTE throughput.}\n\t\\label{fig_rfft_tput}\n\\end{figure}\nSubsequently, in this evaluation, we study the delay characteristics of R-FFT\ndeployment in comparison to existing R-PHY deployments. In addition, \nwe also present the impact of cable traffic on the LTE fronthaul link when\nthe same cable infrastructure is shared between LTE and DOCSIS.\nWe developed a simulation framework in a discreet event \nsimulator OMNET++ to implement the DCCAP cable architecture of \nthe HFC network. A remote node (RPHY or RFFT node) is connected \nto the headend through an optical connectivity separated by a \ndistance $d$ and data rate of $R_o=10$~Gbps. \n200~Cable Modems (CMs) are connected to the remote node through\na analog broadcast cable connectivity. We assume the DOCSIS~3.1 protocol\nto coordinate the cable transmissions in the cable broadcast medium \nwith the bidirectional cable data rate $R_c=1$~Gbps.\nWe implemented a Double Phase Polling (DPP) protocol\\cite{cho2009dou}\nfor the scheduling of 200~CMs over the cable segment. In case of \nRPHY, DOCSIS PHY frames are digitized and transported over Upstream \nExternal PHY Interface (UEPI) where the REQ packets were prioritized. \nWhereas for the RFFT, the upstream cable \ndata is converted to frequency I\/Q and transported as a generic UDP packet.\nFFT size of $4K$ which corresponds to $T_D = 40~\\mu s$ and QAM size of $12$~bits,\nand the code rate of $9\/10$ are used in the \nprocess of converting data to frequency domain I\/Qs.\nEach complex number representing an I\/Q symbol\nis digitized with $20$~bits.\nWe assume the deployment of an LTE RRH at the site of \nremote node in conjunction to the operation of RFFT or RPHY \nnode sharing the optical connectivity \nof data rate $R_o$ to the headend where the BBU CRAN is implemented.\nWe note that our study is focused on deployment of LTE networks\nover the HFC cable infrastructures. Therefore, we implement and\nevaluate the performance of a cable network when an\nLTE fronthaul traffic is simultaneously injected to the shared optical link.\nWe have modeled a typical FIFO queue at the remote node to froward the \nLTE packets to CRAN\/BBU. \nWe model the self similar traffic generation \nat each CM with varying burst levels controlled \nby the hurst parameter $H$ with the average packet size of $472$~KB. \nThe hurst parameter $H=0.5$ corresponds to the Poisson traffic, and the\nburstiness increases as the value $H$ is increased.\n\nFigure~\\ref{fig_rfft_perf} presents the performance evaluation of the\nRFFT and RPHY network in presence of the LTE fronthaul traffic. \nIn Fig.~\\ref{fig_rfft_perf}(a) we show the mean packet delay of the DOCSIS\nas a function of LTE fronthaul traffic intensity $\\rho_L$ which corresponds \nto the LTE data rate $R_L = \\rho_L \\times R_o$ for different optical distance \n$d$ and traffic levels burstiness $H$, with fixed cable \ntraffic intensity of $\\rho_c = 0.2$ which corresponds to the \ntraffic of $\\rho_c \\times R_c = 0.2 \\times 1$~Gbps $=200$~Mbps. \nDelay is considered for an end-to-end connection of the cable \nconnectivity i.e., from the CM to headend.\nFrom Fig~\\ref{fig_rfft_perf}(a) we observe for the \nshorter optical distance which is typically deployed for an LTE\nfronthaul link has the minimal impact on the cable infrastructure for both\nRFFT and RPHY deployments. More specifically, \nfor $H=0.5$ in RFFT (and RPHY) for distance $10$~km and $50$~km \nwe observe the delay difference of roughly $0.2$~ms \n(delay of $3.73$~ms for $10$~km RFFT $H=0.5$ and \nthe delay of $3.93$~ms for $50$~km RFFT $H=0.5$)\nwhen the distance is changed from $10$~km to $50$~km. \nThis difference in the delay is from the propagation\ndelay in the optical fiber which is approximately \n$(d_2-d_1)\/c = 40~\\text{km}\/2\\times10^8~\\text{m\/s}=0.2$~ms.  \n\nWe also observe a minute difference in the\ndelay between RPHY and RFFT implementation at the remote node for \nthe Poisson traffic, but a larger difference for higher burstiness\ntraffic. For example, at $\\rho_L=0.5$ \nwe observe the mean packet\ndelay values of $3.75$~ms and $3.73$~ms for \nRFFT and RPHY respectively.\nThe difference in the delay of nearly $0.02$~ms \nis due to the average transmission delay of \nthe cable data which is inflated by the process of \nI\/Q conversion and digitization. For the \ncode rate of $9\/10$ and the QAM size of $12$~bits \nthe cable data would be inflated by \nfactor of $(10\/9) \\times (1\/12) \\times 2 \\times 10=1.85$. Thus,\nthe RFFT packets undergo an additional delay \nat the remote node for the \ncomplete transmission of data to headend. \nHowever, this effect is pronounced for $H=0.8$\nbecause of the traffic with the higher \nburstiness. \nThe behavior of the DOCSIS delay for RPHY and RFFT \nremains constant until $\\rho_L=0.96$ for $H=0.5$ and $\\rho_L=0.8$ \nfor $H=0.8$, as\nthe optical link is not congested. \nHowever for $\\rho_L>0.8$,\nthe effective throughput over the optical link \nreaches more than the capacity $R_o$ crossing \nthe stability limit of the system. For e.g., \nthe instantaneous cable traffic of RFFT on the \noptical link for $H=0.5$ and \n$\\rho_c=0.2$ would be approximately \n$200~$Mbps$\\times 1.85=370$~Mbps or $0.37$~Gbps.\nThus, when $\\rho_L>0.963$ the effective\ntraffic would be more than $R_o$ \nresulting in the very large value of delay due \nto instability of the system. \n\nAs seen in Fig.~\\ref{fig_rfft_perf}(b) which plots the \nmean packet delay for the LTE fronthaul traffic for RFFT and RPHY \nfor different optical distances of $10$ and $10$~km. LTE fronthaul delay\nthe directly impacted by deployed cable technology when the traffic from\nLTE and cable are aggregated at the remote node. \nRFFT induces more delay on the LTE fronthaul traffic compared to the RPHY, \nwhich is mainly resulting from the increase in the effective data rate\nover the optical link. Also, we can observe the \neffect of stability limit of the optical link which is \nin both Fig.~\\ref{fig_rfft_perf}(a) and (b) [see for e.g., \nthe overshoot of the delay \nwhen $\\rho_L=0.96$ for RFFT, $H=0.5$ and $d=50$~km \nat Fig.~\\ref{fig_rfft_perf}(a) and (b)]. \nTherefore, we can infer that RFFT and RPHY has the similar \ndelay characteristics for cable traffic as well as \nthe LTE fronthaul traffic.    \n\nTo further study effect of cable traffic load, in \nFig.~\\ref{fig_rfft_perf}(c) we evaluate the behavior of mean packet delay\nfor DOCSIS when cable load is $\\rho_c = 0.6$. Fig.~\\ref{fig_rfft_perf}(c) plots the\nmean packet delay as the function of LTE fronthaul traffic $\\rho_L$. \nSimilar to Fig.~\\ref{fig_rfft_perf}(a), we observe a narrow difference in the delay \nperformance between RFFT and RPHY for the Poisson traffic ($H=0.5$). But, for the \nself similar traffic with $H=0.8$, we observe the difference of more \nthan $10$~ms between RFFT and RPHY which is partly due to the \ntemporary cable link saturation resulting from the \ninstantaneous load reaching more than the cable link \ncapacity $R_i$ $\\rho_c$ and $20$~\\% which is reserved \nfor cable maintenance. This difference is even \nfurther pronounced by the \ncombined effect from the data inflation at the remote node. \nAs the LTE fronthaul traffic $\\rho_L$ approach $0.8$ and $0.94$ \nfor RFFT and RPHY respectively, the mean packet delay packet \nincreases to a very large value resulted by the\noptical link saturation. For $\\rho_c=0.6$, the effective rate over the \noptical link is $\\rho_L + (1.85 \\times \\rho_c)$, therefore, \nwhen $\\rho_c = 0.6$, the optical link is saturated for \n$(1 -1.85 \\times 0.6) = 0.89$ which results in the \nlarge value of delay at $\\rho_L = 0.8$ and $0.94$ for \nRFFT and RPHY with $H=0.5$. The throughput behavior \nis verified in Fig.~\\ref{fig_rfft_tput} which plots the \ncable and LTE throughput as a function of LTE traffic \nintensity $\\rho_L$. We can compare Fig.~\\ref{fig_rfft_perf}(c) and \nFig.~\\ref{fig_rfft_tput} to see that delay and the average\nthroughput reaching stability at $\\rho_L = 0.8$ and $0.94$ for \nRFFT and RPHY with $H=0.5$. On the other hand, when the distance is increased from $10$~km \nto $50$~km, we see the pronounced effect in \ndelay difference between the RPHY and RFFT \nin Fig.~\\ref{fig_rfft_perf}(c). \nThis effect is because of the \ncombined impact of propagation delay and transmission delay \ndue to data inflation of RFFT on the DOCSIS scheduler.\n\nTo understand the effect of \ntraffic burstiness along with the cable load\non the LTE fronthaul traffic,\nwe show the mean packet delay as a function of LTE fronthaul \ntraffic $\\rho_L$ in Fig.~\\ref{fig_rfft_perf}(d). \nWe can relate the results in \nFig.~\\ref{fig_rfft_perf}(c) to Fig.~\\ref{fig_rfft_perf}(d) \nas such the traffic with higher burstiness $H=0.8$ \nincreases delay of both cable and LTE fronthaul traffic. \nAdditionally, the large fronthaul delay values at  $\\rho_L = 0.8$ and $0.94$\nindicates the stability limits for the Poisson traffic ($H=0.5$) \nfor RPHY and RFFT. However,\nthe fronthaul delay is more impacted the higher burstiness ($H=0.5$) \nas compared to the Poission traffic ($H=0.8$) in the RFFT deployment.\nFor example, even when the fronthaul load is $\\rho_L=0.5$ \n(i.e, below the stability limit), \nthe fronthaul delay is almost increased\nby $4$ times from $0.27~$ to $2.04~$ms for $H=0.8$ in the RFFT deployments. \nTherefore, if the functional split of LTE requires a dedicated \nQoS requirements, then a resource allocation \nmechanism of optical link may be employed for the dedicated assignments. \n\n\\section{\\MakeUppercase{Conclusions}}\nWe have presented a mechanism for Time-Frequency resource elements (QAM) caching along with the \nFFT sharing for LTE and DOCSIS coexistence. We have comprehensively evaluated the FFT \ninterleaving procedure through the simulation evaluation.\nAdditionally we have also provided the \nperformance comparison between the RFFT and RPHY remote nodes while \ncoexisting with LTE networks.\nOur study  numerically evaluate and show that \nRFFT performance is very close to RPHY. As \nRFFT achieves more flexibility and more \nscalability RFFT can be preferred for deployment. Additionally we also \ndiscuss the impact of functional split cable traffic on \nthe LTE fronthaul traffic. \n\n\n\n\n\\chapter{\\MakeUppercase{Future Work Outline: The SDN-LayBack: An SDN-Based Layered Backhaul\n\tArchitecture for Dense Wireless Networks}}\n\n\\section{\\MakeUppercase{Introduction}}\n\n\\subsection{Need for a New Backhaul Architecture for Wireless Networks}\n\nA plethora of wireless devices running a wide range\nof applications connect to radio access networks (RANs), as illustrated in Figure~\\ref{Bottleneck}.\nRANs provide end device to radio node (base station) connectivity.\nThe radio nodes are connected to the core networks by technology-specific backhaul access\nnetworks, such as LTE or WiMax backhaul access networks.\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=1\\textwidth]{layback_commag\/Bottleneck2.pdf}\n \n    \\caption{Bottleneck in the wireless Internet access chain.}\n    \\label{Bottleneck}\n\\end{figure}\nPresent day hand-held wireless end devices have very high\nprocessing and memory capabilities, supporting various\napplications, including resource demanding ones such as live 4K video streaming.\nAdvanced RANs, such as LTE-Advanced,\nsupport up to several hundred Mb\/s in downstream\nby exploiting a range of physical wireless layer techniques, such as\nmulti-carrier aggregation, opportunistic utilization of unlicensed\nbands, and millimeter wave technologies.\nA dense infrastructure of small cells with appropriate interference\nmanagement is a critical technique for advanced RANs~\\cite{nak2013tre}.\nAt the same time, the core networks already employ high-capacity optical links\nproviding abundant transmission capacity.\n\nHowever, the backhaul access networks have emerged as a critical\nbottleneck in wireless Internet\naccess~\\cite{and2014ove,ghi2015rev,rat2010fem,wan2015bac}.\nToday's backhaul access networks typically require\nhighly-priced bulky network equipment with proprietary locked-in control\nsoftware~\\cite{ber2015sof}.\nThey also need to be highly reliable and have high\navailability.\nIt is therefore very difficult to\nadd new network functionalities, reconfigure operational states,\nor upgrade hardware as technology advances.\nThese aspects have combined to stifle progress in backhaul access\nnetworks, causing them to become a critical bottleneck that can stall\nthe progress of wireless networking and undermine the advances in the\ncomplementary RANs and core networks. It can therefore be concluded that there is a need for innovative approaches, such as a new architecture,\nto enable technological progress in terms of backhaul access networks. \n\n\n\\subsection{SDN-LayBack - an Advanced Wireless Network Backhaul Architecture}\nThis paper proposes \\emph{SDN-LayBack, a fundamentally novel backhaul architecture\nfor wireless backhaul access networks based on software defined networking\n(SDN)}~\\cite{Bradai,moh2015con}.\nIn contrast to other recently proposed backhaul architectures, such as\nCROWD \\cite{seb2015dyn},\niJOIN \\cite{Dongyao},\nU-WN \\cite{Shengli}, and Xhaul~\\cite{oli2015xha},\nour SDN-LayBack architecture\nconsistently decouples the wireless RAN technology\n(such as LTE or WiFi) from the backhaul access network.\nIn addition, our SDN-LayBack architecture flexibly accommodates\nhighly heterogeneous RANs,\nranging from sparse cell deployments in rural areas to\nextremely high density cells in crowded stadiums.\nFurther, to the best of our knowledge, SDN-LayBack is the first\narchitecture to simultaneously  provide a new clean-slate platform for\ncellular backhaul and support the coexistence with current cellular access backhaul technologies, such as LTE (4G\/3G),\nWCDMA (3G), and GSM (2G).\nPrevious clean-slate architecture proposals,\nsuch as the one proposed by Ameigeiras et al.~\\cite{Ameigeiras}, do \\textit{not}\nsupport existing cellular backhaul technologies, and\nhence have limited functionalities. SDN-LayBack is layered on top of the existing backhaul networks and flexibly supports a wide range of RANs through a network of SDN switches and a hierarchical SDN controller pool.\n\nThis paper also introduces \\emph{an innovative four-step intra-LayBack handover protocol} within a given\ngateway in the SDN-LayBack architecture. Evaluations indicate 60~\\%\nreductions of the signalling load in comparison with conventional LTE handover.\nFinally, the article discusses future research directions within the\nSDN-LayBack architecture framework, focusing on\ninterference mitigation, device-to-device (D2D) communication, and video streaming~\\cite{seeling2012video, seema2011towards, chikkerur2011objective,\nvan2009implications,van2008traffic, seeling2004network, kangasharju2002distributing, fitzek2001mpeg}.\n\n\\section{\\MakeUppercase{SDN-LayBack: a Novel SDN-Based Layered Backhaul Architecture}}  \\label{arch:sec}\n\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=1\\textwidth]{layback_commag\/ProposedNewArchitecture.pdf}\n    \\caption{SDN-LayBack architecture.}\n    \\label{LA:fig}\n\\end{figure}\n\n\\subsection{Overview}\nAs already mentioned, the proposed SDN-LayBack is a novel SDN-based layered backhaul access network\narchitecture which can be layered on top of an existing backhaul access network (e.g., SAE-3GPP-LTE) in an evolutionary manner. Fig.~\\ref{LA:fig} illustrates SDN-LayBack's deployment on top of the classic LTE architecture.\n\nConventional backhaul access networks include technology-specific\nfunctional blocks for different RAN technologies.  For instance, in the LTE architecture, macro cells are\nserved by an LTE Serving-Gateway (S-GW), while pico\/femto cells can be\nserved by an Home-eNodeB-Gateway (HeNB-GW) (or S-GW), as illustrated in\nthe bottom left of Fig.~\\ref{LA:fig}.\nRAN and backhaul are thus coupled in conventional wireless networks, e.g., a\nWiMax radio node cannot be used within an LTE architecture. Only with\nan additional dedicated network entity, an evolved packet data\ngateway (ePDG), can a WiFi radio node be connected to LTE backhaul\nwith IPSec tunneling, as illustrated in the bottom middle of\nFig.~\\ref{LA:fig}.\n\nIn contrast, our proposed SDN-LayBack architecture decouples the RAN from\nthe backhaul, as illustrated in Fig.~\\ref{LA:fig}.\nThe figure also indicates SDN-LayBack's four architectural layers:\n\\emph{RAN Layer}, \\emph{SDN Backhaul Layer}, \\emph{Core Networks Layer},\n and \\emph{Evolutionary Internetworking Layer}.\n\nIn the SDN-LayBack architecture, heterogeneous RAN technologies, such as LTE\neNBs, pico\/femto LTE, WiFi, and WiMax, can be flexibly\ndeployed and accommodated in the \\emph{RAN Layer}. The \\emph{RAN Layer} interacts\nwith the \\emph{SDN-LayBack Backhaul Layer} through the network of flexible SDN (OpenFlow, OF)\nswitches and onwards through the programmable gateway pool to the Internet with the \\emph{Core\nNetworks Layer}.  The data plane of the SDN-LayBack backhaul network consists of\nthe network of SDN switches and the gateway pool. This data plane is\ncentrally controlled by the SDN controller pool and the applications (with corresponding databases). SDN-LayBack interfaces with the classic backhaul access network architecture through \\emph{the Evolutionary Internetworking Layer}.\n\n\\subsection{Outline of Layers in SDN-LayBack Architecture}\nThis section outlines the key components and functionalities of the\nproposed LayBack architecture and contrasts the LayBack architecture with previous designs. \n\n\\subsubsection{Radio Access Network (RAN) Layer} \n\n\\paragraph{Wireless End Devices}  \n\nMobile wireless end devices are heterogeneous and have a wide range of\nrequirements. Providing reasonable quality-oriented services to\nevery device on the network is a key challenge to the\nwireless network design.\nThe proposed SDN-LayBack architecture takes a unique approach to provide\nrequirement-specific network connectivity to every device that is\nconnected to the SDN-LayBack network.\nFuture devices will likely be\nhighly application-specific, such as a speed sensor on a race track\nor a health monitoring biosensor.\nThe SDN-LayBack architecture is designed to be adaptive\nto the different environments, such as an air port terminal, public park, or\nuniversity\/school. This enables SDN-LayBack to\nsupport a wide range of device requirements, such as\nreal-time D2D video streaming at a large sporting event or music concert.\n\n\\paragraph{Radio Nodes}  \nRadio nodes, such as the evolved NodeB (eNB) in LTE or an\naccess point (AP) in WiFi, provide RAN services to\nthe end devices. Aside from LTE and WiFi, there exists a wide range of\nwireless access technologies (and protocols), including\nWiMax, Zig-Bee, Bluetooth,\nand near field communication (NFC).\nThese wireless protocols have unique advantages and serve unique purposes.\nA fluidly flexible backhaul that homogeneously supports\nmultiple different types of radio nodes does not yet exist, but is\nhighly desirable.\nTherefore, our SDN-LayBack architecture is\nfundamentally designed to work with multiple RAN and communication technologies\nby isolating the RAN layer from the backhaul network.\n\nIn existing backhaul access networks, a given\nRAN technology requires a corresponding specific backhaul. For instance, the\nLTE radio access network can only operate with LTE\nbackhaul network entities. This restriction of\nspecific RAN technologies to specific backhaul\ntechnologies limits the usage of other RAN technologies on\nLTE backhaul networks and vice-versa.\nIn addition, femto cells are expected to operate\non multiple RAN technologies~\\cite{Bennis}.\nTo address this restrictive structure of present RAN-backhaul\ninter-networking, we propose our SDN-LayBack architecture to flexibly\nsupport multiple types of radio nodes, including software defined\nradios integrated with SDN enabled radio nodes.  \nIn an interesting approach, the reconfigurable antennas \n\\cite{saeed2012new, costantine2013new, saeed2015flexible, \n\tsaeed2016reflection, saeed2016radiation, costantine2014mimo, saeed2016inkjet} can be also \nbe further extended SDN to achieve more flexibility in the RAN design.\n\nMoreover, the system modeling for the RAN can be borrowed from the \ncontrol literature. Techniques to design the systems for multiple specifications have been extensively discussed in \n\\cite{puttannaiah_acc_2016,puttannaiah_acc_2015,puttannaiah_cdc_2015,puttannaiah_echols_gnc_2015,puttannaiah_2013_msthesis}. \nOptimization problems can be formulated that address multiple\nobjectives subject to constraints, by exploiting convex optimization ideas.\nProblems on a set of nonlinear hybrid systems, involving mixed-integer\nproblems, can be solved using convexification\n\\cite{puttannaiah_nandola_ecc_2013}. \nHigh fidelity numerical simulators and system identification techniques can\nbe used to efficiently solve optimization problems \\cite{puttannaiah_cartagena_acc_2016} using SDN.\n\n\\subsubsection{SDN Backhaul Layer}  \nThe SDN backhaul layer encompasses the SDN switches (with hypervisor), a\ncontroller pool, and a gateway pool, as well as the applications and corresponding databases.\nNext, the main aspects of the SDN switches, controller pool, applications, and gateways are briefly outlined in this section.\n\n\\paragraph{SDN (OpenFlow) Switches} \nSDN capable switches are today typically based on the OpenFlow (OF)\nprotocol\nand are therefore sometimes referred to as\nOpenFlow switches or OF switches.\nWe present SDN-LayBack in the context of OpenFlow switches, but emphasize that\nthe SDN-LayBack architecture applies to any type of SDN-capable switch.\nOF-switches are capable of a wide range of functions, such as forwarding a\npacket to any port, duplicating a packet on multiple ports, modifying the\ncontent inside a packet, or dropping the packet.\nThe switches can be connected in a mesh to allow disjoint flow paths\nfor load balancing.\n\n\\paragraph{Controller Pool} \nThe SDN-LayBack controller pool configures the entire OF-switch fabric\n(network of OF switches) by adding flow table entries on every switch.\nThe controller pool is also able to dynamically configure and program\nthe gateway pool functions, such as queuing policies, caching,\nbuffering, charging, as well as QoS \\cite{guck2016function, fitzek2002providing} and IP assignment for the end devices.\nThe SDN-LayBack controller pool coordinates with existing (legacy)\ncellular network entities, such as the mobility management entity (MME) and\nother 3GPP entities,\nto coexist with present architectures.\nConnections from the SDN-LayBack controller to the 3GPP entities can be\nprovided by extending the existing tunneling interfaces at the 3GPP entities.\nWe define extensions of the tunneling interfaces as a part of\nthe \\emph{Evolutionary Internetworking Layer} in Fig.~\\ref{LA:fig}.\nThus, the proposed SDN-LayBack architecture can enable communication\nbetween, the new\nflexible RAN architecture in the top left of Fig.~\\ref{LA:fig} as well as\nlegacy technology-specific RAN and backhaul networks, see bottom\nof Fig.~\\ref{LA:fig}. \n\n\\paragraph{Applications} \nApplications are programs executed by the controller for\ndelegating instructions to individual SDN switches. The controller\napplications realize all the network functions required for Internet\nconnectivity, such as authentication and\npolicy enforcement, through the switch flows.\nRadio nodes require RAN-specific interfaces for their operation\n(e.g., the X2 interface in LTE),\nWe realize these interfaces through controller applications on the network of\nSDN-switches. Therefore, for LTE in SDN-LayBack, eNBs are enabled with\ninter-radio node X2 interfaces along with other required interfaces.\nCellular networks have many network functions, such as the\nAutomatic Network Discovery and Selection Function\n(ANDSF)\nof 3GPP. As an example, to replace ANDSF\nin LayBack, a dedicated controller application will be responsible for\ndelegating network access policies to the devices.\nDatabases assist the SDN controller applications in their operations. \n\n\\paragraph{Gateways} \nGateway functions are programmed by the SDN controller\nto perform multiple network functionalities in order\nto simultaneously establish connectivity to heterogeneous RANs, such as\nLTE, Wi-Fi, and Wi-Max.\nFor example, the gateway in SDN-LayBack functions as\nboth S-GW and P-GW for an LTE radio node.\n\n\\subsection{SDN-LayBack Hierarchical Micro-Architectures}\n\\label{ma:sec}\nThe SDN-LayBack architecture can provide a wide range of\nheterogeneous services to localized regions through\nhierarchical SDN-LayBack micro-architectures.\nA micro-architecture hierarchy consists of a global (root) controller\nand multiple environment-specific local controllers located in the\nrespective environments, as illustrated in Fig.~\\ref{fig:microarchitectures}.\nThe local controllers can tailor applications for a specific\nenvironment, e.g., a park with a sparse user population that may be\nlocated adjacent to a stadium with a very high user density.\nAccordingly, micro-architectures allow to\nclassify applications specific to an environment\nas local applications and more common applications as global applications.\nFor example, WiFi offloading and adaptive video streaming\ncan be considered as local\napplications, whereas applications that serve a larger purpose, such as\ninterference management, can be considered as global applications.\n\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[width=1\\textwidth]{layback_commag\/MicroArchitecture2.pdf}\n\t\\caption{Environment-specific micro-architectures and controller\n\t\thierarchy.}\n\t\\label{fig:microarchitectures}\n\\end{figure}\n\n\\section{\\MakeUppercase{Intra-LayBack Handover: A Radically Simplified Handover Protocol}}\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=1\\textwidth]{layback_commag\/HOSigFlow.pdf}\n\\caption{Proposed simplified intra-LayBack handover protocol.}\n\t\\label{fig:handover}\n\\end{figure}\n\n\\subsection{Motivation: Signalling Load Due to Frequent Handovers}\nThe LTE protocol has generally higher signalling overhead\ncompared to legacy 3G protocols.\nA recent forecast from Oracle Corp.~\\cite{OracleSignalling}\nexpects a large increase of the global LTE signalling traffic, which is\npredicted to surpass the total global IP traffic growth by 2019.\nThe increase of signalling traffic can result in\nservice interruptions to customers and financial losses to cellular operators.\nTherefore, new architectures\nwith simplified signalling mechanisms are necessary to tackle the\ngrowth of signalling in the cellular networks.\n\nSignalling in the cellular backhaul is required for a wide range of\nactions, including initial attach and handovers.\nInitial attach procedures are completed only when a device requests\nconnectivity for the very first time. The initial attach procedure can result in\nlarge signalling overhead; however, it is executed only once.\nHandovers are initiated when a device measures and reports\npoor signal quality to the network.\nDepending on the configuration, the neighboring cell signal\nquality along with the quality of serving cell signal may be\nconsidered for handover decisions.\nHandovers require large\namounts of signalling in the backhaul and occur very frequently.\nEspecially in ultra dense networks (UDNs), where the cell\nrange is reduced to relatively small areas on the order of several\nmeters (e.g., 20~m in femto cells), handovers are very frequent.\nEven slow movements of devices\ncan cause device moves between multiple cells, resulting in handovers.\nHence, especially in UDNs, handovers are the source of serious bottlenecks in the backhaul entities.\n\n\\subsection{Proposed 4-Step Intra-LayBack Handover} \nWe propose \\emph{Intra Lay-Back Handover}, as the first mechanism\nto exploit the traffic duplication\nproperty of SDN to radically simplify the handover procedure in\ncellular backhaul.\nIn comparison to X2 based LTE handover \\textit{within} the same S-GW,\nour proposed method for handover within the same LayBack gateway\n(Intra-LayBack Handover) achieves:\n1) 100\\% reduction of the signalling load at the gateway,\ni.e., completely eliminates the handover signalling load on the LTE\ngateways,\n2) 69\\% reduction of signalling load compared to the LTE MME, and\n3) 60\\% reduction in the overall signalling cost.\n\nWe employ four steps in the handover protocol,\nas illustrated in Figure~\\ref{fig:handover}.\n\n\\emph{Step 1 Handover Preparation}\n\nWhen a measurement report satisfying\nthe handover conditions arrives at the serving eNB (SeNB), the device\nbearer context information is forwarded to the target eNB (TeNB) and a\nmessage requesting handover is sent from the SeNB to the SDN-LayBack controller.\n\n\\emph{Step 2 Enable Traffic Duplication}\n\nLayBack controllers enable the traffic\nduplication at the SDN switch through an OpenFlow\nmessage for all the downlink traffic related to the device waiting to\nbe handed over. The SDN switch continuously changes the headers (tunnel\nIDs) such that traffic related to the device reaches at TeNB. The bearer\ninformation pertaining to the device which has already been\nreceived by the TeNB from the SeNB is used to receive the duplicated traffic\nat the TeNB. Once the duplicated traffic related the device is\nreceived at the TeNB, a message is sent to the SeNB indicating the readiness\nfor the handover. The SeNB then sends a handover command to the device\n(i.e., RRC reconfiguration in LTE).\n\n\\emph{Step 3 Perform Wireless Handover}\n\nUpon receiving the handover command, the device\nbreaks the current wireless link (a hard handover)\nand establishes a new wireless connection to the TeNB (through reserved\nRACH preambles in LTE) \\cite{tyagi2014impact, tya2015con, gur2017hyb, vilgelm2017latmapa}. \nBy then, duplicated traffic arrived prior to wireless\nconnection reestablishment is waiting to be forwarded to the device in the\nnew connection. Downlink traffic is then forwarded to\nthe device as soon as the wireless connection is established.\n\n\\emph{Step 4 Stop Traffic Duplication}\n\nWhen the device traffic\nflow path through the new wireless connection\nis successful in both the uplink and downlink, then the\nTeNB sends a handover complete message to the LayBack controller.\nThe controller sends\nan OpenFlow message to the SDN switch to stop the duplication by\ndisabling the device traffic flow to the SeNB.\n\nOther handover types, such as inter-LayBack gateway\nhandover between different LayBack gateways, can be accomplished\nthrough extensions of the presented intra-LayBack handover.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=1\\textwidth]{layback_commag\/HOTable.pdf}\n\t\\caption{Handover cost and load comparison in LayBack and LTE backhaul}\n\t\\label{tab:handover}\n\\end{figure}\n\\subsection{Signalling Overhead Evaluation} \nTable~\\ref{tab:handover} characterizes the amount of required\nsignalling messages\nfor X2-based handovers in LTE and for intra-LayBack handovers\nwithin the same gateway.\nWe observe from Table~\\ref{tab:handover} that the overall signalling\ncost is reduced from 15 messages in LTE to 6 messages in the proposed\nintra-LayBack handover, a 60~\\% reduction of signalling overhead.\nWe note that we count an OpenFlow message sent from the controller to an SDN\nswitch as a signalling message in this evaluation.\nWe further observe from Table~\\ref{tab:handover} that the load\nof processing 9 handover signalling messages at the MME in LTE is reduced\nto processing 4 handover signalling messages at the controller in LayBack.\nMoreover, the P\/S-GWs in LTE need to process 4 signalling messages for\na handover, whereas the LayBack gateway is completely oblivious to the\nhandover mechanism. That is, the LayBack handover completely eliminates the\nhandover signalling load on the gateway.\n\nThe LTE handover changes the bearers (UE context) on the gateway,\nrequiring overall a high handover signalling load, whereby large\nsignalling loads have to be processed at the LTE MME and P\/S-GW.\nIn contrast, the LayBack handover reduces the overall signalling load\nand employs the SDN switches to change the packet flow path.\nEssentially, the LayBack handover moves the handover burden from\nthe  gateway to the network SDN switches.\nThe SDN switches can share the handover burden, avoiding bottlenecks.\n\n\\section{\\MakeUppercase{Summary and Future Directions}}  \n\\subsection{Summary of SDN-LayBack Architecture and Handover Protocol}\n\nWe have introduced SDN-LayBack,\na novel layered backhaul architecture for wireless networks.\nSDN-LayBack consistently decouples the radio access networks (RANs)\nfrom the backhaul, permitting the flexible backhauling\nof highly heterogeneous RANs.\nWithin the SDN-LayBack architectural framework, we have introduced\na first protocol mechanism, namely a 4-step handover protocol for\nhandovers within the scope of an SDN-LayBack gateway.\nThe proposed\nLayBack handover mechanism relieves the signalling bottleneck at the\nbackhaul gateways (e.g., LTE S-GW and P-GW)\nby (a) traffic duplication at SDN switches, and\n(b) bearer information forwarding from the currently serving eNB to the\ntarget eNB.\nMoreover, the LayBack handover reduces the signalling load on the\nbackhaul control entities (e.g., MME) by (a) reducing the required\nsignalling messages, and (b) moving some of the signalling messages\nfrom mobility control entities to the SDN controller.\nOur comparison of the handover signalling load indicates that\nLayBack achieves a 60~\\% reduction of the overall handover signalling load\ncompared to the conventional\nLTE handover. Moreover, LayBack completely avoids handover processing at\nthe gateway and distributes the handover processing load over a network\nof SDN switches, avoiding processing bottlenecks.\n\n\\subsection{Future Direction: Interference Management Protocol} \n\nThere are many exciting avenues for future protocol developments and\nevaluations within the SDN-LayBack architectural framework.\nSince interference is one of the main limitations for the deployment\nof small cells, an important future research direction is to\nexploit the capabilities of the SDN-LayBack architecture for\ninterference mitigation protocols.\nWe can exploit the central control (with a global holistic\nperspective) in SDN-LayBack for coordinating\nthe long-term wireless configurations, such as center\nfrequency, bandwidth, and time-sharing patterns,\nas well as protocol-specific configurations (e.g., random access channel\nparameters in LTE).\nWe emphasize that the goal of SDN-LayBack interference management\nwould \\textit{not} be to manage the physical\nresources, such as sub-frame power control in LTE, at the micro level.\nInstead, SDN-LayBack interference management assigns high-level configurations\nin a coordinated manner to the radio nodes so as to mitigate interference.\n\n\\subsection{Future Direction: D2D Communication Protocol}  \nDevice-to-Device (D2D) communication will likely play an important role in\nthe future of real-time content sharing and is another promising\ndirection for SDN-LayBack protocol development.\nThe LayBack architecture can be exploited for D2D communication so as\nto avoid flooding on the access network.\nMore specifically, the SDN controller can\nmaintain a subscriber (location) data base to coordinate the connection\nestablishment and routing between D2D communication partners.\n\n\\subsection{Future Direction: Video Streaming Protocol}  \nAnother direction is to develop and evaluate collaborative video streaming\nprotocols within the micro-architecture framework\n(Section~\\ref{ma:sec}) of SDN-LayBack.\nFor densely populated areas,\ne.g., airport terminals and stadiums, it is highly likely that\nmultiple RAN technologies, e.g., LTE and WiFi, are deployed,\nand multiple simultaneous (unicast) video streams are being\ntransmitted to (or from) devices within the coverage area of the\nconsidered micro-architecture, which is controlled by a local controller.\nA LayBack video streaming protocol can exploit the multiple ongoing video\nstreams and multiple RANs to collaboratively trade off the video traffic\ncharacteristics (e.g., traffic bursts from complex video scenes)\nwith RAN transmission conditions to achieve improved video experience\nacross an ensemble of video streams. In addition to existing techniques\nsuch \\cite{suresh2007superresolution, suresh2007robust, suresh2006robust, suresh2006discontinuity, aravind2012multispinning, suresh2014hog} can be applied to the SDN LayBack framework.\n\n\n\\chapter{\\MakeUppercase{Software Defined Optical Networks (SDONs): A Comprehensive Survey}}\n\n\\section{\\MakeUppercase{Introduction}} \\label{Sec1_Intro}\nAt least a decade ago \\cite{ComerNetMgmtBook} it was recognized that new\nnetwork abstraction layers for network control functions needed to be\ndeveloped to both simplify and automate network management. Software\nDefined Networking (SDN)~\\cite{HuHB14,jai2013b4,Vau11} is the design\nprinciple that emerged to structure the development of those new\nabstraction layers.\nFundamentally, SDN is defined by three architectural\nprinciples~\\cite{SDNarch11,KreRVR15}:\n$(i)$ the separation of control plane functions\nand data plane functions, $(ii)$ the logical centralization of control, and\n$(iii)$ programmability of network functions. The first two\narchitectural principles are related in that they combine to allow for network\ncontrol functions to have a wider perspective on the network. The idea is that\nnetworks can be made easier to manage (i.e., control and monitor) with a\nmove away from significantly distributed control. A tradeoff is then\nconsidered that balances ease of management arising from control\ncentralization and scalability issues that naturally arise from that\ncentralization.\n\nThe SDN abstraction layering consists of three generally accepted\nlayers~\\cite{SDNarch11} inspired by computing systems,\nfrom the bottom layer to the top layer:\n$(i)$ the \\textit{infrastructure} layer, $(ii)$ the \\textit{control} layer, and\n($iii$) the \\textit{application} layer,\nas illustrated in Fig.~\\ref{fig:sdnlayers}.\nThe interface between the\napplication layer and the control layer is referred to as the\nNorthBound Interface (NBI), while the interface between the control layer\nand the infrastructure layer is referred to as the SouthBound\nInterface (SBI). There are a variety of standards emerging for these\ninterfaces, e.g., the OpenFlow protocol~\\cite{LaKR14} for the SBI.\n\\begin{figure*}[t!]\t\\centering\n\t\\includegraphics[width=6in]{finman_sdon\/fig1.pdf}\n    \\caption{Illustration of Software Defined Networking (SDN) abstraction\n     layers.}\n\t\\label{fig:sdnlayers}\n\\end{figure*}\n\nThe \\textit{application} layer is modeled after software applications\nthat utilize computing resources to complete tasks.\nThe \\textit{control} layer is modeled after a computer's Operating System\n(OS) that manages computer resources (e.g., processors and memory),\nprovides an abstraction layer to simplify interfacing with the\ncomputer's devices, and provides a common set of services that all\napplications can leverage. Device drivers in a computer's OS hide the\ndetails of interfacing with many different devices from the\napplications by offering a simple and unified interface for various\ndevice types. In the SDN model both the unified SBI\nas well as the control layer functionality provide\nthe equivalent of a device driver for interfacing with devices in the\n\\textit{infrastructure} layer, e.g., packet switches.\n\nOptical networks play an important role in our modern information\ntechnology due to their high transmission capacities.\nAt the same time, the specific optical (photonic) transmission\nand switching characteristics, such as circuit, burst, and packet\nswitching on wavelength channels, pose challenges for controlling\noptical networks.\nThis article presents a comprehensive survey of Software\nDefined Optical Networks (SDONs).\nSDONs seek to leverage the flexibility of\nSDN control for supporting networking applications with an\nunderlying optical network infrastructure.\nThis survey comprehensively covers SDN related mechanisms that have\nbeen studied to date for optical networks.\n\n\\subsection{Related Work}\nThe general principles of SDN have been extensively covered in several\nsurveys, see for instance,~\\cite{aky2014roa,Chen2015a,cui2016big,far2015soft, FeRZ14, HuHB14, JaP13,\n  jar2014sur, JaZH14, kha2015jons, KreRVR15, LaKR14, li2014sof, Lopes2015,MiCK13,\n  NuMNO14,Racherla,Trois2016,van2014sca, wic2015sof, XiWF14}.\nSDN security has been surveyed in~\\cite{ahm2015sec,sco2015sur}, while\nmanagement of SDN networks has been surveyed in~\\cite{wic2015sof} and\nSDN-based satellite networking is considered in~\\cite{ber2015sof}.\n\nTo date, there have been relatively few overview and survey articles\non SDONs.\nZhang et al.~\\cite{zha2013surflexi} have presented a thorough survey\non flexible optical networking based on Orthogonal Frequency\nDivision Multiplexing (OFDM) in core (backbone) networks. The survey\nbriefly notes how OFDM-based elastic networking can facilitate\nnetwork virtualization and surveys a few studies on OFDM-based\nnetwork virtualization in core networks.\n\nBhaumik et al.~\\cite{BhZCS14} have presented an overview of\nSDN and network virtualization concepts and outlined principles\nfor extending SDN and network virtualization concepts to\nthe field of optical networking.\nTheir focus has been mainly on industry efforts, reviewing white papers\non SDN strategies from leading networking companies, such as\nCisco, Juniper, Hewlett-Packard, Alcatel-Lucent, and Huawei.\nA few selected academic research projects on general SDN optical\nnetworks, namely projects reported in the journal\narticles~\\cite{ChNS13,liu2013field} and a few related conference papers,\nhave also been reviewed by Bhaumik et al.~\\cite{BhZCS14}.\nIn contrast to Bhaumik et al.~\\cite{BhZCS14}, we provide a comprehensive\nup-to-date review of academic research on\nSDONs. Whereas Bhaumik et al.~\\cite{BhZCS14}\npresented a small sampling of SDON research organized by research projects,\nwe present a comprehensive SDON survey that is organized according\nto the SDN infrastructure, control, and application layer architecture.\n\nFor the SDON sub-domain of access networks,\nCvijetic~\\cite{Cvi14} has given an overview of access network challenges\nthat can be addressed with SDN.\nThese challenges include lack of support for on-demand modifications of\ntraffic transmission policies and rules and limitations to\nvendor-proprietary policies, rules, and software.\nCvijetic~\\cite{Cvi14} also offers a very brief overview of\nresearch progress for SDN-based optical access networks, mainly focusing on\nstudies on the physical (photonics) infrastructure layer.\nCvijetic~\\cite{CviSept14} has further expanded the overview of\nSDON challenges by considering the incorporation of 5G wireless\nsystems. Cvijetic~\\cite{CviSept14} has noted that\nSDN access networks are highly promising for low-latency and high-bandwidth\nback-hauling from 5G cell base stations and briefly surveyed\nthe requirements and areas of future research required for\nintegrating 5G with SDON access networks.\nA related overview of general software defined access networks\nbased on a variety of physical transmission media, including copper\nDigital Subscriber Line (DSL)~\\cite{dslbook}\nand Passive Optical Networks (PONs), has been\npresented by Kerpez et al.~\\cite{KeCGG14}.\n\nBitar~\\cite{Bi14} has surveyed use cases for SDN controlled broadband access,\nsuch as on-demand bandwidth boost, dynamic service\nre-provisioning, as well as value-add services and service protection.\nBitar~\\cite{Bi14} has discussed the commercial perspective of the\naccess networks that are enhanced with SDN to add\ncost-value to the network operation.\nAlmeida Amazonas et al.~\\cite{AmSS14} have surveyed the key issues\nof incorporating SDN in optical and wireless access networks.\nThey briefly outlined the obstacles posed by\n the different specific physical characteristics of\noptical and wireless access networks.\n\nAlthough our focus is on optical networks, for completeness we note that\nfor the field of wireless and mobile networks, SDN based networking mechanisms\nhave been surveyed\nin~\\cite{ArSR15,BeDSB14,haq2016wir,jag2014sof,SaCKA15,soo2015sof,yan2014sof}\nwhile network virtualization has been surveyed in~\\cite{LiY15} for\ngeneral wireless networks and in~\\cite{kha2015wir,lia2015wir} for\nwireless sensor networks.\nSDN and virtualization strategies for LTE wireless cellular networks\nhave been surveyed in~\\cite{ngu2015sdn}. SDN-based 5G wireless\nnetwork developments for mobile networks have been outlined\nin~\\cite{Huawei14,PeLZW14,TrGVS15,YaKO14}.\n\n\\subsection{Survey Organization}\nWe have mainly organize our survey according to the three-layer SDN architecture\nillustrated in Fig.~\\ref{fig:sdnlayers}.\nIn particular, we have organized the survey in a bottom-up manner,\nsurveying first SDON studies focused on the\ninfrastructure layer in Section~\\ref{sdninfra:sec}.\nSubsequently, we survey SDON studies focused on the control layer\nin Section~\\ref{sdnctl:sec}.\nThe virtualization of optical networks is commonly closely related to\nthe SDN control layer. Therefore, we survey SDON studies\nfocused on virtualization in Section~\\ref{virt:sec},\nright after the SDON control layer section.\nResuming the journey up the layers in Fig.~\\ref{fig:sdnlayers},\nwe survey SDON studies focused on the application layer in\nSection~\\ref{sdnapp:sec}.\nWe survey mechanisms for the overarching orchestration of the application\nlayer and lower layers, possibly across multiple network domains\n(see Fig.~\\ref{fig_control_orch}),\nin Section~\\ref{orch:sec}.\nFinally, we outline open challenges and future research directions in\nSection~\\ref{sec:open} and conclude the survey in Section~\\ref{sec:conclusion}.\n\n\n\\section{\\MakeUppercase{Background}}\n\\label{bg:sec}\nThis section first provides background on\nSoftware Defined Networking (SDN), followed by background on virtualization\nand optical networking.\nSDN, as defined by the Internet Engineering\nTask Force (IETF)~\\cite{rfc7426},\nis a networking paradigm enabling the programmability of networks.\nSDN abstracts and separates the data forwarding\nplane from the control plane, allowing faster\ntechnological development both in data and control planes.\nWe provide background on the SDN architecture, including its architectural\nlayers in Subsection~\\ref{sdnarch:sec}.\nThe network programmability provides the flexibility to dynamically initialize,\ncontrol, manipulate, and manage the end-to-end network behavior via open\ninterfaces, which are reviewed in Subsection~\\ref{sdnint:sec}.\nSubsequently, we provide background on network virtualization in\nSubsection~\\ref{bgnetvit:sec}\nand on optical networking in Subsection~\\ref{bg_access:sec}.\n\n\\subsection{Software Defined Networking (SDN) Architectural Layers}\n\\label{sdnarch:sec}\nSDN offers a simplified view of the underlying network infrastructure\nfor the network control and monitoring applications\nthrough the abstraction of each independent network layer.\nFig.~\\ref{fig:sdnlayers} illustrates the\nthree-layer SDN architecture model consisting of application,\ncontrol, and infrastructure layers as defined by the\nOpen Networking Foundation (ONF)~\\cite{SDNarch11}. The ONF is\nthe organization that is responsible for\nthe publication of specifications for the OpenFlow protocol.\nThe OpenFlow protocol~\\cite{HuHao2014,keo2008of,LaKR14}\nhas been the first protocol for the SouthBound Interface (SBI,\nalso referred to as Data-Controller Plane Interface (D-CPI))\nbetween the control and infrastructure layers.\nEach layer operates independently, allowing multiple solutions\nto coexist within each layer, e.g., the\ninfrastructure layer can be built from any\nprogrammable devices, which are commonly referred to as\nnetwork elements~\\cite{SDNarch11_521} or network devices~\\cite{rfc7426}\n(or sometimes as forwarding elements~\\cite{rfc3746}).\nWe will use the terminology network element throughout this survey.\nThe SouthBound Interface (SBI) and the NorthBound\nInterface (NBI, also referred to as Application-Controller Plane\nInterface (A-CPI)) are defined as the primary interfaces\ninterconnecting the SDN layers through abstractions.\nAn SDN network architecture can coexist with both\nconcurrent SDN architectures and non-SDN legacy network architectures.\nAdditional interfaces are defined namely the\nEastBound Interface (EBI) and the WestBound\nInterface (WBI)~\\cite{JaZH14} to interconnect the SDN architecture\nwith external network architectures\n(the EBI and WBI are also collectively\nreferred to as Intermediate-Controller Plane Interfaces (I-CPIs)).\nGenerally, EBIs establish communication links to\nlegacy network architectures (i.e., non-SDN networks); whereas,\nlinks to concurrent (side-by-side) SDN architectures are facilitated\nby the WBIs.\n\n\\subsubsection{Infrastructure Layer}\nThe infrastructure layer includes an environment for\n(payload) data traffic forwarding (data plane)\neither in virtual or actual hardware.\nThe data plane comprises a network of network elements,\nwhich expose their capabilities through the SBI\nto the control plane. In traditional networking, control mechanisms are\nembedded within an infrastructure,\ni.e., decision making capabilities are embedded within the\ninfrastructure to perform network actions, such as switching or routing.\nAdditionally, these forwarding actions in the traditional network elements\nare autonomously\nestablished based on self-evaluated topology information that is often\nobtained through proprietary vendor-specific algorithms.\nTherefore, the configuration setups of traditional network elements\nare generally not reconfigurable without a service disruption,\nlimiting the network flexibility.\nIn contrast, SDN decouples the autonomous control functions, such as\nforwarding algorithms and neighbor discovery of\nthe network nodes, and moves these control functions out of the infrastructure\nto a centrally controlled logical node, the controller.\nIn doing so, the network elements act only as dumb switches which\nact upon the instructions of the controller. This decoupling reduces the\nnetwork element complexity and improves reconfigurability.\n\nIn addition to decoupling the\ncontrol and data planes, packet modification capabilities at the line-rates\nof network elements have been significantly improved with SDN.\nP4~\\cite{bosshart2014p4} is a programmable\nprotocol-independent packet processor, that can arbitrarily\nmatch the fields within any formatted packet\nand is capable of applying any arbitrary actions (as programmed)\non the packet before forwarding. A similar forwarding mechanism,\nProtocol-oblivious Forwarding (PoF) has been\nproposed by Huawei Technologies~\\cite{Song2013}.\n\n\\subsubsection{Control Layer}\nThe control layer is responsible for programming (configuring)\nthe network elements (switches) via the SBIs.\nThe SDN controller is a logical entity that identifies the south bound\ninstructions to configure the network infrastructure\nbased on application layer requirements.\nTo efficiently manage the network, SDN controllers can\nrequest information from the SDN infrastructures,\nsuch as flow statistics, topology information, neighbor relations,\nand link status from the network elements (nodes).\nThe software entity that implements the SDN\ncontroller is often referred to as \\textit{Network Operating System (NOS)}.\nGenerally, a NOS can be implemented independently of\nSDN, i.e., without supporting SDN.\nOn the other hand, in addition to supporting SDN operations,\na NOS can provide advanced capabilities, such as virtualization,\napplication scheduling, and database management.\nThe Open Network Operating System (ONOS)~\\cite{onos}\nis an example of an SDN based NOS\nwith a distributed control architecture designed to operate over\nWide Area Networks (WANs).\nFurthermore, Cisco has recently developed the one Platform Kit\n(onePK)~\\cite{onepk}, which consists of a set of\nApplication Program Interfaces (APIs)\nthat allow the network applications to control Cisco network devices\nwithout a command line interface.\nThe onePK libraries act as an SBI for Cisco ONE controllers and\nare based on C and Java compilers.\n\n\\subsubsection{Application Layer}\nThe application layer comprises network applications and services\nthat utilize the control plane to realize\nnetwork functions over the physical or virtual infrastructure.\nExamples of network applications include\nnetwork topology discovery, provisioning, and fault restoration.\nThe SDN controller presents an abstracted view of the\nnetwork to the SDN applications to facilitate the realization of\napplication functionalities.\nThe applications can also include higher levels of network management,\nsuch as network data analytics, or specialized functions requiring processing\nin large data centers. For instance, the Central Office Re-architected as a\nData center (CORD)~\\cite{cord} is an SDN application based\non ONOS~\\cite{onos},\nthat implements the typical central office network functions, such as\noptical line termination, as well as BaseBand Unit (BBU) and\nData Over Cable Interface (DOCSIS)~\\cite{fel2001doc} processing as\nvirtualized software entities, i.e., as SDN applications.\n\n\\begin{figure}[t!]\n\t\\centering \t\\vspace{0cm}\n\t\\includegraphics[width=3.4in]{finman_sdon\/fig2.pdf}\n    \\caption{Overview of SDN orchestrator and SDN controllers.}\n\t\\label{fig_control_orch}\n\\end{figure}\n\\subsubsection{Orchestration Layer}  \\label{intro:orch:sec}\nAlthough the orchestration layer is commonly not considered one of the\nmain SDN architectural layers illustrated in Fig.~\\ref{fig:sdnlayers},\nas SDN systems become more complex, orchestration becomes increasingly\nimportant. We introduce therefore the orchestration layer as\nan important SDN architectural layer in this background section.\nTypically, an SDN orchestrator is the entity that coordinates\nsoftware modules within a single SDN controller,\na hierarchical structure of multiple SDN controllers, or a set of\nmultiple SDN controllers in a ``flat'' arrangement (i.e., without\na hierarchy) as illustrated in Fig.~\\ref{fig_control_orch}.\nAn SDN controller in contrast\ncan be viewed as a logically centralized single\ncontrol entity. This logically centralized single control entity appears as the\ndirectly controlling entity to the network elements.\nThe SDN controller is responsible for signaling the control actions or rules\nthat are typically predefined (e.g., through OpenFlow) to the network elements.\nIn contrast, the SDN orchestrator makes control\ndecisions that are generally not predefined.\nMore specifically, the SDN orchestrator could make an automated decision\nwith the help of SDN applications or seek a manual\nrecommendation from user inputs; therefore,\nresults are generally not predefined. These\norchestrator decisions (actions\/configurations)\nare then delegated via the SDN controllers\nand the SBIs to the network elements.\n\nIntuitively speaking, SDN orchestration can be viewed as a distinct\nabstracted (higher) layer for coordination and management that is\npositioned above the SDN control and application layers.  Therefore,\nwe generalize the term SDN orchestrator as an entity that realizes a\nwider, more general (more encompassing) network functionality as\ncompared to the SDN controllers. For instance, a cloud SDN\norchestrator can instantiate and tear down Virtual Machines (VMs)\naccording to the cloud\nworkload, i.e., make decisions that span across multiple network\ndomains and layers.  In contrast, SDN controllers realize more\nspecific network functions, such as routing and path computation.\n\n\\subsection{SDN Interfaces}  \\label{sdnint:sec}\n\\subsubsection{Northbound Interfaces (NBIs)}\nA logical interface that interconnects the SDN controller and a\nsoftware entity operating at the application layer is commonly\nreferred to as a NorthBound Interface (NBI), or as Application-Controller Plane\nInterface (A-CPI).\n\n\\paragraph{REST}\nREpresentational State Transfer (REST)~\\cite{REST15} is generally defined as\na software architectural style that supports flexibility, interoperability,\nand scalability.\nIn the context of the SDN NBI, REST is commonly defined as\nan API that meets the REST architectural style~\\cite{LiChou2016},\ni.e., is a so-called RESTful API:\n\\begin{itemize}\n\\item Client-Sever: Two software entities should follow the\n  client-server model.  In SDN, a controller can be a server and the\n  application can be the client.  This allows multiple\n  heterogeneous SDN applications to coexist and operate over a common\n  SDN controller.\n\\item Stateless: The client is responsible for managing all the states and\nthe server acts upon the client's request.\nIn SDN, the applications collect and maintain the states of the network, while\nthe controller follows the instructions from the applications.\n\\item Caching: The client has to support the temporary local storage\nof information such that interactions between the\nclient and server are reduced so as to improve performance and scalability.\n\\item Uniform\/Interface Contract:  An overarching technical\ninterface must be followed across all services using the REST API.\nFor example, the same data format, such as Java Script Object Notation (JSON)\nor eXtended Markup Language (XML), has to be followed for all\ninteractions sharing the common interface.\n\\item Layered System: In a multilayered architectural solution,\n  the interface should only be concerned with the next immediate node\n  and not beyond. Thus, allowing more layers to be inserted, modified,\n  or removed without affecting the rest of the system.\n\\end{itemize}\n\n\\subsubsection{Southbound Interfaces (SBIs)}\nA logical interface that interconnects the SDN controller and the network\nelement operating on the infrastructure layer (data plane) is commonly\nreferred to as a SouthBound Interface (SBI), or as the\nData-Controller Plane Interface (D-CPI).\nAlthough a higher level connection, such as a UDP or TCP connection, is\nsufficient for enabling the communication between two entities of the SDN\narchitecture, e.g., the controller and the network elements,\nspecific SBI protocols have been proposed.\nThese SBI protocols are typically not interoperable\nand thus are limited to work with SBI protocol-specific network elements\n(e.g., an OpenFlow switch does not work with the NETCONF protocol).\n\n\\paragraph{OpenFlow Protocol}\nThe interaction between an OpenFlow switching element\n(data plane) and an OpenFlow controller (control plane) is carried\nout through the OpenFlow protocol~\\cite{keo2008of,LaKR14}. This SBI (or D-CPI)\nis therefore also sometimes referred to as the OpenFlow control channel.\nSDN mainly operates through packet flows that are identified through\nmatches on prescribed packet fields that are specified in the\nOpenFlow protocol specification. For matched packets, SDN switches\nthen take prescribed actions, e.g., process the flow's packets in a\nparticular way, such as dropping the packet, duplicating it on a different\nport or modifying the header information.\n\n\\paragraph{Path Computation Element Protocol (PCEP)}\nThe PCEP enables communication between the Path Computation Client (PCC) of\nthe network elements and the Path Computation Element (PCE) residing within\nthe controller. The PCE centrally computes the paths based on constraints\nreceived from the network elements. Computed paths are then forwarded to the\nindividual network elements through the PCEP protocol~\\cite{rfc4655,rfc5440}.\n\n\\paragraph{Network Configuration Protocol (NETCONF) Protocol}\nThe NETCONF protocol~\\cite{rfc6241}\nprovides mechanisms to configure, modify, and delete configurations\non a network device.\nConfiguration of the data and protocol messages are encoded in the NETCONF\nprotocol using an eXtensible Markup Language (XML).\nRemote procedure calls are used to realize the NETCONF protocol\noperations. Therefore, only devices that are enabled\nwith required remote procedure calls allow the NETCONF protocol to remotely\nmodify device configurations.\n\n\\paragraph{Border Gateway Protocol Link State Distribution (BGP-LS) Protocol}\nThe central controller needs a topology information database,\nalso known as Traffic Engineering Database (TED), for\noptimized end-to-end path computation.\nThe controller has to request the information for building the TED,\nsuch as topology and bandwidth utilization, via the SBIs\nfrom the network elements.\nThis information can be gathered by a BGP extension,\nwhich is referred to as BGP-LS.\n\n\\subsection{Network Virtualization} \\label{bgnetvit:sec}\nAnalogously to the virtualization of computing\nresources~\\cite{gold1974sur,Douglis2013a},\nnetwork virtualization abstracts the underlying physical network\ninfrastructure so that one or multiple virtual networks\ncan operate on a given physical\nnetwork~\\cite{bel2012res,duan2012sur,fis2013vir,han2015net,leo2003vir,mij2015net,pen2015gue,RajJain2013,wan2013net}.\nVirtual networks can span over a\nsingle or multiple physical infrastructures (e.g., geographically\nseparated WAN segments).\nNetwork Virtualization (NV) can flexibly create independent virtual\nnetworks (slices) for distinct\nusers over a given physical infrastructure.  Each network slice\ncan be created with prescribed resource allocations.\nWhen no longer required, a\nslice can be deleted, freeing up the reserved physical resources.\n\nNetwork hypervisors~\\cite{Khan2012,she2009flo}\nare the network elements that abstract the\nphysical network infrastructure (including network elements,\ncommunication links, and control functions) into logically isolated\nvirtual network slices.\nIn particular, in the case of an underlying physical SDN network,\nan SDN hypervisor can create multiple isolated virtual SDN\nnetworks~\\cite{ble2015sur,dru2013sca}.\nThrough hypervisors, NV supports the implementation of a wide\nrange of network services belonging to the link and network protocol\nlayers (L2 and L3), such as switching and routing.\nAdditionally, virtualized infrastructures can also support\nhigher layer services, such as load-balancing of servers and firewalls.\nThe implementation of such higher layer services in a virtualized environment\nis commonly referred to as Network Function Virtualization\n(NFV)~\\cite{haw2014nfv,LiChen2015,lin2016dem,mat2015tow,ye2016joi}.\nNFV can be viewed as a special case of NV in which network\nfunctions, such as address translation and intrusion detection functions,\nare implemented in a virtualized environment. That is,\nthe virtualized functions are implemented in the form of software\nentities (modules) running on a data center (DC) or\nthe cloud~\\cite{Mijumbi2016}.\nIn contrast, the term NV emphasizes the virtualization of\nthe network resources, such as communication links and network nodes.\n\n\n\\subsection{Optical Networking Background}   \\label{bg_access:sec}\n\n\\subsubsection{Optical Switching Paradigms}\nOptical networks are networks that either maintain signals in the\noptical domain or at least utilize transmission channels that carry\nsignals in the optical domain. In optical networks that\nmaintain signals in the optical domain, switching can be\nperformed at the \\textit{circuit, packet, or burst} granularities.\n\n\\paragraph{Circuit Switching}\nOptical \\textit{circuit} switching can be performed in space,\nwaveband, wavelength, or time. The optical spectrum is divided into\nwavelengths either on a fixed wavelength grid or on a flexible\nwavelength grid. Spectrally adjacent wavelengths can be coalesced into\nwavebands. The fixed wavelength grid standard (ITU-T G.694.1)\nspecifies specific center frequencies that are either 12.5~GHz, 25~GHz,\n50~GHz, or 100~GHz apart. The flexible DWDM grid (flexi-grid) standard (ITU-T\nG.694.1)~\\cite{gon2015opt,jue2014sof,tom2014tut,zha2013surflexi}\nallows the center frequency to be any multiple of 6.25~GHz\naway from 193.1~THz and the spectral width to be any multiple of\n12.5~GHz. Elastic Optical Networks (EONs)~\\cite{cha2015rou,tal2014spe,yu2014spe}\nthat take advantage of the flexible\ngrid can make more efficient use of the optical spectrum but can cause\nspectral fragmentation, as lightpaths are set up and torn down, the spectral\nfragmentation counteracts the more efficient spectrum\nutilization~\\cite{Ger2012}.\n\n\\paragraph{Packet Switching}\nOptical \\textit{packet} switching performs packet-by-packet switching\nusing header fields in the optical domain as much as possible. An\nall-optical packet switch requires~\\cite{ram2009opt}:\n\\begin{itemize}\n  \\item Optical synchronization, demultiplexing, and multiplexing\n  \\item Optical packet forwarding table computation\n  \\item Optical packet forwarding table lookup\n  \\item Optical switch fabric\n  \\item Optical buffering\n\\end{itemize}\nOptical packet switches typically relegate some of these design\nelements to the electrical domain. Most commonly the packet forwarding\ntable computation and lookup is performed electrically. When there is\ncontention for a destination port, a packet needs to be buffered\noptically, this buffering can be accomplished with rather impractical\nfiber delay lines. Fiber delay lines are fiber optic cables whose lengths are\nconfigured to provide a certain time delay of the optical signal; e.g.,\n100 meters of fiber provides 500~ns of delay. An alternative to\nbuffering is to either drop the packet or to use deflection routing,\nwhereby a packet is routed to a different output that may or may not\nlead to the desired destination.\n\n\\paragraph{Burst Switching}\nOptical \\textit{burst} switching alleviates the requirements of\noptical packet forwarding table computation, forwarding table lookup,\nas well as buffering while accommodating bursty traffic that would\nlead to poor utilization of optical circuits. In essence, it permits\nthe rapid establishment of short-lived optical circuits to support the\ntransfer of one or more packets coalesced into a burst. A control\npacket is sent through the network that establishes the lightpath for\nthe burst and then the burst is transmitted on the short-lived circuit\nwith no packet lookup or buffering required along the path~\\cite{ram2009opt}.\nSince the circuit is only established for the length of the burst, network\nresources are not wasted during idle periods. To avoid any buffering\nof the burst in the optical network, the burst transmission can begin\nonce the lightpath establishment has been confirmed (tell-and-wait) or\na short time period after the control packet is sent\n(just-enough-time). \\textit{Note}: Sending the burst immediately after\nthe control packet (tell-and-go) would require some buffering of the\noptical burst at the switching nodes.\n\n\\subsubsection{Optical Network Structure}  \\label{optnetstruct:sec}\nOptical networks are typically structured into three main tiers, namely\naccess networks, metropolitan (metro) area networks,\nand backbone (core) networks~\\cite{sim2014opt}.\n\n\\paragraph{Access Networks}\nIn the area of optical access networks~\\cite{for2015nex}, so-called Passive\nOptical Networks (PONs), in particular, Ethernet PONs (EPONs) and\nGigabit PONs (GPONs)~\\cite{haj2006epo,sku2009com},\nhave been widely studied.\nA PON has typically an inverse tree structure with a central\nOptical Line Terminal (OLT) connecting multiple distributed Optical\nNetwork Units (ONUs; also referred to as Optical Network Terminals, ONTs)\nto metro networks.\nIn the downstream (OLT to ONUs) direction, the OLT broadcasts transmissions.\nHowever, in the upstream (ONUs to OLT) direction, the transmissions of the\ndistributed ONUs need to be coordinated to avoid collisions on the\nshared upstream wavelength channel.\nTypically, a cyclic polling based Medium Access Control (MAC) protocol,\ne.g., based on the MultiPoint Control Protocol (MPCP, IEEE 802.3ah),\nis employed.\nThe ONUs report their bandwidth demands to the OLT and the OLT then\nassigns upstream transmission windows according to a Dynamic\nBandwidth Allocation (DBA)\nalgorithm~\\cite{kan2012ban,mcg2010sho,mcg2012inv,zhe2009sur}.\nConventional PONs cover distances up to 20~km, while so-called\nLong-Reach (LR) PONs cover distances up to\naround 100~km~\\cite{mer2013off,nag2016n,son2010lon}.\n\nRecently, hybrid access networks that combine multiple transmission\nmedia, such as Fiber-Wireless (FiWi)\nnetworks~\\cite{aur2014fiw,gha2011fib,liu2016new,sar2015arc,tsa2011sur} and\nPON-DSL networks~\\cite{gur2014pon}, have been explored to take\nadvantage of the respective strengths of the different transmission\nmedia.\n\n\n\\paragraph{Networks Connected to Access Networks}\nOptical access networks provide Internet connectivity for a wide range\nof peripheral networks. Residential (home) wired or wireless\nlocal area networks~\\cite{che2014sur}\ntypically interconnect individual end devices (hosts) in a home or small\nbusiness and may connect directly with an optical access network.\nCellular wireless networks provide Internet access to a wide range of\nmobile devices~\\cite{cap2013dow,dam2011sur,sch2013pus}.\nSpecialized cellular backhaul\nnetworks~\\cite{LiPCYW14,LiZZW13,park2014fro,PeWLP15,raz2013bri,tip2011evo,YaLJS13} relay the traffic\nto\/from base stations of wireless cellular\nnetworks to either wireless access\nnetworks~\\cite{aky2005wir,alo2012sur,ben2012wir,kur2007sur,pat2011sur,vij2013dis} or optical access networks.\nMoreover, optical access networks are often employed to connect\nData Center (DC) networks to the Internet. DC networks interconnect highly\nspecialized server units\n  that process and store large data amounts with specialized\nnetworking technologies~\\cite{cai2013sur,kac2012sur,sam2016sof,yan2016sud,zha2013sur}.\nData centers are\ntypically employed to provide the so-called  ``cloud'' services for\n commercial and social media applications.\n\n\\paragraph{Metropolitan Area Networks}\nOptical Metropolitan (metro) Area Networks (MANs) interconnect the optical\naccess networks in a metropolitan area with each other and with\nwide-area (backbone, core) networks.\nMANs have typically a ring or star\ntopology~\\cite{bia2013cos,bia2015sho,cha2013tow,mai2003hyb,rot2013rou,sch2003wav}\nand commonly employ optical networking technologies.\n\n\\paragraph{Backbone Networks}\nOptical backbone (wide area) networks interconnect the individual MANs\non a national\nor international scale. Backbone networks have typically a mesh structure\nand employ very high speed optical transmission links.\n\\begin{figure*}[t!]\n\\footnotesize\n\\setlength{\\unitlength}{0.10in}\n\\centering\n\\begin{picture}(40,35)\n\\put(-4,33){\\textbf{SDN Controlled Photonic Communication Infrastructure Layer, Sec.~\\ref{sdninfra:sec}}}\n\\put(-7,30){\\line(1,0){50}}\n\\put(19,30){\\line(0,1){2}}\n\\put(-7,30){\\vector(0,-1){2}}\n\n\\put(-9,27){\\makebox(0,0)[lt]{\\shortstack[l]{\t\t\t\n\t\\textbf{Bandwidth Variable Transceivers (BVTs)},\\\\ Sec.~\\ref{transc:sec}\t\\\\\t\\\\ \t\t\n\tSingle-Flow BVTs~\\cite{aut2013eva,ElA12,gri2010fle}, Sec.~\\ref{SF-BVT:sec}\\\\\n\tMach-Zehnder Modulator Based~\\cite{Choi2013,Liu2013c}\\\\\n\tBVTs for PONs~\\cite{LaAVKS14,IiSSK12,yeh2010usi,yu2008cen,VaBPF13,bol2014dig,Bolea2015a}\\\\\n\tBVTs for DC Netw.~\\cite{Malacarne2014} \\\\ \\\\\n\tSliceable Multi-Flow BVTs~\\cite{jin2012mul}, \\\\ Sec.~\\ref{MF-BVT:sec}\t\\\\\t\t\t\t\n\tEncoder Based~\\cite{Sambo2014a,sam2015nex,sam2014sli} \\\\\t\t\t\t\t\n\tDSP Based~\\cite{Moreolo2016} \\\\\t\t\t\t\t\n\tSubcar. $+$ Mod. Pool Based~\\cite{Ou2016} \\\\\t\t\t\t\t\n\tHYDRA~\\cite{mat2015hyd} \n\n\n}}}\n\n\\put(18,30){\\vector(0,-1){26}}\n\n\\put(9,3){\\makebox(0,0)[lt]{\\shortstack[l]{\t\t\t\n\t\t\t\\textbf{Space Div. Multipl.} \\\\\n\t\t\t\\textbf{(SDM)-SDN}, \\\\ Sec.~\\ref{SDM_SDN:sec}\n\t\t\t\\cite{ama2013ful,ama2014sof,Galve2016}\n}}}\n\n\\put(25,30){\\vector(0,-1){2}}\n\\put(20,27){\\makebox(0,0)[lt]{\\shortstack[l]{\t\t\t\t\t\t\n\t\t\t\\textbf{Switching}, Sec.~\\ref{infra_sw:sec} \\\\ \\\\ \t\t\t\t\n\t\t\tSwitching Elements, Sec.~\\ref{sw_elem:sec}\t\\\\\n\t\t\tROADMs~\\cite{Co13,ama2013int,rof2013all,you2013eng,Way2013wav,Garrich2015}\t\\\\\t\t\t\n\t\t\tOpen Transport Switch (OTS)~\\cite{SaSPL13}\t\\\\\t\t\t\t\t\n\t\t\tLogical xBar~\\cite{PaSMK13}\t\\\\\t\t\t\t\t\n\t\t\tOptical White Box~\\cite{Nejabati2015}\t\\\\\t\t\t\t\t\n\t\t\tGPON Virt. Sw.~\\cite{Lee2016,gu2014sof,Amokrane2014,amo2015dyn,Yeh2015}\t\\\\\t\t\t\t\n\t\t\tFlexi Access Netw. Node~\\cite{FoG13,kon2015sdn} \\\\ \\\\\n\t\t\tSwitching Paradigm, Sec.~\\ref{ws_para:sec}\t\\\\\t\t\t\n\t\t\tConverged Pkt-Cir. Sw.~\\cite{DaPM09,DaPMSG10,VeBB13,AzNEJ11,ShJKG12,KaT12,CeLR13}\t\\\\\t\t\t\t\t\n\t\t\tR-LR-UFAN~\\cite{yin2013ult,ShYD14,Yin2015}\t\\\\\n\t\t\tFlexi-grid~\\cite{Cv13,CvTJSM14,OlSCH13,ZhZYY13}\t\n}}}\t\n\\put(43,30){\\vector(0,-1){26}}\n\\put(25, 3){\\makebox(0,0)[lt]{\\shortstack[l]{\t\t\t\t\t\n\t\t\t\\textbf{Opt. Perf. Monitoring}, Sec.~\\ref{opm:sec} \\\\ \\\\\n\t\t\tCognitive Netw. Infra.~\\cite{MiDJF13,Oliveira2015,Giglio2015} \\\\\n\t\t\tWavelength Selective Switch\/Amplifier \\\\ \\ \\ Control~\\cite{Moura2015,pao2015sup,Carvalho2015,Wang2015f}\n}}}\t\n\\end{picture}\n\\vspace{1cm}\n\\caption{Classification of physical infrastructure layer SDON studies.}\n\\label{infra_class:fig}\n\\end{figure*}\n\\section{\\MakeUppercase{SDN Controlled Photonic Communication Infrastructure Layer}}\n\\label{sdninfra:sec}\nThis section surveys mechanisms for controlling physical layer\naspects of the optical (photonic) communication infrastructure through SDN.\nEnabling the SDN control down to the photonic level operation of\noptical communications allows for flexible adaptation of the\nphotonic components supporting optical networking\nfunctionalities~\\cite{ChNS13,gri2013ext,jin2013vir,RoD08}.\nAs illustrated in Fig.~\\ref{infra_class:fig},\nthis section first surveys transmitters and receivers (collectively\nreferred to as transceivers or transponders) that permit SDN control of the\noptical signal transmission characteristics, such as modulation format.\nWe also survey SDN controlled space division\nmultiplexing (SDM), which provides an emerging avenue for highly\nefficient optical transmissions.\nThen, we survey SDN controlled optical switching, covering first\nswitching elements and then overall switching paradigms, such as converged\npacket and circuit switching.\nFinally, we survey cognitive photonic communication infrastructures that\nmonitor the optical signal quality. The optical signal quality\ninformation can be used to dynamically control the transceivers\nas well as the filters in switching elements.\n\n\\subsection{Transceivers}  \\label{transc:sec}\nSoftware defined optical transceivers are optical transmitters and\nreceivers that can be flexibly configured by SDN to transmit or\nreceive a wide range of optical signals~\\cite{Hillerkuss2016}.\nGenerally, software defined optical transceivers vary the modulation\nformat~\\cite{win2006adv} of the transmitted optical signal by\nadjusting the transmitter and receiver operation through Digital\nSignal Processing (DSP)\ntechniques~\\cite{cha2014adv,sch2010rea,yos2013dsp}. These\ntransceivers have evolved in recent years from Bandwidth Variable\nTransceivers (BVTs) generating a single signal flow to sliceable\nmulti-flow BVTs. Single-flow BVTs permit SDN control to adjust the\ntransmission bandwidth of the single generated signal flow. In\ncontrast, sliceable multi-flow BVTs allow for the independent SDN\ncontrol of multiple communication traffic flows generated by a\nsingle BVT.\n\n\\subsubsection{Single-Flow Bandwidth Variable Transceivers (BVTs)}\n\\label{SF-BVT:sec} Software defined optical transceivers have\ninitially been examined in the context of adjusting a single optical\nsignal flow for flexible WDM\nnetworking~\\cite{aut2013eva,ElA12,gri2010fle}. The goal has been to\nmake the photonic transmission characteristics of a given\ntransmitter fully programmable. We proceed to review a\nrepresentative single-flow BVT design for general optical mesh\nnetworks in detail and then summarize related single-flow BVTs for\nPONs and data center networks.\n\n\\paragraph{Mach-Zehnder Modulator Based Flexible Transmitter}\nChoi and Liu et al.~\\cite{Choi2013,Liu2013c} have demonstrated a\nflexible transmitter based on Mach-Zehnder Modulators\n(MZMs)~\\cite{bar2003wid} and a corresponding flexible receiver for\nSDN control in a general mesh network. The flexible transceiver\nemploys a single dual-drive MZM that is fed by two binary electric\nsignals as well as a parallel arrangement of two MZMs which are fed\nby two additional electrical signals. Through adjusting the direct\ncurrent bias voltages and amplitudes of drive signals the\ncombination of MZMs can vary the amplitude and phase of the\ngenerated optical signal~\\cite{cho2012ber}. Thus, modulation formats\nranging from Binary Phase Shift Keying (BPSK) to Quadrature Phase\nShift Keying (QPSK) as well as 8 and 16 quadrature amplitude\nmodulation~\\cite{win2006adv} can be generated. The amplitudes and\nbias voltages of the drive signals can be signaled through an SDN OpenFlow\ncontrol plane to achieve the different modulation formats. The\ncorresponding flexible receiver consists of a polarization filter\nthat feeds four parallel photodetectors, each followed by an\nAnalog-to-Digital Converter (ADC). The outputs of the four parallel\nADCs are then processed with DSP techniques to automatically\n(without SDN control) detect the modulation format. Experiments\nin~\\cite{Choi2013,Liu2013c} have evaluated the bit error rates and\ntransmission capacities of the different modulation formats and have\ndemonstrated the SDN control.\n\n\\paragraph{Single-Flow BVTs for PONs}\nFlexible optical networking with real-time bandwidth adjustments\nis also highly desirable for PON access and metro networks,\nalbeit the BVT technologies for access and metro networks should\nhave low cost and complexity~\\cite{LaAVKS14}.\nIiyama et al.~\\cite{IiSSK12} have developed a DSP based approach\nthat employs SDN to coordinate the downstream PON transmission of\nOn-Off Keying (OOK) modulation~\\cite{yeh2010usi} and\nQuadrature Amplitude Modulation (QAM)~\\cite{yu2008cen} signals.\nThe OOK-QAM-SDN scheme involves a\nnovel multiplexing method, wherein all the data\nare simultaneously sent from the OLT to the ONUs and the ONUs filter\nthe data they need.\nThe experimental setup in~\\cite{IiSSK12} also demonstrated\ndigital software ONUs that concurrently transmit data by exploiting\nthe coexistence of OOK and QAM.\nThe OOK-QAM-SDN evaluations demonstrated the control of the receiving\nsensitivity which is very useful for a wide range of transmission environments.\n\nIn a related study, Vacondio et al.~\\cite{VaBPF13} have examined\nSoftware-Defined Coherent Transponders (SDCT)\nfor TDMA PON access networks.\nThe proposed SDCT digitally processes the burst transmissions to\nachieve improved burst mode transmissions according to the distance of a\nuser from the OLT.\nThe performance results indicate that the proposed flexible\napproach more than doubles the average transmission capacity\nper user compared to  a static approach.\n\n\\begin{figure*}[t]\n    \\centering\n    \\vspace{0cm}\n    \\includegraphics[width=6in]{finman_sdon\/fig4.pdf}\n    \\vspace{0cm}\n    \\caption{Illustration of DSP reconfigurable ONU and OLT\n    designs~\\cite{Bolea2015a}.}\n    \\label{fig:bol2015}\n\\end{figure*}\nBolea et al.~\\cite{bol2014dig,Bolea2015a} have recently\ndeveloped low-complexity DSP reconfigurable ONU and OLT designs for\nSDN-controlled PON communication.\nThe proposed communication is based on carrierless amplitude and phase\nmodulation~\\cite{rod2011car} enhanced with optical Orthogonal\nfrequency Division Multiplexing (OFDM)~\\cite{bol2014dig}.\nThe different OFDM channels are manipulated through DSP filtering.\nAs illustrated in Fig.~\\ref{fig:bol2015}, the ONU consists of a DSP controller\nthat controls the filter coefficients of the shaping filter.\nThe filter output is then passed through a Digital-to-Analog Converter (DAC)\nand intensity modulator for electric-optical conversion.\nAt the OLT, a photo diode converts the optical signal to an electrical signal,\nwhich then passes through an Analog-to-Digital Converter (ADC).\nThe SDN controlled OLT DSP controller\nsets the filter coefficients in the matching filter to\ncorrespond to the filtering in the sending ONU.\nThe OLT DSP controller is also responsible for ensuring the\northogonality of all the ONU filters in the PON.\nThe performance evaluations in~\\cite{Bolea2015a} indicate that\nthe proposed DSP reconfigurable ONU and OLT system\nachieves ONU signal bitrates around 3.7~Gb\/s for eight ONUs transmitting\nupstream over a 25~km PON.\nThe performance evaluations also illustrate that long DSP filter lengths,\nwhich increase the filter complexity, improve performance.\n\n\\paragraph{Single-Flow BVTs for Data Center Networks}\nMalacarne et al.~\\cite{Malacarne2014} have developed a low-complexity\nand low-cost bandwidth adaptable transmitter for data center\nnetworking.\nThe transmitter can multiplex Amplitude Shift Keying (ASK),\nspecifically On-Off Keying (OOK), and Phase Shift Keying (PSK)\non the same optical carrier signal without any\nspecial synchronization or temporal alignment mechanism.\nIn particular, the transmitter design~\\cite{Malacarne2014}\nuses the OOK electronic signal to drive a Mach-Zehnder Modulator (MZM)\nthat is fed by the optical pulse modulated signal.\nSDN control can activate (or de-activate) the OOK signal stream, i.e.,\nadapt from transmitting only the PSK signal to transmitting\nboth the PSK and OOK signal and thus providing a higher transmission bit rate.\n\n\\subsubsection{Sliceable Multi-Flow Bandwidth Variable Transceivers}\n\\label{MF-BVT:sec} Whereas the single-flow transceivers surveyed in\nSection~\\ref{SF-BVT:sec} generate a single optical signal flow,\nparallelization efforts have resulted in multi-flow transceivers\n(transponders)~\\cite{jin2012mul}. Multi-flow transceivers can\ngenerate multiple parallel optical signal flows and thus form the\ninfrastructure basis for network virtualization.\n\n\\paragraph{Encoder Based Programmable Transponder}\nSambo et al.~\\cite{Sambo2014a,sam2015nex} have developed an\nSDN-programmable bandwidth-variable multi-flow transmitter and\ncorresponding SDN-programmable multi-flow bandwidth variable\nreceiver, referred to jointly as programmable bandwidth-variable\ntransponder. The transmitter mainly consists of a programmable\nencoder and multiple parallel Polarization-Multiplexing Quadrature\nPhase Shift Keying (PM-QPSK~\\cite{win2006adv}) laser transmitters,\nwhose signals are multiplexed by a coupler. The encoder is\nSDN-controlled to implement Low-Density Parity-Check (LDPC)\ncoding~\\cite{bon2011low} with different code rates. At the receiver,\nthe SDN control sets the local oscillators and LDPC decoder. The\ndeveloped transponder allows the setting of the number of\nsubcarriers, the subcarrier bitrate, and the LDPC coding rate\nthrough SDN. Related frequency conversion and defragmentation issues\nhave been examined in \\cite{Sambo2015}. In~\\cite{sam2014sli}, a\nlow-cost version of the SDN programmable transponder with a\nmultiwavelength source has been developed. The multiwavelength\nsource is based on a micro-ring resonator~\\cite{ras2009dem} that\ngenerates multiple signal carriers with only a single laser.\nAutomated configuration procedures for the comprehensive set of\ntransmission parameters, including modulation format, coding configuration,\nand carriers have been explored in~\\cite{cug2016tow}.\n\n\\paragraph{DSP Based Sliceable BVT}\nMoreolo et al.~\\cite{Moreolo2016} have developed an SDN controlled\nsliceable BVT based on adaptive Digital Signal Processing (DSP) of\nmultiple parallel signal subcarriers. Each subcarrier is fed by a\nDSP module that configures the modulation format, including the bit\nrate setting, and the power level of the carrier by adapting a gain\ncoefficient. The output of the DSP module is then passed through\ndigital to analog conversion that drives laser sources. The parallel\nflows can be combined with a wavelength selective switch; the\ncombined flow can be sliced into multiple distinct sub-flows for\ndistinct destinations. The functionality of the developed DSP based\nBVT has been verified for a metropolitan area network with links\nreaching up to 150~km.\n\n\\begin{figure}[t]\n    \\centering\n    \\vspace{0cm}\n    \\includegraphics[width=3.2in]{finman_sdon\/fig5.pdf}\n    \\vspace{0cm}\n    \\caption{Illustration of Subcarrier and Modulator Pool Based\n    Virtualizable Bandwidth Variable Transceiver (V-BVT)~\\cite{Ou2016}.}\n    \\label{fig:ou2016}\n\\end{figure}\n\\paragraph{Subcarrier and Modulator Pool Based Virtualizable BVT}\nOu et al.~\\cite{Ou2016,ou2015onl} have developed a Virtualizable BVT\n(V-BVT) based on a combination of an optical subcarriers pool\nwith an independent optical modulators pool, as illustrated in\nFig.~\\ref{fig:ou2016}.\nThe emphasis of the design is on implementing\nVirtual Optical Networks (VONs) at the transceiver level.\nThe optical subcarriers pool contains multiple optical carriers,\nwhereby channel spacing and central frequency (wavelength channel)\ncan be selected.\nThe optical modulators pool contains optical modulators that can\ngenerate a wide variety of modulation formats.\nThe SDN control interacts with a V-BVT Manager that implements a\nvirtualization algorithm. The virtualization algorithm\ngenerates a transceiver slice by combining a particular\nset of subcarriers (with specific number of subcarriers, channel\nspacing, and central frequencies) from the optical subcarriers pool\nwith a particular modulation (with specific number of modulators and\nmodulation formats) from the optical modulators pool.\nThe evaluations in~\\cite{Ou2016} have evaluated the proposed V-BVT in\na network testbed with path lengths up to 200~km with 20~GHz channel\nspacing and a variety of modulation formats, including\nBPSK as well as 16QAM and 32QAM.\n\n\\paragraph{S-BVT Based Hybrid Long-Reach Fiber Access Network (HYDRA)}\nHYDRA~\\cite{mat2015hyd} is a novel hybrid long-reach fiber access\nnetwork architecture based on sliceable BVTs. HYDRA supports\nlow-cost end-user ONUs through an Active Remote Node (ARN) that\ndirectly connects via a distribution fiber segment, a passive remote\nnode, and a trunk fiber segment to the core (backbone) network,\nbypassing the conventional metro network. The ARN is based on an SDN\ncontrolled S-BVT to optimize the modulation format. With the\nmodulation format optimization, the ARN can optimize the\ntransmission capacity for the given distance (via the distribution\nand trunk fiber segments) to the core network. The evaluations\nin~\\cite{mat2015hyd} demonstrate good bit error rate performance of\nrepresentative HYDRA scenarios with a 200~km trunk fiber segment and\ndistribution fiber lengths up to 100~km. In particular, distribution\nfiber lengths up to around 70~km can be supported without Forward\nError Correction (FEC), whereas distribution fiber lengths above\n70~km would require standard FEC. The consolidation of the access\nand metro network infrastructure~\\cite{wan2016mig} achieved through\nthe optimized S-BVT transmissions can significantly reduce the\nnetwork cost and power consumption.\n\n\\subsection{Space Division Multiplexing (SDM)-SDN}  \\label{SDM_SDN:sec}\nAmaya et al.~\\cite{ama2013ful,ama2014sof} have\ndemonstrated SDN control of\nSpace Division Multiplexing (SDM)~\\cite{ric2013spa} in optical networks.\nMore specifically, Amaya et al.~employ SDN to control the\nphysical layer so as to achieve a bandwidth-flexible and programmable\nSDM optical network. The SDN control can perform network slicing,\nresulting in sliceable superchannels.\nA superchannel consists of multiple spatial carriers to support\ndynamic bandwidth and QoS provisioning.\n\nGalve et al.~\\cite{Galve2016} have built on the flexible SDN controlled\nSDM communication principles to develop a reconfigurable Radio\nAccess Network (RAN). The RAN connects the BaseBand processing Units (BBUs)\nin a shared central office with the corresponding distributed\nRemote Radio Heads (RRHs) located at Base Stations (BSs).\nA multicore fiber operated with SDM~\\cite{ric2013spa} connects the\nRRHs to the BBUs in the central office.\nGalve et al.~introduce a radio over fiber operation mode where\nSDN controlled switching maps the subcarriers dynamically to\nspatial output ports.\nA complementary digitized radio over fiber operating mode maintains a\nBBU pool. Virtual BBUs are dynamically allocated to the cores of\nthe SDM operated multicore fiber.\n\n\n\\subsection{SDN-Controlled Switching} \\label{infra_sw:sec}\n\\subsubsection{Switching Elements}   \\label{sw_elem:sec}\n\\paragraph{ROADM} \\label{ROADMs:par}\nThe Reconfigurable Optical Add-Drop Multiplexer (ROADM) is an\nimportant photonic switching device for optical networks. Through\nwavelength selective optical switches, a ROADM can drop (or add) one\nor multiple wavelength channels carrying optical data signals from\n(to) a fiber without requiring the conversion of the optical signal\nto electric signals~\\cite{he2014sur}. The ROADM thus provides an\nelementary switching functionality in the optical wavelength domain.\nInitial ROADM based node architectures for cost-effectively\nsupporting flexible SDN networks have been presented\nin~\\cite{Co13}. Conventional ROADM networks have\ntypically statically configured wavelength channels that transport\ntraffic along a pre-configured route. Changes of wavelength channels\nor routes in the statically configured networks incur presently high\noperational costs due to required physical interventions and are\ntherefore typically avoided. New ROADM node designs allow changes of\nwavelength channels and routes through a management control plane.\nDue to these two flexibility dimensions (wavelength and route),\nthese new ROADM nodes are referred to as ``colorless'' and\n``directionless''. First designs for such colorless and\ndirectionless ROADM nodes have been outlined in~\\cite{Co13} and\nfurther elaborated in~\\cite{ama2013int,rof2013all}.\nIn addition to the colorless and directionless properties, the\ncontentionless property has emerged for ROADMs~\\cite{gri2010fle}.\nContentionless ROADM operation means that any port can be\nrouted on any wavelength (color) in any direction without causing\nresource contention.\nDesigns for such Colorless-Directionless-Contentionless (CDC)\nROADMs have been proposed in~\\cite{you2013eng,Way2013wav}.\nIn general, the ROADM designs consist of an express bank that\ninterconnects the input and output ports coming from\/leading to other ROADMs,\nand an add-drop bank that connects the express bank with the local\nreceivers for dropped wavelength channels or transmitters for added\nwavelength channels. The recent designs have focused on\nthe add-drop bank and explored different arrangements\nof wavelength selective switches and multicast switches to\nprovide add-drop bank functionality with the CDC\nproperty~\\cite{you2013eng,Way2013wav}.\n\nGarrich et al.~\\cite{Garrich2015} have recently designed\nand demonstrated a CDC ROADM with an add-drop bank based on\nan Optical Cross-Connect (OXC) backplane~\\cite{wan2012mul}.\nThe OXC backplane allows for highly flexible add\/drop configurations\nimplemented through SDN control.\nThe backplane based ROADM has been analytically\ncompared with prior designs based on\nwavelength selective and multicast switches and has been\nshown to achieve higher flexibility and lower losses.\nAn experimental evaluation has tested the backplane based ROADM for\na metropolitan area mesh network extending over 100~km\nwith an aggregate traffic load of  close to 9~Tb\/s.\n\n\\paragraph{Open Transport Switch (OTS)}\nThe Open Transport Switch (OTS) \\cite{SaSPL13} is an OpenFlow enabled\noptical virtual switch design. The OTS design abstracts the details\nof the underlying physical switching layer (which could be packet\nswitching or circuit switching) to a virtual switch element. The OTS\ndesign introduces three agent modules (discovery, control, and data\nplane) to interface with the physical switching hardware. These\nagent modules are controlled from an SDN controller through extended\nOpenFlow messages. Performance measurements for an example testbed\nnetwork setup indicate that the circuit path computation latencies\non the order of 2--3 s that can be reduced through faster processing\nin the controller.\n\n\\paragraph{Logical xBar}\nThe logical xBar~\\cite{PaSMK13} has been defined to represent a\nprogrammable switch. An elementary (small) xBar could consist of a\nsingle OpenFlow switch. Multiple small xBars can be recursively\nmerged to form a single large xBar with a single forwarding table.\nThe xBar concept envisions that xBars are the building blocks for\nforming large networks. Moreover, labels based on SDN and MPLS are\nenvisioned for managing the xBar data plane forwarding.\nThe xBar concepts have been further advanced in the Orion\nstudy~\\cite{fu2014ori} to achieve low computational complexity of\nthe SDN control plane.\n\n\\paragraph{Optical White Box}\nNejabati et al.~\\cite{Nejabati2015} have proposed an optical white\nbox switch design as a building block for a completely softwarized\noptical network. The optical white box design combines a\nprogrammable backplane with programmable switching node elements.\nMore specifically, the backplane consists of two slivers, namely an\noptical backplane sliver and an electronic backplane sliver. These\nslivers are set up to allow for flexible arbitrary connections\nbetween the switch node elements. The switch node elements include\nprogrammable interfaces that build on SDN-controlled BVTs (see\nSection~\\ref{transc:sec}), protocol agnostic switching, and DSP elements.\nThe protocol agnostic switching\nelement is envisioned to support both wavelength channel and time\nslot switching in the optical backplane as well as programmable\nswitching with a high-speed packet processor in the electronic\nbackplane. The DSP elements support both the network processing and\nthe signal processing for executing a wide range of network\nfunctions. A prototype of the optical white box has been built with\nonly a optical backplane sliver consisting of a $192 \\times 192$\noptical space switch. Experiments have indicated that the creation of\na virtual switching node with the OpenDayLight SDN controller takes\nroughly 400~ms.\n\n\\paragraph{GPON Virtual Switch}\nLee et al.~\\cite{Lee2016} have developed a GPON virtual switch\ndesign that makes the GPON fully programmable similar to a\nconventional OpenFlow switch. Preliminary steps towards the GPON\nvirtual switch design have been taken by Gu et al.~\\cite{gu2014sof}\nwho developed components for SDN control of a PON in a data center\nand Amokrane et al.~\\cite{Amokrane2014,amo2015dyn} who developed a\nmodule for mapping OpenFlow flow control requests into PON configuration\ncommands. Lee et al.~\\cite{Lee2016} have expanded on this groundwork\nto abstract the entire GPON into a virtual OpenFlow switch. More\nspecifically, Lee et al. have comprehensively designed a hardware\narchitecture and a software architecture to allow SDN control to\ninterface with the virtual GPON as if it were a standard OpenFlow switch.\nThe experimental performance evaluation of the designed GPON virtual\nswitch measured response times for flow entry modifications from an\nONU port (where a subscriber connects to the virtual GPON switch) to\nan SDN external port around 0.6~ms, which compares to 0.2~ms for a\ncorresponding flow entry modification in a conventional OFsoftswitch\nand 1.7~ms in a EdgeCore AS4600 switch. In a related study on SDN\ncontrolled switching in a PON, Yeh et al.~\\cite{Yeh2015} have\ndesigned an ONU with an optical switch that selects OFDM subchannels\nin a TWDM-PON. The switch in the ONU allows for flexible dynamic\nadaption of the downstream bandwidth through SDN.\nGu et al.~\\cite{gu2016eff} have examined the flexible SDN controlled\nre-arrangement of ONUs to OLTs so as to efficiently support PON\nservice with network coding~\\cite{bas2013net}.\n\n\\paragraph{Flexi Access Network Node}\nA flexi-node for an access network that flexibly aggregates\ntraffic flows from a wide range of networks,\nsuch as local area networks and base stations of wireless\nnetworks has been proposed in~\\cite{FoG13}.\nThe flexi-node design is motivated by the shortcomings of the\ncurrently deployed core\/metro\nnetwork architectures that attempt to consolidate the access and\nmetro networks.\nThis consolidation forces all traffic in the access network to\ntraverse the metro network, even if the traffic is destined to\ndestination nodes in the coverage area of an access network.\nIn contrast, the proposed flexi-node encompasses\nelectrical and optical forwarding capabilities that can be\ncontrolled through SDN. The flexi-node can thus serve as an\neffective aggregation node in access-metro networks.\nTraffic that is destined to other nodes in the coverage area of\nan access network can be sent directly to the access network.\n\nKondepu et al. have similarly presented an SDN based PON aggregation\nnode~\\cite{kon2015sdn}.\nIn their architecture, multiple ONUs\ncommunicate with the SDN controller within the aggregation node\nto request the scheduling of upstream transmission resources.\nONUs are then serviced by multiple Optical Service Units (OSUs)\nwhich exist\nwithin the aggregation node alongside with the SDN controller.\nOSUs are then configured by the controller based on Time and Wavelength\nDivision Multiplexed (TWDM) PON. The OSUs step between normal and sleep-mode\ndepending on the traffic loads, thus saving power.\n\n\\subsubsection{Switching Paradigms}  \\label{ws_para:sec}\n\\paragraph{Converged Packet-Circuit Switching}\nHybrid packet-circuit optical network infrastructures controlled by SDN\nhave been explored in a few studies.\nDas et al.~\\cite{DaPM09} have described how to unify the control and\nmanagement of circuit- and packet-switched networks using OpenFlow.\nSince packet- and circuit-switched networking are extensively employed\nin optical networks, examining their integration is an important research\ndirection.\nDas et al.~have given a high-level overview of a flow abstraction for\neach type of switched network and a common control paradigm.\nIn their follow-up work, Das et al.~\\cite{DaPMSG10}\nhave described how a packet and circuit switching\nnetwork can be implemented in the context of an OpenFlow-protocol based testbed. The testbed is a standard Ethernet network that could generally\nbe employed in any access network with Time Division Multiplexing (TDM).\nVeisllari et al.~\\cite{VeBB13} studied packet\/circuit hybrid optical\nlong-haul metro access networks.\nAlthough Veisllari et al.~indicated that SDN can be used for load balancing in\nthe proposed packet\/circuit network, no detailed study of\nsuch an SDN-based load balancing has been conducted in~\\cite{VeBB13}.\nRelated switching paradigms that integrate SDN with Generalized\nMultiple Protocol Label Switching (GMPLS)\nhave been examined in~\\cite{AzNEJ11,ShJKG12},\nwhile data center specific aspects have been surveyed in~\\cite{KaT12}.\n\nCerroni et al.~\\cite{CeLR13} have further developed the concept of\nunifying circuit- and packet-switching networks with OpenFlow,\nwhich was initiated by Das et al.~\\cite{DaPM09,DaPMSG10}.\nThe unification is accomplished with SDN on the network layer and\ncan be used in core networks. Specifically, Cerroni et\nal.~\\cite{CeLR13} have described an extension of the OpenFlow flow\nconcept to support hybrid networks. OpenFlow message format\nextensions to include matching rules and flow entries have also been\nprovided. The matching rules can represent different transport\nfunctions, such as a channel on which a packet is received in\noptical circuit-switched WDM networks, time slots in TDM networks,\nor transport class services (such as guaranteed circuit service or\nbest effort packet service). Cerroni et al.~\\cite{CeLR13} have\npresented a testbed setup and reported performance results for\nthroughput (in bit\/s and packets\/s) to demonstrate the feasibility\nof the proposed unified OpenFlow switching network.\n\n\\paragraph{R-LR-UFAN}  \\label{R_LR_UFAN:sec}\nThe Reconfigurable Long-Reach UltraFlow Access Network\n(R-LR-UFAN)~\\cite{yin2013ult,ShYD14} provides flexible dual-mode transport\nservice based on either the Internet Protocol (IP) or\nOptical Flow Switching (OFS).\nOFS~\\cite{chan2012opt} provides dedicated end-to-end network paths\nthrough purely optical switching, i.e., there is no electronic\nprocessing or buffering at intermediate network nodes. The R-LR-UFAN\narchitecture employs multiple feeder fibers to form subnets within\nthe network.\nUltraFlow coexists alongside the conventional PON OLT and ONUs. The\nR-LR-UFAN introduces new entities, namely the Optical Flow Network\nUnit (OFNU) and the SDN-controlled Optical Flow Line Terminal\n(OFLT). A Quasi-PAssive Reconfigurable (QPAR) node~\\cite{Yin2015} is\nintroduced between the OFNU and OFLT. The QPAR node can re-route\nintra PON traffic between OFNUs without having to pass through the\nOLFTs. The optically rerouted intra-PON channels can be used for\ncommunication between wireless base stations supporting inter cell\ndevice-to-device communication. The testbed evaluations indicate\nthat for an intra-PON traffic ratio of 0.3, the QPAR strategy\nachieves power savings up to 24\\%.\n\n\\paragraph{Flexi-grid}\nThe principle of flexi-grid (elastic) optical\nnetworking~\\cite{cha2015rou,gon2015opt,jue2014sof,she2016sur,tal2014spe,tom2014tut,yu2014spe,zha2013surflexi}\nhas been explored in several SDN infrastructure studies.\nGenerally, flexi-grid networking strives to enhance the efficiency\nof the optical transmissions by adapting physical (photonic)\ntransmission parameters, such as modulation format, symbol rate,\nnumber and spacing of subcarrier wavelength channels, as well as the\nratio of forward error correction to payload.\nFlexi-grid transmissions have become feasible with\nhigh-capacity flexible transceivers.\nFlexi-grid transmissions use narrower\nfrequency slots (e.g., 12.5~GHz) than classical Wavelength Division\nMultiplexing (WDM, with typically 50~GHz frequency slots for WDM)\nand can flexibly form optical transmission channels that span multiple\ncontiguous frequency slots.\n\nCvijetic~\\cite{Cv13} has proposed a hierarchical flexi-grid infrastructure\nfor multiservice broadband optical access utilizing\ncentralized software-reconfigurable resource management and digital signal\nprocessing. The proposed flexi-grid infrastructure incorporates mobile\nbackhaul, as well as SDN controlled transceivers~\\ref{transc:sec}.\nIn follow-up work, Cvijetic et al.~\\cite{CvTJSM14} have designed a\ndynamic flexi-grid optical access and aggregation network.\nThey employ SDN to control tunable lasers in the OLT for flexible\ndownstream transmissions.\nFlexi-grid wavelength selective switches are controlled through SDN\nto dynamically tune the passband for the upstream transmissions\narriving at the OLT.\nCvijetic et al.~\\cite{CvTJSM14} obtained good results for the upstream and\ndownstream bit error rate\nand were able to provide 150~Mb\/s per wireless network cell.\n\nOliveira et al.~\\cite{OlSCH13} have demonstrated a testbed for a\nReconfigurable Flexible Optical Network (RFON), which was one of the\nfirst physical layer SDN-based testbeds. The RFON testbed is\ncomprised of 4 ROADMs with flexi-grid Wavelength Selective Switching\n(WSS) modules, optical amplifiers, optical channel monitors and\nsupervisor boards. The controller daemon implements a node\nabstraction layer and provides configuration details for an overall\nview of the network. Also, virtualization of the GMPLS control plane\nwith topology discovery and Traffic Engineering (TE)-link\ninstantiation have been incorporated.\nInstead of using OpenFlow, the\nRFON testbed uses the controller language YANG~\\cite{Schonwalder2010}\nto obtain the topology information and collect monitoring data  for the\nlightpaths.\n\nZhao et al.~\\cite{ZhZYY13} have presented an architecture with\nOpenFlow-based optical interconnects for intra-data center networking\nand OpenFlow-based flexi-grid optical networks for inter-data center\nnetworking.\nZhao et al.~focus on the SDN benefits for inter-data center networking\nwith heterogeneous networks.\nThe proposed architecture includes a service controller, an IP controller,\nan and optical controller based on the Father Network Operating System\n(F-NOX)~\\cite{gud2008nox,zha2013uni}.\nThe performance evaluations in~\\cite{ZhZYY13} include results for\nblocking probability, release latency, and bandwidth spectrum characteristics.\n\n\\subsection{Optical Performance Monitoring}  \\label{opm:sec}\n\\label{sec:sdmonitoring}\n\\subsubsection{Cognitive Network Infrastructure}\nA Cognitive Heterogeneous Reconfigurable Dynamic Optical Network (CHRON)\narchitecture has been outlined in~\\cite{MiDJF13,cab2014cog,dur2016exp}.\nCHRON senses the current network\nconditions and adapts the network operation accordingly.\nThe three main components of CHRON are monitoring elements, software\nadaptable elements, and cognitive processes. The monitoring elements\nobserve two main types of optical transmission impairments, namely\nnon-catastrophic impairments and catastrophic impairments.\nNon-catastrophic impairments include the photonic impairments that\ndegrade the Optical Signal to Noise Ratio (OSNR), such as the\nvarious forms of dispersion, cross-talk, and non-linear propagation\neffects, but do not completely disrupt the communication. In\ncontrast, a catastrophic impairment, such as a fiber cut or\nmalfunctioning switch, can completely disrupt the communication.\nAdvances in optical performance monitoring allow for in-band OSNR\nmonitoring~\\cite{dah2011opt,sui2010osn,sch2011osn,sai2012ban} at\nmidpoints in the communication path, e.g., at optical amplifiers and\nROADMs.\n\nThe cognitive processes involve the collection of the monitoring\ninformation in the controller, executing control algorithms, and\ninstructing the software adaptable components to implement the\ncontrol decisions. SDN can provide the framework for implementing\nthese cognitive processes.\nTwo main types of software adaptable components have been considered\nso far~\\cite{Oliveira2015,Giglio2015}, namely control of transceivers and\ncontrol of wavelength selective switches\/amplifiers.\nFor transceiver control,\nthe cognitive control adjusts the transmission parameters.\nFor instance, transmission bit\nrates can be adjusted through varying the modulation format or the\nnumber of signal carriers in multicarrier communication\n(see Section~\\ref{transc:sec}).\n\n\\subsubsection{Wavelength Selective Switch\/Amplifier Control}\nIn general, ROADMs (see Section~\\ref{ROADMs:par}) employ wavelength\nselective switches based on filters to add or drop wavelength\nchannels for routing through an optical network. Detrimental\nnon-ideal filtering effects accumulate and impair the\nOSNR~\\cite{pao2015sup}. At the same time, Erbium Doped Fiber\nAmplifiers (EDFAs)~\\cite{zim2004amp} are widely deployed in optical\nnetworks to boost optical signal power that has been depleted\nthrough attenuation in fibers and ROADMs. However, depending on\ntheir operating points, EDFAs can introduce significant noise. Moura\net al.~\\cite{Moura2015,mou2016cog} have explored SDN based adaptation\nstrategies for EDFA operating points to increase the OSNR. In a\ncomplementary study, Paolucci et al.~\\cite{pao2015sup} have\nexploited SDN control to reduce the detrimental filtering effects.\nPaolucci group wavelength channels that jointly traverse a sequence\nof filters at successive switching nodes. Instead of passing these\nwavelength channels through individual (per-wavelength channel)\nfilters, the group of wavelength channels is jointly passed through\na superfilter that encompasses all grouped wavelength channels. This\njoint filtering significantly improves the OSNR.\n\nWhile the studies~\\cite{Moura2015,mou2016cog,pao2015sup} have focused on\neither the EDFA or the filters, Carvalho et al.~\\cite{Carvalho2015} and\nWang et al.~\\cite{Wang2015f} have jointly considered\nthe EDFA and filter control.\nMore specifically, the EDFA gain and the filter attenuation (and\nsignal equalization) profile were adapted to improve the OSNR.\nCarvalho et al.~\\cite{Carvalho2015} propose and evaluate a specific\njoint EDFA and filter optimization approach that exploits the global\nperspective of the SDN controller. The global optimization achieves\nONSR improvements close to 5~dB for a testbed consisting of four ROADMs\nwith 100~km fiber links.\nWang et al.~\\cite{Wang2015f} explore different combinations of\nEDFA gain control strategies and filter equalization strategies for a\nsimulated network with 14 nodes and 100~km fiber links.\nThey find mutual interactions between the EDFA gain control and the\nfilter equalization control as well as an additional wavelength assignment\nmodule.\nThey conclude that global SDN control is highly useful for synchronizing\nthe EDFA gain and filter equalization in conjunction with wavelength\nassignments so as to achieve improved OSNR.\n\n\\subsection{Infrastructure Layer: Summary and Discussion}\nThe research to date on the SDN controlled infrastructure layer has\nresulted in a variety of SDN controlled transceivers as well as a few designs\nof SDN controlled switching elements.\nMoreover, the SDN control of switching paradigms and optical\nperformance monitoring have been examined.\nThe SDN infrastructure studies have paid close attention to the physical\n(photonic) communication aspects. Principles of isolation of control plane and\ndata plane with the goals of simplifying network management and\nmaking the networks more flexible have been explored.\nThe completed SDN infrastructure layer studies have indicated\nthat the SDN control of the infrastructure layer can reduce costs, facilitate\nflexible reconfigurable resource management, increase utilizations,\nand lower latency.\nHowever, detailed comprehensive optimizations of the infrastructure components\nand paradigms\nthat minimize capital and operational expenditures are an important\narea for future research. Also, further refinements of the optical\ncomponents and switching paradigms are needed to ease the\ndeployment of SDONs and make the networks operating on\nthe SDON infrastructures more efficient.\nMoreover, the cost reduction of implementations, easy adoption by network\nproviders, flexible upgrades to adopt new technologies, and reduced\ncomplexity require thorough future research.\n\nMost SDON infrastructure studies have\nfocused on a particular network component or networking aspect, e.g.,\na transceiver or the hybrid\npacket-circuit switching paradigm, or a particular application context, e.g.,\ndata center networking. Future research should comprehensively\nexamine SDON infrastructure components and\nparadigms to optimize their interactions\nfor a wide set of networking scenarios and application contexts.\n\nThe SDON infrastructure studies to date\nhave primarily focused on the optical transmission medium.\nFuture research should explore complementary infrastructure\ncomponents and paradigms to\nsupport transmissions in hybrid fiber-wireless and other hybrid fiber-$X$\nnetworks, such as\nfiber-Digital Subscriber Line (DSL) or fiber-coax cable\nnetworks~\\cite{Fuentes2014,gur2014pon,luo2013act}.\nGenerally, the flexible SDN control can\nbe very advantageous for hybrid networks composed of heterogeneous\nnetwork segments. The OpenFlow protocol can facilitate the topology\nabstraction of the heterogeneous physical transmission media, which\nin turn facilitates control and optimization at the higher network\nprotocol layers.\n\n\\begin{figure*}[t!]\n\\footnotesize\n\\setlength{\\unitlength}{0.10in}\n\\centering\n\\begin{picture}(40,33)\n\\put(15,33){\\textbf{SDN Control Layer, Sec.~\\ref{sdnctl:sec}}}\n\\put(-7,30){\\line(1,0){49}}\n\\put(21,30){\\line(0,1){2}}\n\\put(-7,30){\\vector(0,-1){2}}\n\\put(-9,27){\\makebox(0,0)[lt]{\\shortstack[l]{\t\t\t\n\\textbf{Ctl. of Infra. Comp.}, \\\\ \\ Sec.~\\ref{ctlinfra:sec}  \\\\ \\\\\n\t\tTransceiver Ctl.\\\\~\\cite{YuZZG14,Chen2014,JiXia2014,liu2013field}    \\\\\n\t\tCircuit Sw. Ctl. \\\\ \\cite{ChNS13,DaPMSG10,DaPM09}, \\\\\n\t\t\t\\cite{DaPM09Jul,OFCircuitSwitch,ChNF13,Baik2014} \\\\\n\t\tPkt. $+$ Burst Sw. Ctl.~\\\\ \\cite{Cao2015,Harai2014,liu2013field}\n\t\t}}}\n\t\\put(5,30){\\vector(0,-1){20}}\n\t\\put(-2.5,9){\\makebox(0,0)[lt]{\\shortstack[l]{\t\t\t\n              \\textbf{Retro-fitting Devices},\\\\\n                 Sec.~\\ref{retfitctl:sec} \\\\\n              \\cite{ChNF13,ChNS13,Cao2015,liu2013field},\\\\\n\t\t\\cite{Alvizu2014,LiTMG11,LiuTMGW11,LiCTM12,LiZTV12,ClSLT14}\n\t}}}\n\t\\put(15,30){\\vector(0,-1){2}}\n\t\\put(12,27){\\makebox(0,0)[lt]{\\shortstack[l]{\t\t\t\n              \\textbf{Ctl. of Opt. Netw. Ops.}, \\\\\n                  Sec.~\\ref{ctlops:sec} \\\\ \\\\\n\t\tPON Ctl.~\\cite{KhRS14,LeK15,PaP13,par2014fut,YaZWZ14,KaCTW13} \\\\\n\t\tSpectrum Defrag.~\\\\\\cite{Ger2012,YuZZG14,Chen2014},\\\\\n\t\t\\cite{Munoz2014a,Zhu2015b,Meloni2016} \\\\ \\\\\n\t\tTandem Netw., Sec.~\\ref{ctltan:sec} \\\\\n\t\tMetro$+$Access~\\cite{wu2014glo,zha2015sof}\t\\\\\t\n\t\tAccess$+$Wireless~\\cite{boj2013adv,TaC14,Costa2015}\t\\\\\t\n\t\tAccess$+$Metro$+$Core~\\cite{SlR14} \\\\\t\t\n\t\tDC~\\cite{MaHSC14}\t   \\\\\t\n\t\tIoT~\\cite{wan2015nov}\n\t}}}\n\t\\put(30,30){\\vector(0,-1){22}}\t\t\n\t\\put(25,7){\\makebox(0,0)[lt]{\\shortstack[l]{\t\t\t\n\t      \\textbf{Hybrid SDN-GMPLS}, \\\\\n               Sec.~\\ref{hybsdnctl:sec}  \\\\\n\t\t\\cite{AzNEJ11,Alvizu2014,Munoz2014a}, \\\\\n\t\t\\cite{LiCTMMM12,Casellas2013b}\n\t}}}\n\t\\put(42,30){\\vector(0,-1){2}}\n\t\\put(35,27){\\makebox(0,0)[lt]{\\shortstack[l]{\t\t\t\n\\textbf{Ctl. Perform.}, Sec.~\\ref{cltperf:sec} \\\\ \\\\\n\t\tSDN vs. GMPLS~\\cite{ChNF13}, \\\\\n\t\t\\cite{LiTM12,ZhaoZYY13,CviAPJT13} \\\\\n\t\tFlow Setup Time~\\cite{VeSBR14,liu2013field} \\\\\n\t\tOut of Band Ctl.~\\cite{Sanchez2013} \\\\\n\t\tClust. Ctl~\\cite{Penna2014}\n\t}}}\t\t\n\t\\end{picture}\t\n\\caption{Classification of SDON control layer studies.}\n                  \\label{ctl_class:fig}\n\\end{figure*}\n\\section{\\MakeUppercase{SDN Control Layer}}\n\\label{sdnctl:sec}\nThis section surveys the SDON studies that are focused on applying the\nSDN principles at the SDN control layer to control the various optical network\nelements and operational aspects.\nThe main challenges of SDON control include extensions of the OpenFlow\nprotocol for specifically controlling the optical transmission\nand switching components surveyed in Section~\\ref{sdninfra:sec}\nand for controlling the optical\nspectrum as well as for controlling optical networks spanning\nmultiple optical network tiers (see Section~\\ref{optnetstruct:sec}).\nAs illustrated in Fig.~\\ref{ctl_class:fig},\nwe first survey SDN control mechanisms and frameworks for controlling\ninfrastructure layer components, namely transceivers as well as\noptical circuit, packet, and burst switches.\nMore specifically, we survey OpenFlow extensions for controlling the\noptical infrastructure components.\nWe then survey mechanisms for retro-fitting non-SDN optical network elements\nso that they can be controlled by OpenFlow.\nThe retro-fitting typically involves the insertion of an abstraction layer\ninto the network elements. The abstraction layer\nmakes the optical hardware controllable by OpenFlow.\nThe retro-fitting studies would also fit into Section~\\ref{sdninfra:sec} as\nthe abstraction layer is inserted into the network elements; however,\nthe abstraction mechanisms closely relate to the OpenFlow extensions for\noptical networking and we include the retro-fitting studies therefore in this\ncontrol layer section.\nWe then survey the various SDN control mechanisms for operational aspects of\noptical networks, including the control\nof tandem networks that include optical segments. Lastly, we survey\nSDON controller performance analysis studies.\n\n\\subsection{SDN Control of Optical Infrastructure Components}\n\\label{ctlinfra:sec}\n\n\\subsubsection{Controlling Optical Transceivers with OpenFlow}\nRecent generations of optical transceivers utilize digital signal\nprocessing techniques that allow many parameters of the transceiver to\nbe software controlled (see Sections~\\ref{SF-BVT:sec} and~\\ref{MF-BVT:sec}).\nThese parameters include modulation scheme, symbol rate, and wavelength.\nYu et al.~\\cite{YuZZG14} and Chen et al.~\\cite{Chen2014} proposed adding a ``modulation format'' field to the OpenFlow cross-connect table entries to support this\nprogrammable feature of some software defined optical transceivers.\n\nJi et al.~\\cite{JiXia2014} created a testbed that places super-channel optical transponders and optical\namplifiers under SDN control. An OpenFlow extension is proposed to control these devices. The modulation\ntechnique and FEC code for each optical subcarrier of the super-channel transponder\nand the optical amplifier power level can be controlled via OpenFlow. Ji et al. do not discuss this explicitly\nbut the transponder subcarriers can be treated as OpenFlow switch ports that can be configured through the\nOpenFlow protocol via port modification messages. It is unclear in \\cite{JiXia2014} how the amplifiers would be\ncontrolled via OpenFlow. However, doing so would allow the SDN controller to adaptively modify amplifiers to\ncompensate for channel impairments while minimizing energy consumption. Ji et al.~\\cite{JiXia2014}\nhave established a testbed demonstrating the placement of transponders and EDFA optical amplifiers under SDN control.\n\nLiu et al.~\\cite{liu2013field} propose configuring optical transponder operation via flow table entries with\nnew transponder specific fields (without providing details). They also propose capturing failure alarms from optical\ntransponders and sending them to the SDN controller via OpenFlow Packet-In messages. These messages are normally\nmeant to establish new flow connections. Alternatively, a new OpenFlow message type could be created for the purpose\nof capturing failure alarms~\\cite{liu2013field}. With failure alarm information, the SDN controller can implement\nprotection switching services.\n\n\\subsubsection{Controlling Optical Circuit Switches with OpenFlow}\nCircuit switching can be enabled by OpenFlow by adding new circuit switching flow table\nentries~\\cite{DaPM09Jul,DaPM09,DaPMSG10,Baik2014}. The OpenFlow circuit switching addendum~\\cite{OFCircuitSwitch}\ndiscusses the addition of cross-connect tables for this purpose.\nThese cross-connect tables are configured via OpenFlow messages inside the circuit switches. According to the addendum, a cross-connect table entry consists of the following\nfields to identify the input:\n\\begin{itemize}\n  \\item Input Port\n  \\item Input Wavelength\n  \\item Input Time Slot\n  \\item Virtual Concatenation Group\n\\end{itemize}\nand the following fields to identify the output:\n\\begin{itemize}\n  \\item Output Port\n  \\item Output Wavelength\n  \\item Output Time Slot\n  \\item Virtual Concatenation Group\n\\end{itemize}\nThese cross-connect tables cover circuit switching in space, fixed-grid wavelength, and time.\n\nChannegowda et al.~\\cite{ChNF13,ChNS13} extend the capabilities of the OpenFlow circuit switching addendum to\nsupport flexible wavelength grid optical switching. Specifically, the wavelength identifier specified in the\ncircuit switching addendum to OpenFlow is replaced with two fields: \\textit{center frequency}, and\n\\textit{slot width}. The \\textit{center frequency} is an integer specifying the multiple of 6.25~GHz the center\nfrequency is away from 193.1~Thz and the \\textit{slot width} is a positive integer specifying the spectral\nwidth in multiples of 12.5~GHz.\n\nAn SDN controlled optical network testbed at the University of Bristol has been established to\ndemonstrate the OpenFlow extensions for flexible grid DWDM~\\cite{ChNS13}. The testbed consists of\nboth fixed-grid and flexible-grid optical switching devices.\nSouth Korea Telekom has also\nbuilt an SDN controlled optical network testbed~\\cite{Shin2014}.\n\n\\subsubsection{Controlling Optical Packet and Burst Switches with OpenFlow}\nOpenFlow flow tables can be utilized in\noptical packet switches for expressing the forwarding table and its computation can be offloaded to an SDN\ncontroller. This offloading can simplify the design of highly complex optical packet switches~\\cite{Cao2015}.\n\nCao et al.~\\cite{Cao2015} extend the OpenFlow protocol to work with Optical Packet Switching (OPS) devices by creating:\n$(i)$ an abstraction layer that converts OpenFlow configuration messages to the native OPS configuration, $(ii)$ a process that\nconverts optical packets that do not match a flow table entry to the electrical domain for forwarding to the SDN\ncontroller, and $(iii)$ a wavelength identifier extension to the flow table entries. To compensate for either the\nlack of any optical buffering or limited optical buffering, an SDN controller, with its global view, can provide more\neffective means to resolve contention that would lead to packet loss in optical packet switches.\nSpecifically, Cao et al. suggest to select the path with the most available resources among multiple available paths between two nodes~\\cite{Cao2015}.\nPaths can be re-computed periodically or on-demand to account for changes\nin traffic conditions. Monitoring messages can be defined to keep the SDN controller updated of network traffic\nconditions.\n\nEngineers with Japan's National Institute of Information and Communications Technology~\\cite{Harai2014}\nhave created an optical circuit and packet switched demonstration system in which the packet portion is SDN\ncontrolled. The optical circuit switching is implemented with Wavelength Selective Switches (WSSs) and the\noptical packet switching is implemented with an Semiconductor Optical Amplifier (SOA) switch.\n\nOpenFlow flow tables can also be used to configure optical burst switching devices~\\cite{liu2013field}. When there is\nno flow table entry for a burst of packets, the optical burst switching device can send the Burst Header Packet (BHP) to\nthe SDN controller to process the addition of the new flow to the network~\\cite{liu2013field} rather than the first\npacket in the burst.\n\n\n\\subsection{Retro-fitting Devices to Support OpenFlow}  \\label{retfitctl:sec}\nAn abstraction layer can be used to turn non-SDN optical switching\ndevices into OpenFlow controllable switching\ndevices~\\cite{ChNF13,ChNS13,liu2013field,Alvizu2014,Cao2015}.\nAs illustrated in Fig.~\\ref{fig:retrofit}, the\nabstraction layer provides a conversion layer between OpenFlow\nconfiguration messages and the optical switching devices' native\nmanagement interface, e.g., the Simple Network Management Protocol\n(SNMP), the Transaction Language 1 (TL1) protocol, or a proprietary\n(vendor-specific) API. Additionally, a virtual OpenFlow switch with\nvirtual interfaces that correspond to physical switching ports on the\nnon-SDN switching device completes the abstraction\nlayer~\\cite{LiTMG11,LiuTMGW11,LiCTM12,LiZTV12,liu2013field}. When a\nflow entry is added between two virtual ports in the virtual OpenFlow\nswitch, the abstraction layer uses the switching devices' native\nmanagement interface to add the flow entry between the two\ncorresponding physical ports.\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=3in]{finman_sdon\/fig7.pdf}\n    \\caption{Traditional non-SDN network elements can be retro-fitted for control by an SDN controller using OpenFlow using\n    a hardware abstraction layer~\\cite{LiTMG11,LiuTMGW11,LiCTM12,LiZTV12,liu2013field}.}\n    \\label{fig:retrofit}\n\\end{figure}\n\nA non-SDN PON OLT can be supplemented with a two-port OpenFlow switch and a hardware abstraction layer that\nconverts OpenFlow forwarding rules to control messages understood by the non-SDN OLT~\\cite{ClSLT14}. Fig.~\\ref{fig:oltretrofit} illustrates this OLT retro-fit for SDN control via OpenFlow. In this way the\nPON has its switching functions controlled by OpenFlow.\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=3.5in]{finman_sdon\/fig8.pdf}\n    \\caption{Non-SDN OLTs can be retro-fitted for control by an SDN controller using OpenFlow~\\cite{ClSLT14}.}\n    \\label{fig:oltretrofit}\n\\end{figure}\n\n\\subsection{SDN Control of Optical Network Operation}  \\label{ctlops:sec}\n\n\\subsubsection{Controlling Passive Optical Networks with OpenFlow}\n\\label{ponctl:sec}\nAn SDN controlled PON can be created by upgrading OLTs to SDN-OLTs that can be\ncontrolled using a Southbound Interface, such as OpenFlow~\\cite{KhRS14,LeK15}. A centralized PON controller,\npotentially executing in a data center, controls one or more SDN-OLTs. The advantage of using SDN is the\nbroadened perspective of the PON controller as well as the potentially reduced cost of the SDN-OLT\ncompared to a non-SDN OLT.\n\nParol and Pawlowski~\\cite{PaP13,par2014fut} define OpenFlowPLUS to extend the OpenFlow SBI for GPON. OpenFlowPLUS extends SDN programmability to both OLT and ONU devices whereby each act\nas an OpenFlow switch through a programmable flow table. Non-switching functions (e.g., ONU registration,\ndynamic bandwidth allocation) are outside the scope of OpenFlowPLUS. OpenFlowPLUS extends OpenFlow\nby channeling OpenFlow messages through the GPON ONU Management and Control Interface (OMCI) control channel and adding PON specific action\ninstructions to flow table entries. The PON specific action instructions defined in OpenFlowPLUS are:\n\\begin{itemize}\n  \\item (new \\textit{gpon} action type): map matching packets to PON specific traffic identifiers, e.g.,\n  GPON Encapsulation Method (GEM) ports and GPON Traffic CONTainers (T-CONTs)\n  \\item (\\textit{output} action type): activate PON specific framing of matching packets\n\\end{itemize}\n\nMany of the OLT functions operate at timescales that are problematic for the controller due\nto the latency between the controller and OLTs. However, Khalili et al.~\\cite{KhRS14} identify ONU\nregistration policy and coarse timescale DBA policy as functions that operate at timescales that allow\neffective offloading to an SDN controller. Yan et al.~\\cite{YaZWZ14} further identify OLT and ONU power\ncontrol for energy savings as a function that can be effectively offloaded to an SDN controller.\n\nThere is also a movement to use PONs in edge networks to provide\nconnectivity inside a multitenant building or on a campus with multiple\nbuildings~\\cite{PaP13,par2014fut}. The use of PONs in this edge scenario\nrequires rapid re-provisioning from the OLT. A software controlled PON\ncan provide this needed rapid reprovisioning~\\cite{PaP13,par2014fut}.\n\nKanonakis et al. \\cite{KaCTW13} propose leveraging the\nbroad perspective that SDN can provide to perform dynamic bandwidth\nallocation across several Virtual PONs (VPONs). The VPONs are\nseparated on a physical PON by the wavelength bands that they\nutilize. Bandwidth allocation is performed at the granularity of OFDMA\nsubcarriers that compose the optical spectrum.\n\n\n\\subsubsection{SDN Control of Optical Spectrum Defragmentation}\nIn a departure from the fixed wavelength grid\n(ITU-T G.694.1), elastic optical networking allows flexible use of the optical spectrum. This flexibility can permit higher spectral efficiency by avoiding consuming an entire fixed-grid\nwavelength channel when unnecessary and avoiding unnecessary guard bands in certain circumstances \\cite{Ger2012}.\nHowever, this flexibility causes fragmentation of the optical spectrum as flexible grid lightpaths are established\nand terminated over time.\n\nSpectrum fragmentation leads to the circumstance in which there is enough spectral capacity to satisfy a demand but\nthat capacity is spread over several fragments rather than being consolidated in adjacent spectrum as required. If\nthe fragmentation is not counter-acted by a periodic defragmentation process than overall spectral utilization will\nsuffer. This resource fragmentation problem appears in computer systems in main memory and long term storage. In those\ncontexts the problem is typically solved by allowing the memory to be allocated using non-adjacent segments. Memory and\nstorage is partitioned into pages and blocks, respectively. The allocations of pages to a process or blocks to a file do\nnot need to be contiguous. With communication spectrum this would mean combining multiple small bandwidth channels\nthrough inverse multiplexing to create a larger channel~\\cite{Munoz2014a}.\n\nAn SDN controller can provide a broad network perspective to empower the periodic optical spectrum defragmentation\nprocess to be more effective~\\cite{Munoz2014a}. In general, optical spectrum defragmentation operations can reduce\nlightpath blocking probabilities from 3\\%~\\cite{YuZZG14} up to as much as 75\\%~\\cite{Chen2014,Zhu2015b}. Multicore\nfibers provide additional spectral resources through additional transmission cores to permit quasi-hitless\ndefragmentation~\\cite{Meloni2016}.\n\n\\subsubsection{SDN Control of Tandem Networks}  \\label{ctltan:sec}\n\n\\paragraph{Metro and Access}\nWu et al.~\\cite{wu2014glo,zha2015sof} propose leveraging the broad perspective that SDN can provide to\nimprove bandwidth allocation. Two cooperating stages of SDN controllers: $(i)$ access stage that controls each SDN\nOLT individually, and $(ii)$ metro stage that controls global bandwidth allocation strategy, can coordinate bandwidth\nallocation across several physical PONs~\\cite{wu2014glo,zha2015sof}. The bandwidth allocation is managed cooperatively\namong the two stages of SDN controllers to optimize the utilization of the access and metro network bandwidth. Simulation\nexperiments indicate a 40\\% increase in network bandwidth utilization as a result of the global coordination compared\nto operating the bandwidth allocation only within the individual PONs~\\cite{wu2014glo,zha2015sof}.\n\n\\paragraph{Access and Wireless}\nBojic et al. \\cite{boj2013adv} expand on the concept of SDN controlled OFDMA enabled VPONs~\\cite{KaCTW13} to\nprovide mobile backhaul service. The backhaul service can be provided for wireless small-cell sites (e.g., micro and femto cells)\nthat utilize millimeter wave frequencies. Each small-cell site contains an OFDMA-PON ONU that provides the backhaul service\nthrough the access network over a VPON. An SDN controller is utilized to assign bandwidth to each small-cell site through\nOFDMA subcarrier assignment in a VPON to the constituent ONU. The SDN controller leverages its broad view of the network\nto provide solutions to the joint bandwidth allocation and routing across several network segments. With this broad\nperspective of the network, the SDN controller can make globally rather than just locally optimal bandwidth allocation and\nrouting decisions. Efficient optimization algorithms, such as genetic algorithms, can be used to provide computationally\nefficient competitive solutions, mitigating computational complexity issues associated with optimization for large networks.\nAdditionally, network partitioning with an SDN controller for each partition can be used to mitigate unreasonable\ncomputational complexity that arises when scaling to large networks. Tanaka and Cvijetic~\\cite{TaC14} presented one\nsuch optimization formulation for maximizing throughput.\n\nCosta-Requena et al.~\\cite{Costa2015} described a proof-of-concept LTE testbed they have constructed\nwhereby the network consists of software defined base stations and various network functions\nexecuting on cloud resources. The testbed is described in broad qualitative terms, no technical\ndetails are provided. There was no mathematical or experimental analysis provided.\n\n\\paragraph{Access, Metro, and Core}\nSlyne and Ruffini~\\cite{SlR14} provide a use case for SDN switching control across network segments: use Layer 2 switching\nacross the access, metro, and core networks. Layer 2 (e.g., Ethernet) switching does not scale well due to a lack of hierarchy\nin its addresses. That lack of hierarchy does not allow for switching rules on aggregates of addresses thereby limiting the\nscaling of these networks. Slyne and Ruffini~\\cite{SlR14} propose using SDN to create hierarchical pseudo-MAC addresses that\npermit a small number of flow table entries to configure the switching of traffic using Layer 2 addresses across network\nsegments. The pseudo-MAC addresses encode information about the device location to permit simple switching rules. At the entry\nof the network, flow table entries are set up to translate from real (non-hierarchical) MAC addresses to hierarchical pseudo-MAC\naddresses. The reverse takes place at the exit point of the network.\n\n\\paragraph{DC Virtual Machine Migration}\nMandal et al.~\\cite{MaHSC14} provided a cloud computing use case for SDN bandwidth allocation across network segments:\nVirtual Machine (VM) migration between data centers. VM migrations require significant network bandwidth. Bandwidth\nallocation that utilizes the broad perspective that SDN can provide is critical for reasonable VM migration latencies\nwithout sacrificing network bandwidth utilization.\n\n\\paragraph{Internet of Things}\nWang et al.~\\cite{wan2015nov} examine another use case for SDN bandwidth allocation across network segments: the Internet\nof Things (IoT). Specifically, Wang et al. have developed a Dynamic Bandwidth Allocation (DBA) protocol that exploits\nSDN control for multicasting and suspending flows. This DBA protocol is studied in the context of a virtualized\nWDM optical access network that provides IoT services through the distributed ONUs to individual devices.\nThe SDN controller employs multicasting and flow suspension to efficiently prioritize the IoT service requests.\nMulticasting allows multiple requests to share resources in the central nodes that are responsible for processing a\nprescribed wavelength in the central office (OLT). Flow suspension allows high-priority requests (e.g., an emergency\ncall) to suspend ongoing low-priority traffic flows (e.g., routine meter readings). Performance results for a real-time\nSDN controller implementation indicate that the proposed bandwidth (resource) allocation with multicast and flow\nsuspension can improve several key performance metrics, such as request serving ratio, revenue, and delays\nby 30--50~\\%~\\cite{wan2015nov}.\n\n\n\\subsection{Hybrid SDN-GMPLS Control}  \\label{hybsdnctl:sec}\n\\subsubsection{Generalized MultiProtocol Label Switching (GMPLS)}\nPrior to SDN, MultiProtocol Label Switching (MPLS) offered a mechanism to separate the control and data planes\nthrough label switching. With MPLS, packets are forwarded in a connection-oriented manner through Label Switched\nPaths (LSPs) traversing Label Switching Routers (LSRs). An entity in the network establishes an LSP through a network\nof LSRs for a particular class of packets and then signals the label-based forwarding table entries to the LSRs. At\neach hop along an LSP, a packet is assigned a label that determines its forwarding rule at the next hop. At the next\nhop, that label determines that packet's output port and label for the next hop; the process repeats until the packet\nreaches the end of the LSP. Several signalling protocols for programming the label-based forwarding table entries\ninside LSRs have been defined, e.g., through the Resource Reservation Protocol (RSVP). Generalized MPLS (GMPLS) extends MPLS to offer circuit switching\ncapability. Although never commercially deployed~\\cite{liu2013field}, GMPLS and a centralized Path Computation Element\n(PCE)~\\cite{mun2014pce,oki2005dyn,pao2013sur,Casellas2013} have been considered for control of optical networks.\n\n\n\\subsubsection{Path Computation Element (PCE)}  \\label{PCE:sec}\nA PCE is a concept developed by the IETF (see RFC 4655) to refer to an entity that computes network paths given a topology\nand some criteria. The PCE concept breaks the path computation action from the forwarding action in switching devices.\nA PCE could be distributed in every switching element in a network domain or there could be a single centralized PCE\nfor an entire network domain. The network domain could be an area of an Autonomous System (AS), an AS, a conglomeration\nof several ASes, or just a group of switching devices relying on one PCE. Some of an SDN controller's functionality\nfalls under the classification of a centralized PCE. However, the PCE concept does not include the external\nconfiguration of forwarding tables. Thus, a centralized PCE device does not necessarily have a means to configure the\nswitching elements to provision a computed path.\n\nWhen the entity requesting path computation is not co-located with the PCE, a PCE Communication Protocol (PCEP) is used\nover TCP port 4189 to facilitate path computation requests and responses. The PCEP consists of the following message\ntypes:\n\\begin{itemize}\n  \\item Session establishment messages (open, keepalive, close)\n  \\item PCReq -- Path computation request\n  \\item PCRep -- Path computation reply\n  \\item PCNtf -- event notification\n  \\item PCErr -- signal a protocol error\n\\end{itemize}\n\nThe path computation request message must include the end points of the path and can optionally include the requested\nbandwidth, the metric to be optimized in the path computation, and a list of links to be included in the path. The Path\ncomputation reply includes the computed path expressed in the Explicit Route Object format (see RFC 3209) or an\nindication that there is no path. See RFC 5440 for more details on PCEP.\n\nA PCE has been proposed as a central entity to manage a GMPLS-enabled optical circuit switched network. Specifically,\nthe PCE maintains the network topology in a structure called the Traffic Engineering Database (TED). The traffic\nengineering modifier (see RFC 2702) signifies that the path computations are made to relieve congestion that is caused\nby the sub-optimal allocation of network resources. This modifier is used extensively in discussions of MPLS\/GMPLS\nbecause their use case is for traffic engineering; in acronym form the modifier is TE (e.g., TE LSP, RSVP-TE).\n\nIf the PCE is stateful with complete control over its network domain, it will also maintain an LSP database recording the provisioned GMPLS lightpaths. A lightpath request can be sent to the PCE, it will use the topology and LSP\ndatabase to find the optimal path and then configure the GMPLS-controlled optical circuit switching nodes using NETCONF\n(see RFC 6241) or proprietary command line interfaces (CLIs)~\\cite{Munoz2014a}. This stateful PCE with instantiation\ncapabilities (capabilities to provision lightpaths) operates similarly to an SDN controller. For that reason, GMPLS with a\ncentralized stateful PCE with instantiation capabilities can provide a baseline for performance analysis of an SDN\ncontroller as well as provide a mechanism to be blended with an SDN controller for hybrid control~\\cite{ChNF13,ChNS13,Alvizu2014}.\n\n\\subsubsection{Approaches to Hybrid SDN-GMPLS Control}\nHybrid GMPLS\/PCE and SDN control can be formed by allowing an SDN controller to leverage a centralized PCE to control a\nportion of the infrastructure using PCEP as the SBI~\\cite{AzNEJ11,Munoz2014a}; see illustration a) in Fig.~\\ref{fig:hybridctl}.\nThe SDN controller builds higher functionality above what the PCE provides and can possibly control a large network that\nutilizes several PCEs as well as OpenFlow controlled network elements.\n\nAlternatively, the SDN controller can leverage a PCE for its path computation abilities with the SDN controller handling\nthe configuration of the network elements to establish a path using an SBI protocol, such as\nOpenFlow~\\cite{LiCTMMM12,Casellas2013b,Alvizu2014};\nsee illustration b) in Fig.~\\ref{fig:hybridctl}.\n\\begin{figure}[t!]\n\t\\centering\n\t\\includegraphics[width=5in]{finman_sdon\/fig9.pdf}\n    \\caption{Hybrid GMPLS\/PCE and SDN network control~\\cite{LiCTMMM12,Casellas2013b,Alvizu2014}.}\n\t\\label{fig:hybridctl}\n\\end{figure}\n\n\\subsection{SDN Performance Analysis}  \\label{cltperf:sec}\n\\subsubsection{SDN vs. GMPLS}\nLiu et al.~\\cite{LiTM12} provided a qualitative comparison of GMPLS,\nGMPLS\/PCE, and SDN OpenFlow for control of wavelength switched optical\nnetworks. Liu et al.~noted that there is an evolution of centralized\ncontrol from GMPLS to GMPLS\/PCE to OpenFlow. Whereas GMPLS offers\ndistributed control, GMPLS\/PCE is commonly regarded as having centralized path\ncomputation but still distributed provisioning\/configuration; while\nOpenFlow centralizes all of the network control. In\nour discussion in Section \\ref{hybsdnctl:sec} we noted that a\nstateful PCE with instantiation capabilities centralizes all\nnetwork control and is therefore very similar to SDN. Liu et al.~have\nalso pointed out that GMPLS\/PCE is more technically mature compared to\nOpenFlow with IETF RFCs for GMPLS (see RFC 3471) and PCE (see RFC\n4655) that date back to 2003 and 2006, respectively. SDN has just\nrecently, in 2014, received standardization attention from the IETF\n(see RFC 7149).\n\nA comparison of GMPLS and OpenFlow has been conducted by Zhao et al.~\\cite{ZhaoZYY13} for large-scale optical networks.\nTwo testbeds were built, based on GMPLS and on Openflow, respectively. Performance metrics, such as\nblocking probability, wavelength utilization, and lightpath setup time were evaluated for a 1000 node topology. The\nresults indicated that GMPLS gives slightly lower blocking probability. However, OpenFlow gives higher wavelength\nutilization and shorter average lightpath setup time. Thus, the results suggest that OpenFlow is overall advantageous\ncompared to GMPLS in large-scale optical networks.\n\nCvijetic et al.~\\cite{CviAPJT13} conducted a numerical analysis to compare the computed shortest path\nlengths for non-SDN, partial-SDN, and full-SDN optical networks. A full-SDN network enables path lengths\nthat are approximately a third of those computed on a non-SDN network. These path lengths can also translate\ninto an energy consumption measure, with shortest paths resulting in reduced energy consumption. An SDN\ncontrolled network can result in smaller computed shortest paths that translates to smaller network latency\nand energy consumption~\\cite{CviAPJT13}.\n\nExperiments conducted on the testbed described in \\cite{ChNF13} show a 4~\\% reduction in lightpath blocking probability\nusing SDN OpenFlow compared to GMPLS for lightpath provisioning. The same experiments show that lightpath setup times\ncan be reduced to nearly half using SDN OpenFlow compared to GMPLS. Finally, the experiments show that an Open vSwitch\nbased controller can process about three times the number of flows per second as a NOX~\\cite{gud2008nox} based controller.\n\n\\subsubsection{SDN Controller Flow Setup}\nVeisllari et al.~\\cite{VeSBR14} evaluated the use of SDN to support both circuit and packet\nswitching in a metropolitan area ring network that interconnects access network segments\nwith a backbone network. This network is assumed to be controlled by a single SDN controller.\nThe objective of the study~\\cite{VeSBR14} was to determine the effect of packet service flow\nsize on the required SDN controller flow service time to meet stability conditions at the\ncontroller. Toward this end, Veisllari et al.~produced a mean arrival rate function of new packet and\ncircuit flows at that controller. This arrival rate function was visualized by varying the length\nof short-lived (``mice'') flows, the fraction of long-lived (``elephant'') flows, and the volume\nof traffic consumed by ``elephant'' flows. Veisllari et al.~discovered, through these visualizations,\nthat the length of ``mice'' flows is the dominating parameter in this model.\n\nVeisllari et al.~translated the arrival rate function analysis to an analysis of the ring MAN network\ndimensions that can be supported by a single SDN controller. The current state-of-the-art Beacon\ncontroller can handle a flow request every 571~ns. Assuming mice flows sizes of 20~kB and average circuit\nlifetimes of 1 second, as the fraction of packet traffic increases from 0.1 to 0.9, the network\ndimension supported by a single Beacon SDN controller decreases from 14 nodes with 92 wavelengths per\nnode to 5 nodes with 10 wavelengths per node.\n\nLiu et al.~\\cite{liu2013field} use a multinational (Japan, China, Spain) NOX:OpenFlow controlled\nfour-wavelength optical circuit and burst switched network to study path setup\/release times as well as path\nrestoration times. The optical transponders that can generate failure alarms were also under NOX:OpenFlow\ncontrol and these alarms were used to trigger protection switching. The single SDN controller was located\nin the Japanese portion of the network. The experiments found the path setup time to vary from 250--600 ms and the path release times to vary from 130--450 ms. Path restoration times varied\nfrom 250--500~ms. Liu et al.~noted that the major contributing factor to these times was the OpenFlow message delivery time~\\cite{liu2013field}.\n\n\\subsubsection{Out of Band Control}\nSanchez et al.~\\cite{Sanchez2013} have qualitatively compared four SDN controlled ring metropolitan network architectures. The architectures vary in whether the SDN control traffic is carried\nin-band with the data traffic or out-of-band separately from the data traffic. In a single wavelength ring\nnetwork, out-of-band control would require a separate physical network that would come at a high cost,\nbut provide reliability of the network control under failure of the ring network. In a multiwavelength\nring network, a separate wavelength can be allocated to carry the control traffic. Sanchez et al.~\\cite{Sanchez2013}\nfocused on a Tunable Transceiver Fixed Receiver (TTFR) WDM ring node architecture. In this architecture each\nnode receives data on a home wavelength channel and has the capability to transmit on any of the available\nwavelengths to reach any other node. The addition of the out-of-band control channel on a separate wavelength\nrequires each node to have an additional fixed receiver, thereby increasing cost. Sanchez et al. identified a clear tradeoff between cost and reliability when comparing the four architectures.\n\n\\subsubsection{Clustered SDN Control}\nPenna et al.~\\cite{Penna2014} described partitioning a wavelength-switched optical network into administrative\ndomains or clusters for control by a single SDN controller. The clustering should meet certain\nperformance criteria for the SDN controller. To permit lightpath establishment across clusters, an inter-cluster\nlightpath establishment protocol is established. Each SDN controller provides a lightpath establishment function\nbetween any two points in its associated cluster. Each SDN controller also keeps a global view of the network\ntopology. When an SDN controller receives a lightpath establishment request whose computed path traverses\nother clusters, the SDN controller requests lightpath establishment within those clusters via a WBI.\n\nThe formation of clusters can be performed such that for a specified number of clusters the average distance to\neach SDN controller is minimized~\\cite{Penna2014}. The lightpath establishment time decreases exponentially\nas the number of clusters increases.\n\n\\subsection{Control Layer: Summary and Discussion}\nA very large body of literature has explored how to expand the OpenFlow protocol to support various optical network\ntechnologies (e.g., optical circuit switching, optical packet switching, passive optical networks). A significant\nbody of literature has investigated methodologies for retro-fitting non-SDN network elements for OpenFlow control\nas well as integrating SDN\/OpenFlow with the GMPLS\/PCE control framework. A variety of SDN controller use cases have\nbeen identified that motivate the benefits of the centralized network control made possible with SDN (e.g., bandwidth\nallocation over large numbers of subscribers, controlling tandem networks).\n\nHowever, analyzing the performance of SDN controllers for optical network applications is still in a state of infancy.\nIt will be important to understand the connection between the implementation of the SDN controller (e.g., processor\ncore architecture, number of threads, operating system) and the network it can effectively control (e.g., network\ntraffic volume, network size) to meet certain performance objectives (e.g., maximum flow setup time). At present there\nare not enough relevant studies to gain an understanding of this connection. With this understanding network service\nproviders will be able to partition their networks into control domains in a manner that meets their performance\nobjectives.\n\n\n\\begin{figure*}[t!]\n\\footnotesize\n\\setlength{\\unitlength}{0.10in}\n\\centering\n\\begin{picture}(40,33)\n\\put(15,33){\\textbf{Virtualization, Sec.~\\ref{virt:sec}}}\n\\put(-8,30){\\line(1,0){42}}\n\\put(25,23){\\line(1,0){20}}\n\\put(27,26.2){\\textbf{Metro\/Core Netw.}, Sec.~\\ref{virt_core:sec}}\n\\put(34,25){\\line(0,-1){2}}\n\\put(21,30){\\line(0,1){2}}\n\\put(-8,30){\\vector(0,-1){2}}\n\\put(-10,27){\\makebox(0,0)[lt]{\\shortstack[l]{\t\t\t\n\\textbf{Access Networks}, Sec.\\ref{virt_access:sec}\t\\\\\t\\\\\nOFDMA PON~\\cite{wei2009pon,wei2009pro,wei2010ada,li2016pro,jin2009vir,zhou2015dem}\\\\\t\t\nVirtualized FiWi~\\cite{QiZSH14,QiGYZ13,ShGYZ13,he2013int,men2014eff,QiZHG14}\\\\\t\t\nWiMAX-VPON~\\cite{dha2010wim,DhPX10}\n}}}\n\n\\put(9,30){\\vector(0,-1){20}}\n\\put(3,10){\\makebox(0,0)[lt]{\\shortstack[l]{\t\t\t\n\\textbf{Data Ctr. Netw.}, \\\\\nSec.~\\ref{virt_dc:sec} \\\\ \\\\\nLIGHTNESS~\\cite{miao2015sdn,pen2015mul,pag2015opt,Saridis2016} \\\\\nCloudnets~\\cite{kan2015res,ahm2014enh,dov2015usi,xie2014dyn,vel2014tow,zha2015ren,tza2014con}\n}}}\t\t\t\t\n\\put(34,30){\\vector(0,-1){2}}\n\\put(25,23){\\vector(0,-1){2}}\n\\put(19,20){\\makebox(0,0)[lt]{\\shortstack[l]{\n\\textbf{Virt. Opt. Netw. Embedding}, Sec.~\\ref{virt_emb:sec} \\\\\nImpairment-Aware Emb.~\\cite{pen2011imp,PeNS13} \\\\\nWDM\/Flexi-grid \\\\ Emb.~\\cite{zha2013net,zha2013vir,gon2014vir,wan2015vir,pag2015opt} \\\\\nSurvivable Emb.~\\cite{hu2013sur,ye2015sur,xie2014sur,che2016cos,jia2015ava,son2011ban,pao2014mul,ass2016net,Hong2015} \\\\\nDynamic Emb.~\\cite{ye2014upg,zha2015dyn} \\\\\nEnergy-efficient Emb.~\\cite{non2015ene,che2016pow,she2014fol,wan2014hie}\n}}}\t\t\t\t\t\t\t\n\\put(45,23){\\vector(0,-1){15}}\t\t\n\\put(30,7){\\makebox(0,0)[lt]{\\shortstack[l]{\t\t\t\n\\textbf{VON Hypervisors}, Sec.~\\ref{virt_hv:sec} \\\\ \\\\\nT-NOS~\\cite{siq2015pro} \\\\\nOpenSlice~\\cite{LiuMCTMM13} \\\\\nOpt. FV~\\cite{Azod2012}\n}}}\n\\end{picture}\t\n\\caption{Classification of SDON virtualization studies.}\n\\label{virt_class:fig}\n\\end{figure*}\n\\section{\\MakeUppercase{Virtualization}}  \\label{virt:sec}\nThis section surveys control layer mechanisms for virtualizing\nSDONs.\nAs optical infrastructures have typically high costs, creating\nmultiple VONs over the optical network infrastructure is\nespecially important for access networks, where the costs need to\nbe amortized over relatively few users. Throughout, accounting\nfor the specific optical transmission and signal propagation characteristics\nis a key challenge for SDON virtualization.\nFollowing the classification structure illustrated in\nFig.~\\ref{virt_class:fig},\nwe initially survey virtualization mechanisms for access networks\nand data center networks, followed by virtualization mechanisms for\noptical core networks.\n\n\\begin{figure*}[t!]\n\t\\centering\n\t\\vspace{0cm}\n\t\\begin{tabular}{c}\n\t\t\\includegraphics[width=3.5in]{finman_sdon\/fig11a.pdf} \\\\\n\t\t\\footnotesize{(a) Overall virtualization structure} \\\\\n\t\t\\includegraphics[width=4in]{finman_sdon\/fig11b.pdf} \\\\\t\t\n\t\t\\footnotesize{(b) Virtualization of OLT}\n\t\\end{tabular}\n\t\\vspace{0cm}\n\t\\caption{Illustration of OFDMA based virtual access\n    network~\\cite{wei2009pon}.}\n\t\\vspace{0cm}\n\t\\label{fig_poniard}\n\\end{figure*}\n\n\\subsection{Access Networks}  \\label{virt_access:sec}\n\\subsubsection{OFDMA Based PON Access Network Virtualization}\nWei et al.~\\cite{wei2009pon,wei2009pro,wei2010ada} have developed\na link virtualization mechanism that can span from\noptical access to backbone networks\nbased on Orthogonal Frequency Division Multiplexing Access (OFDMA).\nSpecifically, for access networks, a Virtual PON (VPON) approach\nbased on multicarrier OFDMA over WDM has been proposed. Distinct\nnetwork slices (VPONs) utilize distinct OFDMA subcarriers, which\nprovide a level of isolation between the VPONs. Thus, different\nVPONs may operate with different MAC standards, e.g., as illustrated\nin Fig.~\\ref{fig_poniard}(a), VPON A may operate as an Ethernet PON\n(EPON) while VPON~B operates as a Gigabit PON (GPON). In addition,\nvirtual MAC queues and processors are isolated to store\n  and process the data from multiple VPONs, thus creating virtual\nMAC protocols, as illustrated in Fig.~\\ref{fig_poniard}(b).\nThe OFDMA transmissions and receptions are processed in a\nDSP module that is controlled by a central SDN control module.\nThe central SDN control module also controls the different\nvirtual MAC processes in Fig.~\\ref{fig_poniard}(b), which feed\/receive\ndata to\/from the DSP module.\nAdditional bandwidth partitioning between VPONs can be achieved through\nTime Division Multiple Access (TDMA).\nSimulation studies compared a static allocation of subcarriers to\nVPONs with a dynamic allocation based on traffic demands.\nThe dynamic allocation achieved significantly higher numbers of\nsupported VPONs on a given network infrastructure as well as lower\npacket delays than the static allocation.\nA similar strategy for flexibly employing different dynamic\nbandwidth allocation modules for different groups of ONU queues has been\nexamined in~\\cite{li2016pro}.\n\nSimilar OFDMA based slicing strategies for supporting\ncloud computing have been examined by Jinno et al.~\\cite{jin2009vir}.\nZhou et al.~\\cite{zhou2015dem} have explored a FlexPON with similar\nvirtualization capabilities. The FlexPON employs OFDM for adaptive\ntransmissions. The isolation of different VPONs is\nmainly achieved through separate MAC processing.\nThe resulting VPONs allow for flexible port\nassignments in ONUs and OLT, which have been demonstrated\nin a testbed~\\cite{zhou2015dem}.\n\n\\subsubsection{FiWi Access Network Virtualization}  \\label{virt_fiwi:sec}\n\\paragraph{Virtualized FiWi Network}\nDai et al.~\\cite{QiZSH14,QiGYZ13,ShGYZ13} have examined the\nvirtualization of FiWi networks~\\cite{Bock2014,Effenberger2015}\nto eliminate the differences between\nthe heterogeneous segments (fiber and wireless). The virtualization\nprovides a unified homogenous (virtual) view of the FiWi network.\nThe unified network view simplifies flow control and other\noperational algorithms for traffic transmissions over the\nheterogeneous network segments. In particular, a virtual resource\nmanager operates the heterogeneous segments. The resource manager\npermits multiple routes from a given source node to a given\ndestination node. Load balancing across the multiple paths has been\nexamined in~\\cite{he2013int,men2014eff}. Simulation results indicate\nthat the virtualized FiWi network with load balancing significantly\nreduces packet delays compared to a conventional FiWi network. An\nexperimental OpenFlow switch testbed of the virtualized FiWi\nnetwork has been presented in~\\cite{QiZHG14}. Testbed measurements\ndemonstrate the seamless networking across the heterogeneous fiber\nand wireless networks segments. Measurements for nodal throughput,\nlink bandwidth utilization, and packet delay indicate performance\nimprovements due to the virtualized FiWi networking approach.\nMoreover, the FiWi testbed performance is measured for a video\nservice scenario indicating that the virtualized FiWi networking approach\nimproves the Quality of Experience\n(QoE)~\\cite{che2014qos,seu2014sur} of the video streaming. A\nmathematical performance model of the virtualized FiWi network has\nbeen developed in~\\cite{QiZHG14}.\n\n\\paragraph{WiMAX-VPON}\nWiMAX-VPON~\\cite{dha2010wim,DhPX10} is a Layer-2 Virtual Private Network\n(VPN) design for FiWi access networks.\nWiMAX-VPON executes a common MAC protocol across the\nwireless and fiber network segments.\nA VPN based admission control mechanism in conjunction with a VPN\nbandwidth allocation ensures per-flow Quality of Service (QoS).\nResults from discrete event simulations demonstrate that the\nproposed WiMAX-VPON achieves favorable performance.\nAlso, Dhaini et al.~\\cite{dha2010wim,DhPX10}\ndemonstrate how the WiMAX-VPON design can be extended\nto different access network types with polling-based wireless and optical\nmedium access control.\n\n\\subsection{Data Centers}  \\label{virt_dc:sec}\n\n\\subsubsection{LIGHTNESS}\nLIGHTNESS~\\cite{miao2015sdn,pen2015mul,pag2015opt,Saridis2016} is a\nEuropean research project examining an optical Data\nCenter Network (DCN) capable of providing dynamic,\nprogrammable, and highly available DCN connectivity services.\nWhereas conventional DCNs have rigid control and management\nplatforms, LIGHTNESS strives to introduce flexible control and\nmanagement through SDN control.\nThe LIGHTNESSS architecture comprises server racks that are\ninterconnected through optical packet\nswitches, optical circuit switches, and hybrid Top-of-the-Rack\n(ToR) switches. The server racks and switches  are all\ncontrolled and managed by an SDN controller.\nLIGHTNESS control consists of an SDN controller above the optical\nphysical layer and OpenFlow agents that interact with the optical\nnetwork and server  elements. The SDN controller in cooperation with\nthe OpenFlow-agents provides a programmable data plane to the\nvirtualization modules.\nThe virtualization creates  multiple\nVirtual Data Centers (VDCs), each with its own virtual computing\nand memory resources, as well as virtual networking resources, based on\na given physical data center.\nThe virtualization is achieved through a VDC planner module and an\nNFV application that directly interact with\nthe SDN controller.\nThe VDC planner composes the VDC slices through mapping of the VDC requests\nto the physical SDN-controlled switches and server racks.\nThe VDC slices are monitored by the NFV application, which interfaces\nwith the VDC planner. Based on monitoring data, the NFV application\nand VDC planner may revise the VDC composition, e.g., transition from\noptical packet switches to optical circuit switches.\n\n\\subsubsection{Cloudnets}\nCloudnets~\\cite{azo2013sdn,ban2013mer,bari2013data,FerLR13,WoRSV11,WoRVS10}\nexploit network virtualization for\npooling resources among distributed data centers. Cloudnets support\nthe migration of virtual machines across networks to achieve\nresource pooling. Cloudnet designs can be supported through\noptical networks~\\cite{ShNS13}.\nKantarci and Mouftah~\\cite{kan2015res} have examined\ndesigns for a virtual cloud backbone network that interconnects\ndistributed backbone nodes, whereby each backbone node is\nassociated with one data center. A network resource manager\nperiodically executes a virtualization algorithm to accommodate\ntraffic demands through appropriate resource provisioning. Kantarci\nand Mouftah~\\cite{kan2015res} have developed and evaluated algorithms for\nthree provisioning objectives: minimize the outage probability of\nthe cloud, minimize the resource provisioning, and minimize a\ntradeoff between resource saving and cloud outage probability. The\nrange of performance characteristics for outage probability,\nresource consumption, and delays of the provisioning approaches have\nbeen evaluated through simulations. The outage probability of\noptical cloud networks has been reduced in~\\cite{ahm2014enh} through\noptimized service re-locations.\n\nSeveral complementary aspects of optical cloudnet networks have\nrecently been investigated.\nA multilayer network architecture with an SDN based\nnetwork management structure for cloud services has been\ndeveloped in~\\cite{dov2015usi}.\nA dynamic variation of the sharing of optical network resources\nfor intra- and inter-data center networking has been examined\nin~\\cite{xie2014dyn}.\nThe dynamic sharing does not statically assign optical network resources\nto virtual optical networks; instead, the network resources are\ndynamically assigned according to the time-varying traffic demands.\nAn SDN based optical transport mode for data center traffic has been\nexplored in~\\cite{vel2014tow}.\nVirtual machine migration mechanisms that take the characteristics\nof renewable energy into account have been examined in~\\cite{zha2015ren}\nwhile general energy efficiency mechanisms for optically networked\ncould computing resources have been examined in~\\cite{tza2014con}.\n\n\n\\subsection{Metro\/Core Networks}  \\label{virt_core:sec}\n\n\\subsubsection{Virtual Optical Network Embedding}  \\label{virt_emb:sec}\nVirtual optical network embedding seeks to map requests for virtual\noptical networks to a given physical optical network infrastructure (substrate).\nA virtual optical network consists of both a set of virtual\nnodes and a set of interconnecting links that need to\nbe mapped to the network substrate.\nThis mapping of virtual networks consisting of both\nnetwork nodes and links is fundamentally different from the\nextensively studied virtual topology design for optical wavelength\nrouted networks~\\cite{dut2000sur}, which only considered network links\n(and did not map nodes).\nVirtual network embedding of both nodes and link has already been\nextensively studied in general network graphs~\\cite{fis2013vir,rah2013svn}.\nHowever, virtual optical network embedding requires additional\nconstraints to account for the special optical transmission characteristics,\nsuch as the wavelength continuity constraint and the transmission reach\nconstraint.\nConsequently, several studies have begun to examine virtual network\nembedding algorithms specifically for optical networks.\n\n\\paragraph{Impairment-Aware Embedding}\nPeng et al.~\\cite{pen2011imp,PeNS13} have modeled the optical\ntransmission impairments to facilitate the embedding of isolated\nVONs in a given\nunderlying physical network infrastructure.\nSpecifically, they model the physical (photonic)\nlayer impairments of both single-line\nrate and mixed-line rates~\\cite{nag2010opt}.\nPeng et al.~\\cite{PeNS13} consider intra-VON impairments\nfrom Amplified Spontaneous Emission (ASE) and inter-VON impairments\nfrom non-linear impairments and four wave mixing.\nThese impairments are captured in a $Q$-factor~\\cite{azo2009sur,sar2009phy},\nwhich is considered in the mapping of\nvirtual links to the underlying physical link resources,\nsuch as wavelengths and wavebands.\n\n\\paragraph{Embedding on WDM and Flexi-grid Networks} \\label{wdm_flexi_emb:sec}\nZhang et al.~\\cite{zha2013net} have considered the embedding of\noverall virtual networks encompassing both virtual nodes and virtual links.\nZhang et al. have considered both conventional WDM networks as well as\nflexi-grid networks.\nFor each network type, they formulate the virtual node and virtual link\nmapping as a mixed integer linear program.\nConcluding that the mixed integer\nlinear program is NP-hard, heuristic solution approaches\nare developed. Specifically, the overall embedding (mapping) problem\nis divided into a node mapping problem and a link mapping problem.\nThe node mapping problem is heuristically solved through a greedy\nMinMapping strategy that maps the largest computing resource demand to the\nnode with the minimum remaining computing capacity (a complementary\nMaxMapping strategy that maps the largest demand to the\nnode with the maximum remaining capacity is also considered).\nAfter the node mapping, the link mapping problem is solved\nwith an extended grooming graph~\\cite{zhu2003nov}.\nComparisons for a small network indicate that\nthe MinMapping strategy approaches the optimal mixed integer linear program\nsolution quite closely;\nwhereas the MaxMapping strategy gives poor results.\nThe evaluations also indicate that the flexi-grid network requires only about\nhalf the spectrum compared to an equivalent WDM network for several\nevaluation scenarios.\n\nThe embedding of virtual optical networks in the context of\nelastic flexi-grid optical networking has been further examined in several\nstudies.\nFor a flexi-grid network based on OFDM~\\cite{zha2013surflexi},\nZhao et al.~\\cite{zha2013vir} have compared a greedy heuristic that maps\nrequests in decreasing order of the required resources with an arbitrary\nfirst-fit benchmark.\nGong et al.~\\cite{gon2014vir} have considered flexi-grid networks with\na similar overall strategy of node mapping followed by link mapping\nas Zhang et al.~\\cite{zha2013net}.\nBased on the local resource constraints at each node,\nGong et al.~have formed a layered auxiliary graph for the node mapping.\nThe link mapping is then solved with a shortest path routing approach.\nWang et al.~\\cite{wan2015vir} have examined an embedding approach\nbased on candidate mapping patterns that could provide the requested\nresources. The VON is then embedded according to a shortest path\nrouting.\nPages et al.~\\cite{pag2015opt} have considered embeddings that\nminimize the required optical transponders.\n\nche2016cos\n\n\\paragraph{Survivable Embedding}  \\label{surv_emb:sec}\nSurvivability of a virtual optical network, i.e., its continued\noperation in the face of physical node or link failures, is\nimportant for many applications that require dependable service. Hu\net al.~\\cite{hu2013sur} developed an embedding that can survive the\nfailure of a single physical node. Ye et al.~\\cite{ye2015sur} have\nexamined the embedding of virtual optical networks so as to survive\nthe failure of a single physical node or a physical link.\nSpecifically, Ye et al. ensure that each virtual\nnode request is mapped to a primary physical node as well as a\ndistinct backup physical node. Similarly, each virtual link is\nmapped to a primary physical route as well as a node-disjoint backup\nphysical route. Ye et al. mathematically formulate an optimization\nproblem for the survivable embedding and then propose a Parallel\nVirtual Infrastructure (VI) Mapping (PAR) algorithm. The PAR\nalgorithm finds distinct candidate physical nodes (with the highest\nremaining resources) for each virtual node request. The candidate\nphysical nodes are then jointly examined with pairs of\n shortest node-disjoint paths.\nThe evaluations in~\\cite{ye2015sur} indicate that the parallel\nPAR algorithm reduces the blocking probabilities of virtual network requests\nby 5--20~\\% compared to a sequential algorithm benchmark.\nA limitation of the survivable embedding~\\cite{ye2015sur} is that it\nprotects only from a single link or node failure.\nAs the optical infrastructure is expected to penetrate\ndeeper in the access network deployments (e.g., mobile backhaul),\nit will become necessary to consider multiple failure points.\nSimilar survivable network embedding algorithms that employ node-disjoint\nshortest paths in conjunction with specific cost metrics for\nnode mappings have been investigated by Xie et al.~\\cite{xie2014sur}\nand Chen et al.~\\cite{che2016cos}.\nJiang et al.~\\cite{jia2015ava} have examined a solution variant based\non maximum-weight maximum clique formation.\n\nThe studies~\\cite{son2011ban,pao2014mul,ass2016net} have examined\nso-called bandwidth squeezed restoration for virtual topologies.\nWith bandwidth squeezing, the back-up path\nbandwidths of the surviving virtual topologies are generally lower than\nthe bandwidths on the working paths.\n\nSurvivable virtual topology design in the context of multidomain optical\nnetworks has been studied by Hong et al.~\\cite{Hong2015}.\nHong et al.~focused on minimizing the total network link cost\nfor a given virtual traffic demand. A heuristic algorithm\nfor partition and contraction\nmechanisms based on cut set theory has been proposed for the\nmapping of virtual links onto multidomain optical networks.\nA hierarchical SDN control plane is split between local controllers\nthat to manage individual domains and a\nglobal controller for the overall management.\nThe partition and contraction\nmechanisms abstract inter- and intra-domain information as a\nmethod of contraction. Survivability conditions are ensured\nindividually for inter- and intra-domains such that survivability is\nmet for the entire network.\nThe evaluations in~\\cite{Hong2015} demonstrate successful virtual\nnetwork mapping at the scale required by commercial Internet\nservice providers and infrastructure providers.\n\n\n\\paragraph{Dynamic Embedding}\nThe embedding approaches surveyed so far have mainly focused on the\noffline embedding of a static set of virtual network requests.\nHowever, in the ongoing network operation the dynamic embedding of\nmodifications (upgrades) of existing virtual networks, or the\naddition of new virtual networks are important. Ye et\nal.~\\cite{ye2014upg} have examined a variety of strategies for\nupgrading existing virtual topologies. Ye et al. have considered\nboth scenarios without advance planning (knowledge) of virtual\nnetwork upgrades and scenarios that plan ahead for possible\n(anticipated) upgrades. For both scenarios, a divide-and-conquer\nstrategy and an integrate-and-cooperate strategy are\nexamined. The divide-and conquer strategy sequentially maps all the\nvirtual nodes\nand then the virtual links. In contrast, the integrate-and-cooperate\nstrategy jointly\nconsiders the virtual node and virtual link mappings. Without\nadvance planning, these strategies are applied sequentially, as the\nvirtual network requests arrive over time, whereas, with planning,\nthe initial and upgrade requests are jointly considered. Evaluation\nresults indicate that the integrate-and-cooperate strategy\nslightly increases a revenue\nmeasure and request acceptance ratio compared to the divide-and-conquer\nstrategy.\nThe results also indicate that planning has the potential to\nsubstantially increase the revenue and acceptance ratio. In a\nrelated study, Zhang et al.~\\cite{zha2015dyn} have examined\nembedding algorithms for virtual network requests that arrive\ndynamically to a multilayer network consisting of electrical and\noptical network substrates.\n\n\\paragraph{Energy-efficient Embedding}\nMotivated by the growing importance of green networking and information\ntechnology~\\cite{BiPCR12}, a few studies have begun to consider\nthe energy efficiency of the embedded virtual optical networks.\nNonde et al.~\\cite{non2015ene} have developed and evaluated\nmechanisms for embedding virtual cloud networks so as to minimize\nthe overall power consumption, i.e., the aggregate of the\npower consumption for communication and computing (in the data centers).\nNonde et al. have incorporated the power consumption of the\ncommunication components, such as transponders and optical switches,\nas well as the power consumption characteristics of data center servers\ninto a mathematical power minimization model.\nNonde et al. then develop a real-time heuristic for energy-optimized\nvirtual network embedding.\nThe heuristic strives to consolidate computing requests in the\nphysical nodes with the least residual computing capacity.\nThis consolidation strategy is motivated by the typical\npower consumption characteristic of a compute server that\nhas a significant idle power consumption and then grows linearly with\nincreasing computing load; thus a fully loaded server is more\nenergy-efficient than a lightly loaded server.\nThe bandwidth demands are then routed between the nodes according to\n a minimum hop algorithm.\nThe energy optimized embedding is compared with a cost optimized\n embedding that only seeks to minimize the number of utilized wavelength\n channels.\n The evaluation results in~\\cite{non2015ene} indicate\n that the energy optimized embedding significantly reduces the overall energy\n consumption for low to moderate loads on the physical infrastructure;\n for high loads, when all physical resources need to be utilized,\n there are no significant savings. Across the entire load range, the\n energy optimized embedding saves on average 20~\\% energy compared to the\n benchmark minimizing the wavelength channels.\n\nChen~\\cite{che2016pow} has examined a similar energy-efficient\nvirtual optical network embedding that considers primary and\nlink-disjoint backup paths, similar to the survivable embeddings in\nSection~\\ref{surv_emb:sec}. More specifically, virtual link requests\nare mapped in decreasing order of their bandwidth requirements to\nthe shortest physical transmission distance paths, i.e., the highest\nvirtual bandwidth demands are allocated to the shortest physical\npaths. Evaluations indicate that this link mapping approach roughly\nhalves the power consumption compared to a random node mapping\nbenchmark. Further studies focused on energy savings have examined\nvirtual link embeddings that maximize the usage of nodes with\nrenewable energy~\\cite{she2014fol} and the traffic\ngrooming~\\cite{wan2014hie} onto sliceable BVTs~\\cite{zha2015ene}.\n\n\\subsubsection{Hypervisors for VONs} \\label{virt_hv:sec}\nThe operation of VONs over a given\nunderlying physical (substrate) optical network requires an\nintermediate hypervisor. The hypervisor presents the physical\nnetwork as multiple isolated VONs to the corresponding VON\ncontrollers (with typically one VON controller per VON). In turn,\nthe hypervisor intercepts the control messages issued by a VON\ncontroller and controls the physical network to effect the control\nactions desired by the VON controller for the corresponding VON.\n\nTowards the development of an optical network hypervisor, Siquera et\nal.~\\cite{siq2015pro} have developed a SDN-based controller for an\noptical transport architecture. The controller implements a\nvirtualized GMPLS control plane with offloading to facilitate the\nimplementation of hypervisor functionalities, namely the creation\noptical virtual private networks, optical network slicing, and\noptical interface management. A major contribution of Siquera et\nal.~\\cite{siq2015pro} is a Transport Network Operating System\n(T-NOS), which abstracts the physical layer for the controller and\ncould be utilized for hypervisor functionalities.\n\nOpenSlice~\\cite{LiuMCTMM13} is a comprehensive\nOpenFlow-based hypervisor that creates VONs over underlying elastic\noptical networks~\\cite{cha2015rou,tal2014spe}. OpenSlice\ndynamically provisions end-to-end paths and offloads IP traffic by\nslicing the optical communications spectrum. The paths are set up\nthrough a handshake protocol that fills in cross-connection table\nentries. The control messages for slicing the optical communications\nspectrum, such as slot width and modulation format, are carried in\nextended OpenFlow protocol messages. OpenSlice relies on special\ndistributed network elements, namely bandwidth variable wavelength\ncross-connects~\\cite{jin2009spe} and multiflow optical\ntransponders~\\cite{jin2012mul} that have been extended for control\nthrough the extended OpenFlow messages. The OpenSlice evaluation\nincludes an experimental demonstration. The evaluation results\ninclude path provisioning latency comparisons with a GMPLS-based\ncontrol plane and indicate that OpenFlow outperforms GMPLS for paths\nwith more than three hops.\nOpenSlice extension and refinements to\nmultilayer and multidomain networks are surveyed in\nSection~\\ref{orch:sec}.\nAn alternate centralized Optical FlowVisor that does not require\nextensions to the distributed network elements has been investigated\nin~\\cite{Azod2012}.\n\n\\subsection{Virtualization: Summary and Discussion}\nThe virtualization studies on access\nnetworks~\\cite{wei2009pon,wei2009pro,wei2010ada,jin2009vir,zhou2015dem,QiZSH14,QiGYZ13,ShGYZ13,he2013int,men2014eff,QiZHG14,dha2010wim,DhPX10} have primarily\nfocused on exploiting and manipulating the specific properties of the\noptical physical layer (e.g., different OFDMA subcarriers)\nand MAC layer (e.g., polling based MAC protocol) of the optical\naccess networks for virtualization.\nIn addition, to virtualization studies on\npurely optical PON access networks, two sets of studies, namely\nsets~\\cite{QiZSH14,QiGYZ13,ShGYZ13,he2013int,men2014eff,QiZHG14}\nand WiMAX-VPON~\\cite{dha2010wim,DhPX10} have examined\nvirtualization for two forms of FiWi access networks.\nFuture research needs to consider virtualization of a wider set of\nFiWi network technologies, i.e., FiWi networks that consider\noptical access networks with a wider variety of wireless access\ntechnologies, such as different forms of cellular access or combinations\nof cellular with other forms of wireless access.\nAlso, virtualization of integrated access and metropolitan area\nnetworks~\\cite{ahm2012rpr,seg2007all,val2015exp,woe2013sdn} is\nan important future research direction.\n\nA set of studies has begun to explore optical networking support for\nSDN-enabled cloudnets that exploit virtualization to\ndynamically pool resources across distributed data centers.\nOne important direction for\nfuture work on cloudnets is to examine moving data center resources\ncloser to the users and the subsequent resource pooling across edge\nnetworks~\\cite{man2013clo}. Also, the exploration of the benefits of\nFiWi networks for decentralized\ncloudlets~\\cite{din2013sur,sat2009cas,ScSF12,ver2012clo} that\nsupport mobile wireless network services is an important future\nresearch direction~\\cite{mai2015inv}.\n\n\\begin{figure*}[t!]\n\t\\footnotesize\n\t\\setlength{\\unitlength}{0.10in}\n\t\\centering\n\t\\begin{picture}(40,33)\n\t\\put(15,33){\\textbf{Applications, Sec.~\\ref{sdnapp:sec}}}\n\t\\put(-6,30){\\line(1,0){51}}\n\t\\put(21,30){\\line(0,1){2}}\n\t\n\t\\put(-6,30){\\vector(0,-1){2}}\n\t\\put(-9,27){\\makebox(0,0)[lt]{\\shortstack[l]{\t\t\t\n\\textbf{QoS}, Sec.~\\ref{app_qos:sec} \\\\ \\\\\n\t\t\t\tLong-term QoS~\\cite{ZhZZ13,kho2016qua} \\\\\n\t\t\t\tShort-term QoS~\\cite{Li2014,PatelJiWang2013} \\\\\n\t\t\t\tVirt. Top. Reconfig.~~\\cite{WetteKarl2013} \\\\\n\t\t\t\tQoS Routing~\\cite{Tariq2015,SgPCVC13,Ilchmann2015,ChangLi2015} \\\\\n\t\t\t\tQoS Management~\\cite{RuBRK14,TeMAD14} \\\\\n\t\t\t\tVideo Appl.~\\cite{chi2015app,Chitimalla2015,Li2014video}\n\t\t\t}}}\n\t\t\t\n\t\t\\put(10,30){\\vector(0,-1){17}}\n\t\t\\put(5,12){\\makebox(0,0)[lt]{\\shortstack[l]{\t\t\n\\textbf{Access Control and} \\\\\n\\textbf{Security}, Sec.~\\ref{app_secur:sec}\\\\ \\\\  \t\n\t\t\t\tFlow-based Access \\\\ Ctl.~\\cite{MaGMT14,nay2009res}\t\\\\\t\t\t\n\t\t\t\tLightpath Hopping Sec.~\\cite{Li2016c}\t\\\\\t\t\t\n\t\t\t\tFlow Timeout~\\cite{ZhuFan2015}\n\t\t\t}}}\n\t\t\t\t\t\n\t\t\\put(31,30){\\vector(0,-1){2}}\n\t\t\\put(23,27){\\makebox(0,0)[lt]{\\shortstack[l]{\t\t\t\n\\textbf{Energy Eff.}, Sec.~\\ref{ene_eff:sec}\t\\\\ \\\\  \t\t\t\n\t\t\t\tAppl. Ctl.~\\cite{Ji2014,yan2013mul,yan2015per,yan2016per}\t\t\t\\\\\t\n\t\t\t\tRouting~\\cite{Tego2014,Wang2015,Yevsieieva2015,Yoon2015,val2015exp}\n\t\t\t}}}\n\t\n\t\t\\put(45,30){\\vector(0,-1){17}}\t\t\n\t\t\\put(33,12){\\makebox(0,0)[lt]{\\shortstack[l]{\t\t\t\n\\textbf{Failure Recov. $+$} \\\\\n\\textbf{Restoration}, Sec.~\\ref{app_fail_rec:sec} \\\\ \\\\\n\t\t\t\tNetw. Reprov.~\\cite{sav2015bac} \\\\\n\t\t\t\tRestoration~\\cite{Giorgetti2015} \\\\\n\t\tReconfig.~\\cite{Aguado2016,aib2016sof,SlKMPR14,Kim2015} \\\\\n\t\t\t\tHierarchical Surv.~\\cite{ZhangSong2014} \\\\\n\t\t\t\tRobust Power Grid~\\cite{Rastegarfar2016}\n\t\t\t}}}\n\t\t\t\t\t\t\t\t\t\n\\end{picture}\t\n\\caption{Classification of application layer SDON studies.}\n\\label{app_class:fig}\n\\end{figure*}\nA fairly extensive set of studies has examined virtual network embedding\nfor metro\/core networks.\nThe virtual network embedding studies have considered the\nspecific limitations and constraints of optical networks and have\nbegun to explore specialized embedding strategies that strive to\nmeet a specific optimization objective,\nsuch as survivability, dynamic adaptability,\nor energy efficiency.\nFuture research should seek to develop a comprehensive framework of\nembedding algorithms that can be tuned with weights to achieve\nprescribed degrees of the different optimization objectives.\n\nA relatively smaller set of studies has developed and refined hypervisors\nfor creating VONs over metro\/core optical networks.\nMuch of the SDON hypervisor research has centered on the OpenSlice\nhypervisor concept~\\cite{LiuMCTMM13}. While OpenSlice accounts\nfor the specific characteristics of the optical transmission medium,\nit is relatively complex as it requires a distributed implementation\nwith specialized optical networking components.\nFuture research should seek to achieve the hypervisor functionalities\nwith a wider set of common optical components so as to reduce cost\nand complexity.\nOverall, SDON hypervisor research should examine the performance-complexity\/cost\ntradeoffs of distributed versus centralized approaches.\nWithin this context of examining the spectrum of distributed to\ncentralized hypervisors,\nfuture hypervisor research should further refine and optimize the\nvirtualization mechanisms so as to achieve strict isolation between\nvirtual network slices, as well as low-complexity hypervisor\ndeployment, operation, and maintenance.\n\n\n\\section{\\MakeUppercase{SDN Application Layer}}  \\label{sdnapp:sec}\nIn the SDN paradigm, applications interact with the controllers to\nimplement network services. We organize the survey of the studies on\napplication layer aspects of SDONs according to the main application\ncategories of quality of service (QoS), access control and security,\nenergy efficiency, and failure recovery, as illustrated in\nFig.~\\ref{app_class:fig}.\n\n\\subsection{QoS}  \\label{app_qos:sec}\n\\subsubsection{Long-term QoS: Time-Aware SDN}\nData Center (DC) networks move data back and forth between\nDCs to balance the computing load and the data storage usage\n(for upload)~\\cite{DeCusatis2014}.\nThese data movements between DCs can span large geographical areas and help\nensure DC service QoS for the end users.\nLoad balancing algorithms can exploit the characteristics\nof the user requests.\nOne such request characteristic is the high degree of time-correlation over\nvarious time scales\nranging from several hours of a day (e.g., due to a sporting event)\nto several days in a year (e.g., due to a political event).\nZhao et al.~\\cite{ZhZZ13} have proposed a time-aware SDN application using\nOpenFlow extensions to dynamically balance the load across the\nDC resources so as to improve the QoS.\nSpecifically, a time correlated PCE algorithm based on flexi-grid optical\ntransport (see Section~\\ref{PCE:sec})\nhas been proposed. An SDN application monitors\nthe DC resources and applies network rules to preserve the QoS.\nEvaluations of the algorithm indicate improvements\nin terms of network blocking probability, global blocking probability, and\nspectrum consumption ratio.\nThis study did not consider short time scale traffic bursts,\nwhich can significantly affect the load conditions.\n\nWe believe that in order to avoid pitfalls in the operation of load balancing\nthrough PCE algorithms implemented with SDN, a wide range of\ntraffic conditions needs to be considered.\nThe considered traffic range should include short and long term traffic\nvariations, which should be traded off with various QoS aspects,\nsuch as type of application\nand delay constraints, as well as the resulting costs and control overheads.\nKhodakarami et al.~\\cite{kho2016qua} have taken steps in this direction by\nforming a traffic forecasting model for both long-term and short-term forecasts\nin a wide-area mesh network.\nOptical lightpaths are then configured based on the overall traffic forecast,\nwhile electronic switching capacities are allocated based on short-term\nforecasts.\n\n\\begin{figure}[t!]\n    \\centering\n    \\vspace{0cm}\n    \\includegraphics[width=4in]{finman_sdon\/fig13.pdf}\n    \\caption{Optical SDN-based QoS-aware burst switching\n      application~\\cite{PatelJiWang2013}.}\n    \\label{fig_app_qos}\n\\end{figure}\n\\subsubsection{Short Term QoS}\nUsers of a high-speed FTTH access network may request\nvery large bandwidths\ndue to simultaneously running applications that require high data rates.\nIn such a scenario,\napplications requiring very high data rates may affect each other.\nFor instance, a video conference running simultaneously with\nthe streaming of a sports video may result in\ncall drops in the video conference application and\nin stalls of the sports video.\nLi et al.~\\cite{Li2014} proposed an SDN based bandwidth provisioning\napplication in the broadband remote access server~\\cite{Dietz2015}\nnetwork. They defined and assigned the minimum bandwidth, which they named\n``sweet point'', required\nfor each application to experience good QoE.\nLi et al. showed that maintaining the ``sweet point'' bandwidth for each\napplication can significantly improve the QoE while other\napplications are being served\naccording to their bandwidth requirements.\n\nIn a similar study, Patel et al.~\\cite{PatelJiWang2013} proposed\na burst switching mechanism based on a software\ndefined optical network. Bursts typically originate\nat the edge nodes and the aggregation\npoints due to statistical multiplexing of high speed\noptical transmissions. To ensure QoS for multiple traffic classes,\nbursts at the edge nodes have to be managed by deciding their end-to-end\npath to meet their QoS requirements, such as minimum delay and data rate.\nIn non-SDN based mechanisms, complicated\ndistributed protocols, such as GMPLS~\\cite{mun2014pce,pao2013sur},\nare used to route the burst traffic.\nIn the proposed application,\nthe centralized unified control plane decides the routing\npath for the burst based on latency and QoS requirements.\nA simplified procedure involves $(i)$ burst evaluation at the edge node,\n$(ii)$ reporting burst information to the SDN controller, and\n$(iii)$ sending of configurations to the optical nodes by the controller\nto set up a lightpath as illustrated in Fig.~\\ref{fig_app_qos}.\nSimulations indicate an increase of performance in terms of\nthroughput, network blocking probability, and latency\nalong with improved QoS when compared to non-SDN GMPLS methods.\n\n\\subsubsection{Virtual Topology Reconfigurations}\nThe QoS experienced by traffic flows greatly depends on their\nroute through a network.\nWette et al.~\\cite{WetteKarl2013} have examined an application algorithm that\nreconfigures WDM network virtual topologies\n(see Section~\\ref{wdm_flexi_emb:sec})\naccording to the traffic levels.\nThe algorithm considers the localized traffic information and\noptical resource availability at the nodes.\nThe algorithm does not require synchronization,\nthus reducing the overhead while simplifying the network design.\nIn the proposed architecture, optical switches are connected to ROADMs.\nThe reconfiguration application manages\nand controls the optical switches through the SDN controller.\nA new WDM controller is introduced to configure the lightpaths taking\n wavelength conversion and lightpath switching at the ROADMs into consideration.\nThe SDN controller operates on the optical network which appears as a\nstatic network, while the WDM controller configures (and re-configures)\nthe ROADMs to create\nmultiple virtual optical networks according to the traffic levels.\nEvaluation results indicate improved utilization and throughput.\nThe results indicate that virtual topologies reconfigurations can\nsignificantly increase the flexibility of the network while achieving\nthe desired QoS.  However, the control overhead and the delay aspects\ndue to virtualization and separation of control and lightwave paths\nneeds to be carefully considered.\n\n\n\\subsubsection{End-to-End QoS Routing}\nInterconnections between DCs involve typically\nmultiple data paths. All the interfaces existing\nbetween DCs can be utilized by MultiPath TCP (MPTCP).\nEnsuring QoS in such an MPTCP setting\nwhile preserving throughput efficiency in a\nreconfigurable underlying burst switching optical network is a\nchallenging task. Tariq et al.~\\cite{Tariq2015} have\nproposed QoS-aware bandwidth reservation for\nMPTCP in an SDON.\nThe bandwidth reservation proceeds in two stages\n$(i)$ path selection for MPTCP, and\n$(ii)$ OBS wavelength reservation to assign the priorities for\nlatency-sensitive flows. Larger portions of a wavelength reservation are\nassigned to high priority flows, resulting in reduced\nburst blocking probability\nwhile achieving the higher MPTCP throughput.\nThe simulation results in~\\cite{Tariq2015} validate the two-stage algorithm\nfor QoS-aware  MPTCP over an SDON, indicating decreased dropping probabilities,\nand increased throughputs.\n\n\\begin{figure}[t!]\n    \\centering\n    \\vspace{0cm}\n    \\includegraphics[width=3.5in]{finman_sdon\/fig14.pdf}\n    \\vspace{0cm}\n    \\caption{Illustration of routing application.}  \n   \\vspace{0cm}\n    \\label{fig_app_i2rs}\n\\end{figure}\nInformation To the Routing System (I2RS)~\\cite{I2RSietf} is a high-level\narchitecture for communicating and interacting with routing systems,\nsuch as BGP routers.\nA routing system may consists of several complex\nfunctional entities, such as a Routing Information Base (RIB),\nan RIB manager, topology and policy databases,\nalong with routing and signalling units.\nThe I2RS provides a programmability platform that enables\naccess and modifications of the configurations of the routing system elements.\nThe I2RS can be extended with SDN principles to achieve global network\nmanagement and reconfiguration~\\cite{har2013sof}.\nSgambelluri et al.~\\cite{SgPCVC13} presented an SDN based routing application\nwithin the I2RS framework to integrate the control of the\naccess, metro, and core networks as illustrated in Fig.~\\ref{fig_app_i2rs}.\nThe SDN controller communicates with the Path Computation\nElements (PCEs) of the core network\nto create Label Switched Paths (LSPs)\nbased on the information received by the OLTs.\nExperimental demonstrations validated the routing optimization\nbased on the current traffic status and previous load as well as\nthe unified control interface for access, metro, and core networks.\n\nIlchmann et al.~\\cite{Ilchmann2015} developed an SDN application that communicates to an SDN controller via\nan HTTP-based REST API. Over time, lightpaths in an optical network can become inefficient for a number of\nreasons (e.g., optical spectrum fragmentation). For this reason, Ilchmann et al. developed an SDN application\nthat evaluates existing lightpaths in an optical network and offers an application user the option to\nre-optimize the lightpath routing to improve various performance metrics (e.g., path length). The application\nis user-interactive in that the user can see the number of proposed lightpath routing changes before they are made\nand can potentially select a subset of the proposed changes to minimize network down-time.\n\nAt the ingress and egress routers of optical networks (e.g., the\nedge routers between access and metro networks), buffers are highly\nnon-economical to implement, as they require large buffers sizes to\naccommodate the channel rates of 40~Mb\/s or more. To reduce the buffer\nrequirements at the edge routers, Chang et al.~\\cite{ChangLi2015}\nhave proposed a backpressure application referred to as Refill and\nSDN-based Random Early Detection (RS-RED).  RS-RED implements a\nrefill queue at the ingress device and a droptail\nqueue at the egress device, whereby both queues are centrally\nmanaged by the RS-RED algorithm running on the SDN\ncontroller. Simulation results showed that at the expense of small delay\nincreases, edge router buffer sizes can be significantly reduced.\n\n\\subsubsection{QoS Management}\nRukert et al.~\\cite{RuBRK14} proposed SDN based controlled home-gateway\nsupporting heterogeneous wired technologies, such as DSL, and\nwireless technologies, such as LTE and WiFi.\nSDN controllers managed by the ISPs optimize the traffic flows\nto each user while accommodating large numbers of users  and ensuring their\nminimum QoS.\nAdditionally, Tego et al.~\\cite{TeMAD14} demonstrated an experimental\nSDN based QoS management setup to optimize the energy utilization.\nGbE links are switched on and off based on the traffic levels.\nThe QoS management reroutes the traffic to avoid congestion and achieve\nefficient throughput. SDN applications conduct active QoS\nprobing to monitor the network QoS characteristics.\nEvaluations have indicated that the SDN based techniques\nachieve significantly higher throughput than non-SDN techniques~\\cite{TeMAD14}.\n\n\\subsubsection{Video Applications}\nThe application-aware SDN-enabled resource allocation\napplication has been introduced by Chitimalla et al.~\\cite{chi2015app}\nto improve the video QoE in a PON access network.\nThe resource allocation application\nuses application level feedback to schedule the optical\nresources. The video resolution is incrementally increased or\ndecreased based on the buffer utilization statistics that the client\nsends to the controller. The scheduler at the OLT schedules the\npackets based on weights calculated by the SDN controller, whereby\nthe video applications at the clients communicate with the\ncontroller to determine the weights. If the network is congested,\nthen the SDN controller communicates to the clients to reduce the\nvideo resolution so as to reduce the stalls and to improve the QoE.\n\n\\begin{figure}[t!]\n    \\centering\n    \\vspace{0cm}\n    \\includegraphics[width=5in]{finman_sdon\/fig15.pdf}\n    \\caption{SDN based video caching application in PON\n        for mobile users~\\cite{Li2014video}.}\n    \\label{fig_app_video} \n\\end{figure}\nCaching of video data close the users is generally beneficial for\nimproving the QoE of video services~\\cite{Ahlgren2012,Choi2012}.\nLi et al.~\\cite{Li2014video} have introduced\ncaching mechanisms for software-defined PONs.\nIn particular, Li et al.~have proposed\njoint provisioning of the bandwidth to service the video and\nthe cache management, as illustrated in Fig.~\\ref{fig_app_video}.\nBased on the request frequency for specific\nvideo content, the Base Station (BS) caches the content with the\nassistance of the SDN controller.\nThe proposed \\textit{push}-based mechanism delivers (pushes) the video\nto the BS caches when the PON is not congested.\nA specific PON transmission sub-band can be used to multicast video\ncontent that needs to be cached at multiple BSs.\nThe simulation evaluation in~\\cite{Li2014video} indicate\nthat up to 30\\% additional videos can be serviced while the service\nresponse delay is reduced to 50\\%.\n\n\n\\subsection{Access Control and Security} \\label{app_secur:sec}\n\\subsubsection{Flow-based Access Control}\nNetwork Access Control (NAC) is a networking application that\nregulates the access to network services~\\cite{cas2007eth,par2014fut}.\nA NAC based on traffic flows has been developed by Matias~\\cite{MaGMT14}.\nFlow-NAC exploits the forwarding rules of OpenFlow switches, which are set by a\ncentral SDN controller, to control the access of traffic flows to\nnetwork services.\nFlowNAC can implement the access control based on various flow identifiers,\nsuch as MAC addresses or IP source and destination addresses.\nPerformance evaluations measured\nthe connections times for flows on a testbed and found average connection\ntimes on the order of 100~ms for completing the flow access control.\n\nIn a related study, Nayak et al.~\\cite{nay2009res} developed the\nResonance flow based access control system for an enterprise\nnetwork. In the Resonance system, the network elements, such as the\nrouters themselves, dynamically enforce access control policies. The\naccess control policies are implemented through real-time alerts and\nflow based information that is exchanged with SDN principles. Nayak\net al. have demonstrated the Resonance system on a production\nnetwork at Georgia Tech. The Resonance design can be readily\nimplemented in SDON networks and can be readily extended to wide\narea networks. Consider for example multiple heterogeneous DCs of\nmultiple organizations that are connected by an optical backbone\nnetwork. The Resonance system can be extended to provide access\ncontrol mechanisms, such as authentication and authorization,\nthrough such a wide area SDON.\n\n\\subsubsection{Lightpath Hopping Security}\n\\begin{figure}[t!]\n    \\centering     \\vspace{0cm}\n    \\includegraphics[width=3.5in]{finman_sdon\/fig16.pdf} \n    \\caption{Overview of optical light path hopping mechanism~\\cite{Li2016c}.} \n    \\label{fig_app_hop}\n\\end{figure}\nThe broad network perspective of SDN controllers facilitates\nthe implementation of security functions that require this broad\nperspective~\\cite{ahm2015sec, sco2015sur, ShJJJ14}.\nHowever, SDN may also be vulnerable to a wide range of attacks and\nvulnerabilities, including unauthorized access, data leakage,\ndata modification, and misconfiguration.\nEavesdropping and jamming are security threats on the physical layer\nand are especially relevant for the optical layer of SDONs.\nIn order to prevent\neavesdropping and jamming in an optical lightpath, Li et al.~\\cite{Li2016c}\nhave proposed an SDN based fast lightpath hopping mechanism.\nAs illustrated in Fig.~\\ref{fig_app_hop}, the hopping mechanism operates\nover multiple lightpath channels. Conventional optical\nlightpath setup times range from several hundreds of milliseconds\nto several seconds and would result in a very low hopping frequency.\nTo avoid the\noptical setup times during each hopping period, an SDN based\nhigh precision time synchronization has been proposed.\nAs a result, a fast hopping\nmechanism can be implemented and executed in a coordinated manner. A hop frame\nis defined and guard periods are added in between hop frames.\nThe experimental evaluations indicate that a maximum hopping frequency\nof~1~MHz can be achieved with a BER of $1 \\times 10^{-3}$.\nHowever, shortcomings of such mechanisms are the secure exchange of hopping\nsequences between the transmitter and\nthe receiver. Although, centralized SDN control provides authenticated\nprovisioning of the hopping sequence, additional mechanisms to secure\nthe hopping sequence from\nbeing obtained through man-in-the-middle attacks should be investigated.\n\n\\subsubsection{Flow Timeout}\nSDN flow actions on the forwarding and switching elements have generally a\nvalidity period.\nUpon expiration of the validity period, i.e., the flow action timeout,\nthe forwarding or switching\nelement drops the flow action from the forwarding information base or the flow\ntable. The switching\nelement  CPU must be able to access the flow action information with very\nlow latency so as to perform switching actions at the line rate.\nTherefore, the flow actions are commonly stored in Ternary Content\nAddressable Memories (TCAMs)~\\cite{Pagiamtzis2006}, which are limited\nto storing on the order of thousands of distinct entries.\nIn SDONs, the optical network elements perform\nthe actions set by the SDN controller. These actions have to be stored in\na finite memory space.  Therefore, it is important to utilize the finite\nmemory space as efficiently as\npossible~\\cite{Bull2015, Liang2015, ngu2016rul, Xie2014d, Zhang2015timeout}.\nIn the dynamic timeout approach~\\cite{ZhuFan2015},\nthe SDN controller tracks the TCAM occupancy levels in the switches and adjusts\ntimeout durations accordingly.\nHowever, a shortcoming of such techniques\nis that the bookkeeping processes at the SDN controllers can become\ncumbersome for a large network.  Therefore, autonomous\ntimeout management techniques that are implemented at the hypervisors\ncan reduce the controller processing load and are\nan important future research direction.\n\n\\subsection{Energy Efficiency}  \\label{ene_eff:sec}\nThe separation of the control plane from the data plane\nand the global network perspective are unique advantages of SDN for\n improving the energy efficiency of networks,\nwhich is an important goal~\\cite{tuc2011gre,zha2010ene}.\n\n\\subsubsection{Power-saving Application Controller}\nJi et al.~\\cite{Ji2014} have proposed an all optical\nenergy-efficient network centered around an application\ncontroller~\\cite{yan2013mul,yan2015per} that monitors power consumption\ncharacteristics and enforces power savings policies.\nJi et al. first\nintroduce energy-efficient variations of Digital-to-Analog\nConverters (DACs) and wavelength selective ROADMs as components for\ntheir energy-efficient network. Second, Jie et al. introduce\nan energy-efficient switch architecture that consists of multiple\nparallel switching planes, whereby each plane consists of three\nstages with optical burst switching employed in the second (central)\nswitching stage.\nThird, Jie et al. detail a multilevel SDN based control architecture\nfor the network built from the introduced components and switch.\nThe control structure accommodates multiple networks domains,\nwhereby each network domain can involve multiple switching\ntechnologies, such as time-based and frequency-based optical switching.\nAll controllers for the various domains and technologies are placed\nunder the control of an application controller.\nDedicated power monitors that are distributed throughout the network\nupdate the SDN based application controller about the energy consumption\ncharacteristics of each network node.\nBased on the received energy consumption updates,\nthe application controller executes power-saving strategies.\nThe resulting control actions are signalled by the application\ncontroller to the various controllers for the different network domains\nand technologies.\nAn extension of this multi-level architecture to cloud-based\nradio access networks has been examined in~\\cite{yan2016per}.\n\n\\subsubsection{Energy-Saving Routing}\nTego et al.~\\cite{Tego2014} have proposed an energy-saving\napplication that switches off under-utilized GbE network links.\nSpecifically, Tego et al. proposed two methods: Fixed Upper Fixed\nLower (FUFL) and Dynamic Upper and Fixed Lower (DLFU). In FUFL, the\nIP routing and the connectivity of the logical topology are\n\\textit{fixed}. The utilization of physical GbE links (whereby\nmultiple parallel physical links form a logical link) is compared\nwith a threshold to determine whether to switch off or on individual\nphysical links (that support a given logical link). The traffic on a\nphysical link that is about to be switched off is rerouted on a\nparallel physical GbE link (within the same logical link). In\ncontrast, in the DLFU approach, the energy saving application\nmonitors the load levels on the virtual links. If the load level on\na given virtual link falls below a threshold value, then the virtual\nlink topology is reconfigured to eliminate the virtual link with the\nlow load. A general pitfall of such link switch-off techniques is\nthat energy savings may be achieved at the expense of deteriorating\nQoS. The QoS should therefore be closely monitored when switching\noff links and re-routing flows.\n\nA similar SDN based routing strategy that strives to save energy while\npreserving the QoS has been examined in the context of a GMPLS\noptical networks in~\\cite{Wang2015}.\nMultipath routing optimizing applications that strive to save\nenergy in an SDN based transport optical network have been presented\nin~\\cite{Yevsieieva2015}.\nA similar SDN based optimization approach\nfor reducing the energy consumption in\ndata centers has been examined by Yoon et al.~\\cite{Yoon2015}.\nYoon et al. formulated a mixed integer linear program that models the switches\nand hosts as queues. Essentially, the optimization decides on the\nswitches and hosts that could be turned off.\nAs the problem is NP-hard,\nannealing algorithms are examined.\nSimulations indicate that energy savings of more than 80\\% are possible for\nlow data center utilization rates, while the energy savings decrease to less\nthan 40\\% for high data center utilization rates.\nTraffic balancing in the metro optical access networks through the\nSDN based reconfiguration of optical subscriber units\nin a TWDM-PON systems for energy\nsavings has been additionally demonstrated in~\\cite{val2015exp}.\n\n\\subsection{Failure Recovery and Restoration}  \\label{app_fail_rec:sec}\n\\subsubsection{Network Reprovisioning}\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=3.5in]{finman_sdon\/fig17.pdf}\n    \\caption{Illustration of application layer\n        for disaster aware networking~\\cite{sav2015bac}.}\n    \\label{fig_app_reprov}\n\\end{figure}\nNetwork disruptions can occur due to various natural and\/or man-made\nfactors.\nNetwork resource reprovisioning is a process to\nchange the network configurations, e.g., the network topology and routes,\nto recover from failures.\nA Backup Reprovisioning with Path Protection (BRPP), based on SDN for\noptical networks has been presented by Savas et al.~\\cite{sav2015bac}.\nAn SDN application framework as illustrated in\nFig.~\\ref{fig_app_reprov} was designed to support the\nreprovisioning with services, such\nas provisioning the new connections, risk assessment, as well as service level\nand backup management.\nWhen new requests are received by the BRPP application\nframework, the statistics module evaluates the network state to find the\nprimary path and a link-disjoint backup path.  The computed backup\npaths are stored as logical links without being\nprovisioned on the physical network. The logical backup module manages\nand recalculates the logical links when a new backup path cannot be\naccommodated or to optimize the existing backup paths (e.g., minimize the\nbackup path distance). Savas et al.~introduce a degraded backup path mechanism\nthat reserves not the full, but a lower (degraded) transmission capacity\non the backup paths, so as to accommodate more requests.\nEmulations of the proposed mechanisms indicate improved\nnetwork utilization while effectively provisioning the backup paths\nfor restoring the network after network failures.\n\nAs a part of DARPA's core\noptical networks CORONET project, a non-SDN based Robust Optical Layer\nEnd-to-end X-connection (ROLEX)\nprotocol has been demonstrated and presented along with the lessons\nlearned~\\cite{VonLehmen2015}.\nROLEX is a distributed protocol for failure recovery\nwhich requires a considerable amount of signaling between nodes\nfor the distributed management.\nTherefore to avoid the pitfall of excessive signalling,\nit may be worthwhile to examine a ROLEX version with\ncentralized SDN control in future research\nto reduce the recovery time and\nsignaling overhead, as well as the costs of restored\npaths while ensuring the user QoS.\n\n\\subsubsection{Restoration Processing}\nDuring a restoration, the network control plane simultaneously triggers backup\nprovisioning of all disrupted paths.\nIn GMPLS restoration, along with signal flooding,\nthere can be contention of signal messages at the network nodes.\nContentions may arise due to spectrum conflicts of the lightpath,\nor node-configuration overrides, i.e., a new configuration request arrives\nwhile a preceding reconfiguration is under way.\nGiorgetti et al.~\\cite{Giorgetti2015} have proposed\ndynamic restoration in the elastic optical\nnetwork to avoid signaling contention in SDN (i.e., of OpenFlow messages).\nTwo SDN restoration mechanisms were presented:\n$(i)$ the independent restoration scheme (SDN-ind),\nand $(ii)$ the bundle restoration scheme (SDN-bund).\nIn SDN-ind, the controller triggers simultaneous independent\nflow modification (Flow-Mod) messages for each backup path\nto the switches involved in the reconfigurations.\nDuring contention, switches enqueue the multiple received Flow-Mod messages\nand process them sequentially. Although SDN-ind achieves reduced\nrecovery time as compared to non-SDN GMPLS, the waiting of messages in\nthe queue incurs a delay. In SDN-bund,\nthe backup path reconfigurations are bundled into a single message,\ni.e., a Bundle Flow-Mod message, and sent to each involved switch.\nEach switch then configures the flow modifications in\none reconfiguration, eliminating the delay incurred by the queuing of\nFlow-Mod messages. A similar OpenFlow enabled restoration\nin Elastic Optical Networks (EONs) has been studied in~\\cite{Liu2015d}.\n\n\\subsubsection{Reconfiguration}\n\\begin{figure}[t!]\n    \\centering\n    \\vspace{0cm}\n    \\includegraphics[width=3in]{finman_sdon\/fig18.pdf}\n    \\vspace{0cm}\n    \\caption{Illustration of Application-Based Network Operation (ABNO)\n         architecture~\\cite{Aguado2016}.}\n    \\label{fig_app_abno}\n\\end{figure}\nAguado et al.~\\cite{Aguado2016} have demonstrated\na failure recovery mechanism as part of the EU FP7 STRAUSS project with\ndynamic virtual reconfigurations using SDN.\nThey considered multidomain hypervisors and\ndomain-specific controllers to virtualize the multidomain networks.\nThe Application-Based Network Operations (ABNO) framework\nillustrated in Fig.~\\ref{fig_app_abno} enables network automation\nand programmability.\nABNO can compute end-to-end optical paths and delegate the\nconfigurations to lower layer domain SDN controllers.\nRequirements for fast recovery from network failures\nwould be in the order of tens of milliseconds, which is challenging to\nachieve in large scale networks. ABNO reduces the recovery\ntimes by pre-computing the backup connections after the first failure,\nwhile the Operation, Administration and Maintenance (OAM)\nmodule~\\cite{Paolucci2015}\ncommunicates with the ABNO controller to configure the new end-to-end\nconnections in response to a failure alarm. Failure alarms are triggered\nby the domain SDN controllers monitoring the traffic via the optical\npower meters when power is below $-20$~dBm.\nIn order to ensure survivability, an adaptive survivability scheme that\ntakes routing as well as spectrum assignment and modulation into\nconsideration has been explored in~\\cite{aib2016sof}.\n\nA similar design for end-to-end protection and failure\nrecovery has been demonstrated by Slyne et al.~\\cite{SlKMPR14} for a long-reach\n(LR) PON. LR-PON failures are highly\nlikely due to physical breaks in the long feeder fibers. Along with the\nhigh impact of connectivity break down or degraded service,\nphysical restoration time can be very long. Therefore, 1:1 protection for\nLR-PONs based on SDN has been proposed, where primary and secondary (backup)\nOLTs are used without traffic duplication.\nMore specifically, Slyne et al. have\ndevised and demonstrated an OpenFlow-Relay located at the switching unit.\nThe OpenFlow-Relay\ndetects and reports a failure along with fast updating of forwarding rules.\nExperimental demonstration show the backup OLT carrying protected traffic\nwithin $7.2$ ms after a failure event.\n\nAn experimental demonstration utilizing multiple paths in optical\ntransport networks for failure recovery has been discussed by\nKim et al.~\\cite{Kim2015}.\nKim et al. have used commercial grade IP WDM network equipment and\nimplemented multipath TCP\nin an SDN framework to emulate inter-DC communication.\nThey developed an SDN application, consisting of an\ncross-layer service manager module and a\ncross-layer multipath transport module to\nreconfigure the optical paths\nfor the recovery from connection impairments.\nTheir evaluations show increased bandwidth\nutilization and reduced cost while being resilient to network impairments\nas the cross-layer multipath transport module does not reserve the backup\npath on the transport network.\n\n\\subsubsection{Hierarchical Survivability}\nNetworks can be made survivable by introducing resource redundancy.\nHowever, the cost of the network increases with increased redundancy.\nZhang et al.~\\cite{ZhangSong2014} have demonstrated a highly survivable\nIP-Optical multilayered transport network.\nHierarchal controllers are placed for multilayer\nresource provisioning. Optical nodes are controlled by\nTransport Controllers (TCs), while higher\nlayers (IP) are controlled by unified controllers (UCs).\nThe UCs communicate with the TCs\nto optimize the routes based on cross-layer information. If a fiber causes a\nservice disruption, TCs may directly set up alternate routes or\nask the UCs for optimized routes. A pitfall of such\nhierarchical control techniques can be long restoration times. However,\nthe cross layer restorations\ncan recover from high degrees of failures, such as\nmultipoint and concurrent failures.\n\n\n\\subsubsection{Robust Power Grid}\nThe lack of a reliable communication infrastructure for power grid\nmanagement was one the many reasons for the widespread blackout in\nthe Northeastern U.S.A. in the year 2003, which affected the lives\nof 50 million people~\\cite{Parandehgheibi2014}. Since then building\na reliable communication infrastructure for the power grid has\nbecome an important priority. Rastegarfar et\nal.~\\cite{Rastegarfar2016} have proposed a communication\ninfrastructure that is focused on monitoring and can react to and recover from\nfailures so as to reliably support power grid applications.\nMore specifically, their\narchitecture was built on SDN based optical networking for\nimplementing robust power grid control applications. Control and\ninfrastructure in the SDN based power grid management exhibits an\ninterdependency i.e., the physical fiber relies on the control plane\nfor its operations and the logical control plane relies on the\nsame physical fiber for its signalling communications. Therefore,\nthey only focus on optical protection switching instead of IP layer\nprotection, for the resilience of the SDN control. Cascaded failure\nmechanisms were modeled and simulated for two geographical\ntopologies (U.S. and E.U.). In addition, the impacts of cascaded\nfailures were studied for two scenarios $(i)$ static optical layer\n(static OL), and $(ii)$ dynamic optical layer (dynamic OL). Results\nfor a static OL illustrated that the failure cascades are persistent\nand are closely dependent on the network topology. However, for a\ndynamic OL (i.e., with reconfiguration of the physical layer),\nfailure cascades were suppressed by an average of 73\\%.\n\n\\subsection{Application Layer: Summary and Discussion}\nThe SDON QoS application studies have mainly examined traffic and\nnetwork management mechanisms that are supported through the\nOpenFlow protocol and the central SDN controller.\nThe studied SDON QoS applications are structurally very similar in\nthat the traffic conditions or\nnetwork states (e.g., congestion levels) are probed\nor monitored by the central SDN controller.\nThe centralized knowledge of the traffic and network is then utilized\nto allocate or configure resources, such as DC resources in~\\cite{ZhZZ13},\napplication bandwidths in~\\cite{Li2014}, and\ntopology configurations or routes\nin~\\cite{WetteKarl2013,Tariq2015,SgPCVC13,ChangLi2015}.\nFuture research\n on SDON QoS needs to further optimize the interactions of the\ncontroller with the network applications and data plane\nto quickly and correctly react to changing user demands and\nnetwork conditions, so as to assure consistent QoS.\nThe specific characteristics and requirements\nof video streaming applications have\nbeen considered in the few studies on video\nQoS~\\cite{chi2015app,Chitimalla2015,Li2014video}.\nFuture SDON QoS research should consider a wider range of\nspecific prominent application traffic types with\nspecific characteristics and requirements, e.g., Voice over IP (VoIP)\ntraffic has relatively low bit rate requirements, but requires low\nend-to-end latency.\n\n\\begin{figure*}[t!]\n\t\\centering\n\t\\includegraphics[width=5in]{finman_sdon\/fig19.pdf}\n\t\\caption{Illustration of SDN orchestration of multilayer networking.}\n\t\\label{fig_control_multilay}\n\\end{figure*}\nVery few studies have considered security and access control for SDONs.\nThe thorough study of the broad topic area of security and privacy is\nan important future research direction in SDONs, as outlined in\nSection~\\ref{futworksec:sec}\nEnergy efficiency is similarly a highly important topic\nwithin the SDON research area that has received relatively little\nattention so far and presents overarching research challenges,\nsee Section~\\ref{futworkene:sec}.\n\nOne common theme of the SDON application layer studies focused\non failure recovery and restoration has been to exploit\nthe global perspective of the SDN control.\nThe global perspective has been exploited for\nfor improved planning of the recovery and\nrestoration~\\cite{sav2015bac,Aguado2016,ZhangSong2014} as well as\nfor improved coordination of the execution of the restoration\nprocesses~\\cite{Giorgetti2015,Liu2015d}.\nGenerally, the existing failure recovery and restoration studies\nhave focused on network (routing) domain that is owned by a particular\norganizational entity. Future research should seek to examine the\ntradeoffs when exploiting the\nglobal perspective of orchestration of multiple routing domains, i.e.,\nthe failure recovery and restoration techniques surveyed in this section\ncould be combined with the multidomain orchestration techniques\nsurveyed in Section~\\ref{orch:sec}.\nOne concrete example of multidomain orchestration could be to coordinate\nthe specific LR-PON access network protection and\nfailure recovery~\\cite{SlKMPR14}\nwith protection and recovery techniques for metropolitan and core\nnetwork domains, e.g.,~\\cite{sav2015bac,Aguado2016,Kim2015,ZhangSong2014},\nfor improved end-to-end protection and recovery.\n\n\\section{\\MakeUppercase{Orchestration}}  \\label{orch:sec}\nAs introduced in Section~\\ref{intro:orch:sec}, orchestration\naccomplishes higher layer abstract coordination of network services\nand operations. In the context of SDONs, orchestration has mainly\nbeen studied in support of multilayer networking.\nMultilayer networking in the context of SDN and network virtualization\ngenerally refers to networking across multiple network layers\nand their respective technologies, such as\nIP, MPLS, and WDM, in combination with networking across multiple\nrouting domains~\\cite{LeZGF11,leo2003vir,rui2011sur,tou2001net,vig2005mul}.\nThe concept of multilayer networking is generally an abstraction of\nproviding network services with multiple networking layers (technologies) and\nmultiple routing domains.\nThe different network layers and their technologies are sometimes classified into\nLayer~0 (e.g., fiber-switch capable), Layer~1 (e.g., lambda switching\ncapable), Layer~1.5 (e.g., TDM SONET\/SDH), Layer~2 (e.g., Ethernet),\nLayer~2.5 (e.g., packet switching capable using MPLS), and Layer~3\n(e.g., packet switching capable using IP routing)~\\cite{YoMKN14}.\nRouting domains are also commonly referred to as\nnetwork domains, routing areas, or levels~\\cite{LeZGF11}.\n\nThe recent multilayer\nnetworking review article~\\cite{LeZGF11} has introduced a range of capability\nplanes to represent the grouping of related functionalities for a\ngiven networking technology. The capability planes include the data\nplane for transmitting and switching data. The control plane and the\nmanagement plane directly interact with the data plane for\ncontrolling and provisioning data plane services as well as for\ntrouble shooting and monitoring the data plane. Furthermore, an\nauthentication and authorization plane, a service plane, and an\napplication plane have been introduced for providing network\nservices to users.\n\nMultilayer networking can involve vertical layering or\nhorizontal layering~\\cite{LeZGF11}, as illustrated in\nFig.~\\ref{fig_control_multilay}. In vertical layering, a given\nlayer, e.g., the routing layer, which may employ a\nparticular technology, e.g., the Internet Protocol (IP),\nuses another (underlying) layer, e.g., the Wavelength\nDivision Multiplexing (WDM) circuit switching layer, to provide\nservices to higher layers. In horizontal layering, services are\nprovided by ``stitching'' together a service path across multiple\nrouting domains.\n\nSDN provides a convenient control framework for these\nflexible multilayer networks~\\cite{LeZGF11}. Several research\nnetworks, such as ESnet, Internet2, GEANT, Science DMZ\n(Demilitarized Zone) have experimented with these multilayer\nnetworking concepts~\\cite{KiSTP13,RoMSS14}.\nIn particular, SDN based multilayer network\narchitectures, e.g.,~\\cite{woe2013sdn, iiz2016mul, mun2016nee},\nare formed by conjoining\nthe layered technology regions\n$(i)$ in vertical fashion i.e., multiple technology layers internetwork\nwithin a single domain, or\n$(ii)$ in horizontal layering fashion across multiple domains,\ni.e., technology layers internetwork across distinct domains.\nHorizontal multilayer networking can be viewed as a generalization of\nvertical multilayer networking in that the horizontal networking\nmay involve the same or different (or even multiple) layers in the\ndistinct domains. As illustrated in\nFig.~\\ref{fig_control_multilay}, the formed SDN based multilayer network\narchitecture is controlled by an SDN orchestrator.\n\\begin{figure*}[t!]\n\\footnotesize\n\\setlength{\\unitlength}{0.10in}\n\\centering\n\\begin{picture}(40,33)\n\\put(13,33){\\textbf{Orchestration, Sec.~\\ref{orch:sec}}}\n\\put(18,30){\\line(0,1){2}}\n\\put(0,30){\\line(1,0){34}}\n\\put(0,30){\\vector(0,-1){2}}\n\\put(-4,26.2){\\textbf{Multilayer Orch.}, Sec.~\\ref{mullayorch:sec} }\n\\put(0,25){\\line(0,-1){2}}\n\n\\put(25,23){\\line(1,0){20}}\n\\put(27,26.2){\\textbf{Multidomain Orch.}, Sec.~\\ref{muldomorch:sec}}\n\\put(34,25){\\line(0,-1){2}}\n\n\n\\put(-8,23){\\line(1,0){18}}\n\\put(-10,23){\\vector(0,-1){2}}\n\\put(10,23){\\vector(0,-1){11}}\n\n\\put(-10,20){\\makebox(0,0)[lt]{\\shortstack[l]{\t\t\t\n\\textbf{Frameworks}, Sec.~\\ref{mullayorchfr:sec} \\\\ \\\\\nHier. Multilay. Ctl.~\\cite{Felix2014,ger2013dem,sal2013inf,sun2014des,ShZBLT12}\\\\\nAppl. Centric Orch.~\\cite{Gerstel2015}\n}}}\n\n\\put(2,10){\\makebox(0,0)[lt]{\\shortstack[l]{\t\t\t\n\\textbf{Appl.-Specific Orch.}, \\\\ Sec.~\\ref{applorch:sec} \\\\ \\\\\nFailure Rec.~\\cite{Khaddam2015} \\\\\nRes. Util.~\\cite{Liu2015e} \\\\\nVirt. Opt. Netws.~\\cite{vil2015net}\n}}}\t\t\t\t\n\\put(34,30){\\vector(0,-1){2}}\n\\put(25,23){\\vector(0,-1){2}}\n\\put(20,20){\\makebox(0,0)[lt]{\\shortstack[l]{\n\\textbf{General Netw.}, Sec.~\\ref{muldomorchgen:sec} \\\\ \\\\\nOpt. Multidom. Multitechn.~\\cite{YoMKN14} \\\\\nHier. Multidom. Ctl.~\\cite{Jing2015} \\\\\nIDP~\\cite{Zhu2015} \\\\\nMultidom. Net. Hyperv.~\\cite{vil2015mul} \\\\\nABNO~\\cite{Munoz2015}\n}}}\t\t\t\t\t\t\t\n\\put(45,23){\\vector(0,-1){11}}\t\t\n\\put(30,10){\\makebox(0,0)[lt]{\\shortstack[l]{\t\t\t\n\\textbf{Data Center Orch.}, Sec.~\\ref{muldomorchdc:sec} \\\\ \\\\\nControl Arch.~\\cite{Liu2015b, may2016sdn} \\\\\nH-PCE~\\cite{cas2015sdn} \\\\\nVirt.-SDN Ctl.~\\cite{mun2015int,Vilalta2016}\n}}}\n\\end{picture}\t\n\\caption{Classification of SDON orchestration studies.}\n\\label{orch_class:fig}\n\\end{figure*}\nAs illustrated in Fig.~\\ref{orch_class:fig} we organize\nthe SDON orchestration studies according to their focus into studies\nthat primarily address the orchestration of vertical multilayer\n(multitechnology) networking,\ni.e., the vertical networking across multiple layers (that typically\nimplement different technologies) within a given domain,\nand into studies that primarily\naddress the orchestration of horizontal multilayer (multidomain)\nnetworking, i.e., the horizontal networking across multiple\nrouting domains (which may possibly involve different or multiple\nvertical layers in the different domains).\nWe subclassify the vertical multilayer studies into general (vertical)\nmultilayer networking frameworks and studies focused on supporting specific\napplications through vertical multilayer networking.\nWe subclassify the multidomain (horizontal multilayer) networking studies into\nstudies on general network domains and studies focused on internetworking\nwith Data Center (DC) network domains.\n\n\n\\subsection{Multilayer Orchestration}  \\label{mullayorch:sec}\n\n\\subsubsection{Multilayer Orchestration Frameworks} \\label{mullayorchfr:sec}\n\n\\paragraph{Hierarchical Multilayer Control}\nFelix et al.~\\cite{Felix2014} presented an hierarchical\nSDN control mechanism for packet optical networks.\nMultilayer optimization techniques\nare employed at the SDN orchestrator\nto integrate the optical transport technology\nwith packet services by provisioning end-to-end Ethernet services.\nTwo aspects are investigated, namely\n$(i)$ bandwidth optimization for the optical transport services,\nand $(ii)$ congestion control for packet network services\nin an integrated packet optical network.\nMore specifically, the SDN controller initially allocates the\nminimum available bandwidth required\nfor the services and then dynamically scales\nallocations based on the availability.\nOptical-Virtual Private Networks (O-VPNs) are created\nover the physical transport network. Services are then mapped\nto O-VPNs based on class of service requirements.\nWhen congestion is detected for a service,\nthe SDN controller switches the service to another O-VPN, thus\nbalancing the traffic to maintain the required class of service.\n\nSimilar steps towards the orchestration of multilayer networks\nhave been taken within the OFELIA\nproject~\\cite{ger2013dem,sal2013inf,sun2014des}. Specifically,\nShirazipour et al.~\\cite{ShZBLT12} have explored\nextensions to OpenFlow version 1.1 actions to enable\nmultitechnology transport layers, including Ethernet transport and\noptical transport. The explorations of the extensions include\njustifications of the use of SDN in circuit-based transport\nnetworks.\n\n\\paragraph{Application Centric Orchestration}\nGerstel et al.~\\cite{Gerstel2015} proposed an application centric network\nservice provisioning approach based on multilayer orchestration.\nThis approach enables the network applications\nto directly interact with the physical layer\nresource allocations to achieve the desired\nservice requirements.\nApplication requirements for\na network service may include\nmaximum end-to-end latency, connection setup and hold times,\nfailure protection, as well as security and encryption.\nIn traditional IP networking, packets from multiple applications\nrequiring heterogeneous services\nare simply aggregated and sent over a common transport link (IP services).\nAs a result, network applications are typically\nassigned to a single (common) transport service within an optical link.\nConsider a failure recovery process with multiple available paths.\nIP networking typically selects the single path\nwith the least end-to-end delay.\nHowever, some applications may tolerate higher\nlatencies and therefore, the traffic can be split\nover multiple restoration paths achieving better traffic management.\nThe orchestrator needs to interact with multiple\nnetwork controllers operating across multiple (vertical) layers\nsupported by north\/south bound interfaces\nto achieve the application centric control.\nDynamic additions of new IP links are demonstrated\nto accommodate the requirements of multiple application services\nwith multiple IP links when\nthe load on the existing IP link was increased.\n\n\\subsubsection{Application-specific Orchestration}  \\label{applorch:sec}\n\\paragraph{Failure Recovery}\nGenerally, network CapEx and OpEx increase as more protection against\nnetwork failures is added.\nKhaddam et al.~\\cite{Khaddam2015} propose\nan SDN based integration of multiple layers, such as WDM and IP, in a failure\nrecovery mechanism to improve the utilization\n(i.e., to eventually reduce CapEx and OpEx while maintaining\nhigh protection levels).\nAn observation study was conducted\nover a five year period to understand the impact of\nnetwork failures on the real deployment\nof backbone networks.\nResults showed $75$ distinct failures following a Pareto distribution,\nin which, $48\\%$ of the total deployed capacity was affected\nby the top (i.e., the highest impact) $20\\%$ of the failures.\nAnd, $10\\%$ of the total deployed\ncapacity was impacted by the top two failure instances.\nThese results emphasize the significance of backup\ncapacities in the optical links for restoration processes.\nHowever, attaining the optimal protection capacities while\nachieving a high utilization of the optical links is challenging.\nA failure recovery mechanism is proposed based\non a ``hybrid'' (i.e., combination of optical transport and IP)\nmultilayer optimization.\nThe hybrid mechanism improved the optical link utilization up to 50~\\%.\nSpecifically, 30~\\% increase of the transport\ncapacity utilization is achieved by dynamically reusing the remainder\ncapacities in the optical links, i.e.,\nthe capacity reserved for failure recoveries.\nThe multilayer optimization technique was validated on an experimental\ntestbed utilizing central path-computation (PCE)~\\cite{rfc5440}\nwithin the SDN framework.\nExperimental verification of failure\nrecovery mechanism resulted in recovery times\non the order of sub-seconds for MPLS restorations and\nseveral seconds for optical WSON restorations.\n\n\\paragraph{Resource Utilization}\nLiu et al.~\\cite{Liu2015e} proposed a method to improve\nresource utilization and to reduce transmission latencies\nthrough the processes of virtualization and service abstraction.\nA centralized SDN control implements the service abstraction\nlayer (to enable SDN orchestrations) in order to integrate the\nnetwork topology management (across both IP and WDM),\nand the spectrum resource allocation in a single control platform.\nThe SDN orchestrator also achieves dynamic and simultaneous\nconnection establishment across both IP and OTN layers\nreducing the transmission latencies. The control plane design is split\nbetween local (child) and root (parent) controllers.\nThe local controller realizes the label switched paths on the optical nodes\nwhile the root controller realizes the forwarding rules\nfor realizing the IP layer.\nExperimental evaluation of average transfer time measurements\nshowed IP layer\nlatencies on the order of several milliseconds, and\nseveral hundreds of milliseconds for the OTN latencies, validating\nthe feasibility of control plane unification for IP over\noptical transport networks.\n\n\\paragraph{Virtual Optical Networks (VONs)}\nVilalta et al.~\\cite{vil2015net} presented\ncontroller orchestration to integrate multiple transport network technologies,\nsuch as IP and GMPLS. The proposed architectural framework devises\nVONs to enable the virtualization\nof the physical resources within each domain.\nVONs are managed by lower level physical controllers (PCs), which\nare hierarchically managed by an SDN network orchestrator (NO). Network\nVirtualization Controllers (NVC) are introduced (on top of the NO) to\nabstract the virtualized multilayers across multiple domains.\nEnd-to-end provisioning of VONs is facilitated through\nhierarchical control interaction over three levels, the\ncustomer controller, the NO\\&NVCs, and the PCs.\nAn experimental evaluation demonstrated average VON\nprovisioning delays on the order of\nseveral seconds (5~s and 10~s), validating the flexibility of\ndynamic VON deployments over the optical transport networks.\nLonger provisioning delays may impact the network\napplication requirements, such as failure recovery processes,\ncongestion control, and traffic engineering.\nGeneral pitfalls of such hierarchical structures are\nincreased control plane complexity, risk of controller failures, and\nmaintenance of reliable communication links between control plane entities.\n\n\\subsection{Multidomain Orchestration}  \\label{muldomorch:sec}\nLarge scale network deployments typically involve multiple domains,\nwhich have often heterogeneous layer technologies.\nAchieve high utilization of the networking resources\nwhile provisioning end-to-end network paths and services across multiple\ndomains and their respective layers and respective technologies is\nhighly challenging~\\cite{Mayoral2015,Munoz2015b,Yu2015a}.\nMultidomain SDN orchestration studies have sought to exploit\nthe unified SDN control plane to aid the resource-efficient\nprovisioning across the multiple domains.\n\n\\subsubsection{General Multidomain Networks}  \\label{muldomorchgen:sec}\n\n\\paragraph{Optical Multitechnologies Across Multiple Domains}\nOptical nodes are becoming\nincreasingly reconfigurable (e.g., through variable BVTs and\nOFDM transceivers, see Section~\\ref{sdninfra:sec}),\nadding flexibility to the switching elements. When a single\nend-to-end service\nestablishment is considered, it is more likely that a service is supported by\ndifferent optical technologies that operate across multiple domains.\nYoshida et al.~\\cite{YoMKN14} have demonstrated SDN based orchestration with\nemphasis on the physical interconnects between multiple domains and multiple\ntechnology specific controllers so as to realize end-to-end services.\nOpenFlow capabilities have been extended for fixed-length variable capacity\noptical packet switching~\\cite{Losada2015}.\nThat is, when an optical switch matches the label on an\nincoming optical packet, if a rule\nexists in the switch (flow entry in the table) for a specific label,\na defined action is performed on the optical packet by the switch.\nOtherwise, the optical packet is dropped\nand the controller is notified. Interconnects between optical packet\nswitching networks and elastic optical networks are enabled\nthrough a novel OPS-EON interface card.\nThe OPS-EON interface is designed as an extension\nto a reconfigurable, programmable\nand flexi-grid EON supporting the OpenFlow protocol.\nThe testbed implementation of OPS-EON interface cards demonstrated\nthe orchestration of multiple domain controllers and the reconfigurability\nof FL-VC OPS across multidomain, multilayer, multitechnology scenarios.\n\n\\paragraph{Hierarchical Multidomain Control}\nJing et al.~\\cite{Jing2015} have also examined\nthe integration of multiple optical transport\ntechnologies from to multiple\nvendors across multiple domains, focusing on\nthe control mechanisms across multiple domains.\nJing et al.~proposed hierarchical SDN orchestration with parent and\ndomain controllers.\nDomain controllers abstract the physical layer by\nvirtualizing the network resources.\nA Parent Controller (PC) encompasses a Connection Controller (CC) and\na Routing Controller (RC) to process the abstracted virtual network.\nWhen a new connection setup request is\nreceived by the PC, the RC (within the PC) evaluates the end-to-end\nrouting mechanisms and forwards the\ninformation to the CC.\nThe CC breaks the end-to-end routing information into shorter\nlink segments belonging to a domain.\nSegmented routes are then sent to the respective domain\ncontrollers for link provisioning over the physical infrastructures.\nThe proposed mechanism was experimentally verified on a testbed\nbuilt with the commercial OTN equipment.\n\n\\paragraph{Inter-Domain Protocol}\n\\begin{figure}[t!]\n\t\\centering\n\t\\vspace{0cm}\n\t\\includegraphics[width=4.2in]{finman_sdon\/fig21.pdf}\n\t\\vspace{0cm}\n\t\\caption{Inter-domain lightpath provisioning mechanism.}\n\n\t\\label{fig_multilay_idp}\n\\end{figure}\nZhu et al.~\\cite{Zhu2015} followed a different approach for the\nSDN multidomain control mechanisms by considering the\nflat arrangement of controllers as shown in Fig.~\\ref{fig_multilay_idp}.\nEach domain is autonomously managed by\nan SDN controller specific to the domain.\nAn Inter-Domain Protocol (IDP) was devised to establish the communication\nbetween domain specific controllers\nto coordinate the lightpath setup across multiple domains.\nZhu et al.~also proposed a Routing and\nSpectrum Allocation (RSA) algorithm for the end-to-end provisioning\nof services in the SD-EONs.\nThe distributed RSA algorithm operates on the domain specific controllers\nusing the IDP protocol. The RSA considers both transparent lightpath\nconnections, i.e., all-optical lightpath,\nand translucent lightpath connections, i.e.,\noptical-electrical-optical connections.\nThe benefit of such techniques is privacy, since\nthe domain specific policies and topology information are not\nshared among other network entities.\nNeighbor discovery is  independently conducted by the domain specific controller\nor can initially be configured.\nA domain appears as an abstracted virtual node to all other domain specific\ncontrollers. Each controller then assigns the shortest path\nrouting within a domain between its border nodes.\nAn experimental setup validating the\nproposed mechanism was demonstrated across\ngeographically-distributed domains in the USA and China.\n\n\\paragraph{Multidomain Network Hypervisors}\n\\begin{figure}[t!]\n\t\\centering\n\t\\includegraphics[width=3in]{finman_sdon\/fig22.pdf}\n\t\\caption{Illustration of multilevel virtualization.} \n\t\\label{fig_multilay_mnh}\n\\end{figure}\nVilalta et al.~\\cite{vil2015mul} presented a mechanism for virtualizing\nmultitechnology optical, multitenant networks.\nThe Multidomain Network Hypervisor (MNH) creates customer specific\nvirtual network slices managed by the customer specific SDN controllers\n(residing at the customers' locations) as illustrated\nin Fig.~\\ref{fig_multilay_mnh}.\nPhysical resources are managed by their domain specific\nphysical SDN controllers.\nThe MNH operates over the network orchestrator and\nphysical SDN controllers for provisioning VONs\non the physical infrastructures.\nThe MNHs abstracts both $(i)$ multiple optical transport technologies,\nsuch as optical packet switching and Elastic Optical Networks (EONs),\nand $(ii)$ multiple control domains, such as GMPLS and OpenFlow.\nExperimental assessments on a testbed achieved VON provisioning\nwithin a few seconds (5~s), and\ncontrol overhead delay on the order of several tens of milliseconds.\nRelated virtualization mechanisms for multidomain optical SDN networks\nwith end-to-end provisioning have been\ninvestigated in~\\cite{SzAEK14,vil2016hie}.\n\n\\paragraph{Application-Based Network Operations}\n\\begin{figure}[t!]\n\t\\centering\n\t\\includegraphics[width=3.5in]{finman_sdon\/fig23.pdf}\n\t\\caption{The application-based network operations (ABNO)~\\cite{Munoz2015}}\n\t\\label{fig_multilay_abno}\n\\end{figure}\nMu{\\~{n}}oz et al.~\\cite{Munoz2015}, have presented an SDN\norchestration mechanism based on the\napplication-based network operations (ABNO) framework, which\nis being defined by the IETF~\\cite{rfc7491}.\nThe ABNO based SDN orchestrator integrates\nOpenFlow and GMPLS in transport networks.\nTwo SDN orchestration designs have been presented: $(i)$ with centralized\nphysical network topology aware path computation\n(illustrated in Fig.~\\ref{fig_multilay_abno}), and\n$(ii)$ with topology abstraction and distributed path computation.\nIn the centralized design, OpenFlow and GMPLS controllers (lower level control)\nexpose the physical\ntopology information to the ABNO-orchestrator (higher level control).\nThe PCE in the ABNO-orchestrator has the global view of the network\nand can compute end-to-end paths with complete knowledge of the network.\nComputed paths are then provisioned through\nthe lower level controllers. The pitfalls of such centralized designs\nare $(i)$ computationally intensive path computations,\n$(ii)$ continuous updates of topology and traffic information,\nand $(iii)$ sharing of confidential network information and policies\nwith other network elements.\nTo reduce the computational load at the orchestrator,\nthe second design implements distributed path computation\nat the lower level controllers (instead of path computation at the\ncentralized orchestrator). However, such distributed mechanisms may\nlead to suboptimal solutions due to the limited network knowledge.\n\n\\subsubsection{Multidomain Data Center Orchestration} \\label{muldomorchdc:sec}\n\\paragraph{Control Architectures}\nGeographically distributed DCs are typically\ninterconnected by links traversing multiple domains.\nThe traversed domains may be homogeneous i.e., have the same type of network\ntechnology, e.g., OpenFlow based ROADMs,\nor may be heterogeneous, i.e., have different types of network\ntechnologies, e.g.,\nOpenFlow based ROADMs and GMPLS based WSON. The SDN control structures\nfor a multidomain network can be broadly classified into the categories of\n$(i)$ single SDN orchestrator\/controller, $(ii)$ multiple mesh SDN\ncontrollers, and\n$(iii)$ multiple hierarchical SDN controllers~\\cite{Liu2015b, may2016sdn}.\nThe single SDN orchestrator\/controller has to support heterogeneous SBIs\nin order to operate with multiple heterogeneous domains, e.g.,\nthe Path Computation Element Protocol (PCEP) for GMPLS network domains\nand the OpenFlow protocol for OpenFlow supported ROADMs.\nAlso, domain specific details, such as topology, as well as\nnetwork statistics and configurations, have to be exposed to an external\nentity, namely the single SDN orchestrator\/controller,\nraising privacy concerns. Furthermore, a single\ncontroller may result in scalability issues.\nMesh SDN control connects the domain-specific controllers side-by-side\nby extending the east\/west bound interfaces.\nAlthough mesh SDN control addresses the scalability and privacy issues,\nthe distributed nature of the control mechanisms may lead to sub-optimal\nsolutions.\nWith hierarchical SDN control, a logically centralized controller\n(parent SDN controller) is placed above the domain-specific controllers\n(child SDN controllers), extending the north\/south bound interfaces.\nDomain-specific controllers virtualize\nthe underlying networks inside their domains, exposing only the abstracted\nview of the domains to the parent controller, which addresses the privacy\nconcerns. Centralized\npath computation at the parent controller can achieve optimal solutions.\nMultiple hierarchical levels can address the scalability issues.\nThese advantages of hierarchal SDN control are achieved at the expense of\nan increased number of network entities,\nresulting in the operational complexities.\n\n\\paragraph{Hierarchical PCE}\n\\begin{figure}[t!]\n\t\\centering\n\t\\includegraphics[width=3.5in]{finman_sdon\/fig24.pdf}\n  \\caption{Illustration of SDN orchestration based on Hierarchical\n  Path Computation Element (H-PCE)~\\cite{cas2015sdn}.}\n\t\\label{fig_multilay_HPCE}\n\\end{figure}\nCasellas et al.~\\cite{cas2015sdn} considered\nDC connectivities involving both intra-DC and inter-DC communications.\nIntra-DC communications enabled through OpenFlow networks\nare supported by an OpenFlow controller.\nThe inter-DC communications\nare enabled by optical transport\nnetworks involving more complex control,\nsuch as GMPLS, as illustrated in Fig.~\\ref{fig_multilay_HPCE}. To\nachieve the desired SDN benefits of flexibility and scalability,\na common centralized control platform spanning across heterogeneous\ncontrol domains is proposed.\nMore specifically, an Hierarchical PCE (H-PCE) aggregates PCE states\nfrom multiple domains.\nThe end-to-end path setup between DCs is orchestrated by a parent-PCE (pPCE)\nelement, while the paths are provisioned\nby the child-PCEs (cPCEs) on the physical resources, i.e.,\nthe OpenFlow and GMPLS domains.\nThe proposed mechanism utilizes existing protocol interfaces,\nsuch as BGP-LS and PCEP, which are extended with OpenFlow to support the H-PCE.\n\n\\paragraph{Virtual-SDN Control}\nMu{\\~n}oz et al.~\\cite{mun2015int,Vilalta2016} proposed a\nmechanism to virtualize the SDN control functions\nin a DC\/cloud by integrating SDN with Network Function Virtualization (NFV).\nIn the considered context, NFV refers to realizing network functions\nby software modules running on\ngeneric computing hardware inside a DC; these network functions were\nconventionally implemented on specialized hardware modules.\nThe orchestration of Virtual Network Functions (VNFs)\nis enabled by an integrated SDN and NFV management which\ndynamically instantiates virtual SDN controllers.\nThe virtual SDN controllers\ncontrol the Virtual Tenant Networks (VTNs), i.e., virtual\nmultidomain and multitechnology networks.\nMultiple VNFs running on a Virtual Machine (VM)\nin a DC are managed by a VNF manger.\nA virtual SDN controller is responsible for creating, managing, and tearing\ndown the VNF achieving the flexibility in the control plane management\nof the multilayer and the multidomain networks.\nAdditionally, as an extension to the proposed mechanism,\nthe virtualization of the control functions of the\nLTE Evolved Packet Core (EPC) has been discussed in~\\cite{mar2016int}.\n\n\\subsection{Orchestration: Summary and Discussion}\nRelatively few SDN orchestration studies to date have focused on\nvertical multilayer networking within a given domain. The few studies\nhave developed two general orchestration frameworks and have examined\na few orchestration strategies for some specific applications.\nMore specifically, one orchestration framework has\nfocused on optimal bandwidth allocation based mainly on\ncongestion~\\cite{Felix2014}, while the other\nframework has focused on exploiting application traffic tolerances for\ndelays for efficiently routing traffic~\\cite{Gerstel2015}.\nSDN orchestration of vertical multilayer optical networking is thus\nstill a relatively little explored area.\nFuture research can develop orchestration frameworks that accommodate\nthe specific optical communication technologies in the various layers and\nrigorously examine their performance-complexity tradeoffs.\nSimilarly, relatively few applications have been\nexamined to date in the application-specific orchestration\nstudies for vertical multilayer\nnetworking~\\cite{Khaddam2015,Liu2015e,vil2015net}.\nThe examination of the wide range of existing applications and\nany newly emerging network application in the context of\nSDN orchestrated vertical multilayer networking presents rich research\nopportunities.\nThe cross-layer perspective of the SDN orchestrator over a given\ndomain could, for instance, be exploited for strengthening security and\nprivacy mechanisms or for accommodating demanding real-time multimedia.\n\nRelatively more SDN orchestration studies to date have\nexamined multidomain networking than multilayer networking (within a single\ndomain). As the completed multidomain orchestration studies\nhave demonstrated, the SDN orchestration can help greatly in coordinating\ncomplex network management decisions across multiple distributed\nrouting domains.\nThe completed studies have illustrated the fundamental\ntradeoff between centralized decision making in a hierarchical\norchestration structure and distributed decision making in a flat\norchestration structure.\nIn particular, most studies have focused on hierarchical\nstructures~\\cite{Jing2015,cas2015sdn,mun2015int}, while only one study\nhas mainly focused on a flat orchestration structure~\\cite{Zhu2015}.\nIn the context of DC internetworking, the\nstudies~\\cite{Liu2015b, may2016sdn} have sought to bring out\nthe tradeoffs between these two structures by examining\na range of structures from centralized to distributed.\nWhile centralized orchestration can make decisions with\na wide knowledge horizon across the states in multiple domains,\ndistributed decision making preserves the privacy of network status\ninformation, reduces control traffic, and can make fast localized\ndecisions.\nFuture research needs to shed further light on these complex\ntradeoffs for a wide range of combinations of optical technologies employed\nin the various domains.\nThroughout, it will be critical to abstract and convey the key\ncharacteristics of optical physical layer components and switching nodes to\nthe overall orchestration protocols. Optimizing each abstraction\nstep as well as the overall orchestration and examining the various\nperformance tradeoffs are important future research directions.\n\n\\section{\\MakeUppercase{Open Challenges and Future SDON Research Directions}} \\label{sec:open}\nWe have outlined open challenges and future\nSoftware Defined Optical Network (SDON) research directions\nfor each sub-category of surveyed SDON studies in the Summary and\nDiscussion subsections in the preceding survey sections.  In this\nsection, we focus on the overall cross-cutting open challenges that\nspan across the preceding considered categories of SDON studies.\nThat is, we focus on open challenges and research directions\nthat span the vertical (inter-layer) and horizontal (inter-domain)\nSDON aspects. The vertical SDON aspects encompass the\nseamless integration of the various (vertical) layers of the\nSDON architecture; especially the optical layer, which is not considered\nin general SDN technology.\nThe horizontal SDON aspects include the integration of SDONs with\nexisting non-SDN optical networking elements, and the internetworking\nwith other domains, which may have  similar or different\nSDN architectures.\nA key challenge for SDON research is to enable the use of SDON concepts\nin operational real-time network infrastructures.\nImportantly, the SDON concepts need to demonstrate performance\ngains and cost reductions to be considered by network and service providers.\nTherefore, we cater some of the open challenges and\nfuture directions towards enabling and demonstrating the successful use\nof SDON in operational networks.\n\nThe SDON research and development effort to date\nhave resulted in insights for making the use of SDN in optical transport\nnetworks feasible and have demonstrated advantages of SDN based optical network\nmanagement.\nHowever, most network and service providers\ndepend on optical transport to integrate with multiple industries to\ncomplete the network infrastructure. Often, network and service providers\nstruggle to integrate hardware components and to provide accessible software\nmanagement to customers. For example, companies that develop hardware\noptical components do not always have a complete associated software stack\nfor the hardware components. Thus, network and service providers using the\nhardware optical components often have to maintain a software development team\nto integrate the various\nhardware components through software based management into their network,\nwhich is often a costly endeavor.\nThus, improving SDN technology so that it seamlessly integrates with\ncomponents of various industries\nand helps the integration of components from various industries\nis an essential underlying theme for future SDON research.\n\n\\subsection{Simplicity and Efficiency}   \\label{simpl_fut:sec}\nOptical network structures typically span heterogeneous devices ranging\nfrom the end user nodes and local area networks via ONUs and OLTs in the\naccess networks to edge routers and metro network nodes and on\nto backbone (core) network infrastructures.\nThese different devices often come from different vendors.\nThe heterogeneity of devices and their vendors often requires\nmanual configuration and maintenance of optical networks.\nMoreover, different communication technologies typically\nrequire the implementation of native functions that are specific\nto the communication technology characteristics, e.g., the transmission and\npropagation properties.\nBy centralizing the optical network control in an\nSDN controller, the SDN networking paradigm creates a unified view\nof the entire optical network.\nThe specific native functions for specific communication devices\ncan be migrated to the software layer and be implemented by a central node,\nrather than through manual node-by-node configurations.\nThe central node would typically be readily accessible and could reduce\nthe required physical accesses to distributed devices at their\non-site locations.\nThis centralization can simplify the network management and\nreduce operational expenditures.\nAn important challenge in this central management is the efficient\nSDN control of components from multiple vendors.\nDetailed vendor contract specifications of open-source middleware may be\nneeded to efficiently control components from different vendors.\n\nThe heterogeneity of devices may reduce the efficiency of\nnetwork infrastructures\ndue to the required multiple software and hardware modules for a\ncomplete networking solution. Future research should\ninvestigate efficient mechanisms for making complete networking solutions\navailable for specific use cases. For example, the use of\nSDON for an access network provider may require multiple SDN controllers\nco-located within the OLT to enable the control of the access network\ninfrastructure from one central location.  While the SDON studies reviewed in\nthis survey have led initial investigations of simple and dynamic\nnetwork management, future research needs to refine these management\nstrategies and optimize their operation across combinations of\nnetwork architecture structures and across various network\nprotocol layers.\nSimplicity is an essential part of this\nchallenge, since overly complicated solutions are generally not deployed\ndue to the risk of high expenditures.\n\n\\subsection{North Bound Interface} \\label{nbi:fut}\nThe NorthBound Interface (NBI) comprises the communication from the\ncontroller to the applications. This is an important area of\nfuture research as applications and their needs are generally\nthe driving force for deploying SDON infrastructures.\nAny application, such as video on demand, VoIP, file\ntransfer, or peer-to-peer networking, is applied from the NBI to the\nSDN controller which consequently conducts the necessary actions to implement\nthe service behaviors on the physical network infrastructure.\nApplications often require specific service behaviors that\nneed to be implemented on the overall network infrastructure.\nFor example,\napplications requiring high data rates and reliability, such as Netflix,\ndepend on data centers and the availability\nof data from servers with highly resilient failure protection mechanisms.\nThe associated management network needs to stack redundant devices as\nto safeguard against outages. Services are provided as\npolicies through the NBI to the SDN controller, which in turn generates flow\nrules for the switching devices. These flow rules can be\nprioritized based on the customer use cases.\nAn important challenge for future NBI research is to\nprovide a simple interface for a wide variety of\nservice deployments without vendor lock-in, as vendor lock-in\ngenerally drives costs up.\nAlso, new forms of communication to the controller, in\naddition to current techniques, such as\nREpresentational State Transfer (REST)~\\cite{REST15} and HTTP, should\nbe researched.\nMoreover, future research should develop an NBI framework that spans\nhorizontally across multiple controllers, so that service customers are not\nrestricted to using only a single controller.\n\nFuture research should examine control mechanisms that optimally\nexploit the central SDN control to provide simple and efficient\nmechanisms for automatic network management and dynamic service\ndeployment~\\cite{ZhYZ14}.  The NBI of SDONs is a\nchallenging facet of research and development because of the multitude\nof interfaces that need to be managed on the physical layer and\ntransport layer. Optical physical layer components and infrastructures\nrequire high capital and\noperational expenditures and their management is generally not associated with\nnetwork or service providers but rather with optical component\/infrastructure\nvendors. Future research should develop novel Application Program Interfaces\n(APIs) for optical layer components and infrastructures that facilitate\nSDN control and are amenable to efficient NBI communication.\nEssentially, the challenge of efficient NBI communication with the\nSDN controller should be\nconsidered when designing the APIs that interface with the\nphysical optical layer components and infrastructures.\n\nOne specific strategy for simplifying network management and operation\ncould be to explore the grouping of control policies of similar\nservice applications, e.g., applications with similar QoS requirements.\nThe grouping can reduce the number of control policies at the\nexpense of slightly coarser granularity of the service offerings.\nThe emerging Intent-Based Networking (IBN) paradigm, which\ndrafts intents for services and policies,\ncan provide a specific avenue for simplifying\ndynamic automatic configuration and virtualization~\\cite{BlDM08,CoBR13}.\nCurrently network applications are deployed based on how the network\nshould behave for a specific action. For example, for inter domain routing,\nthe Border Gateway Protocol (BGP) is used, and the network gateways\nare configured to communicate with the BGP protocol.\nThis complicates the provisioning of\nservices that typically require multiple protocols and limits the\nflexibility of service provisioning. With IBN, the application gives an intent,\nfor example, transferring video across multiple domains.\nThis intent is then\nassociated with automated dynamic configurations of the network elements\nto communicate data over the domains using appropriate protocols.\nThe grouping of service policies, such as intents, can facilitate\neasy and dynamic service provisioning.\nIntent groups can be described in a graph to simplify the compilation of\nservice policies and to resolve conflicts~\\cite{PrLT15}.\n\n\\subsection{Reliability, Security, and Privacy} \\label{futworksec:sec}\nThe SDN paradigm is based on a centrally managed network.\nFaulty behaviors, security infringements, or failures of the\ncontrol would likely result in extensive disruptions and\nperformance losses that are exacerbated by the centralized nature of the\nSDN control. Instances of extensive disruptions and losses due to\nSDN control failures or infringements would likely reduce the trust in\nSDN deployments. Therefore, it is very important to ensure reliable network\noperation~\\cite{rak2016inf}\nand to provision for security and privacy of the communication.\nHence, reliability, security, and privacy are prominent SDON research\nchallenges. Security in SDON techniques is a fairly open research\narea, with only few published findings.\nAs a few reviewed studies (see Section~\\ref{app_fail_rec:sec}) have explored,\nthe central SDN control can facilitate reliable network service through speeding\nup failure recovery.\nThe central SDN control can continuously scan the network and\nthe status messages from the network devices.\nOr, the SDN control can\nredirect the status messages to a monitoring service that analyzes\nthe data network.  Security breaches can be controlled by broadcasting\nmessages from the controller to all affected devices to block traffic\nin a specific direction. Future research should refine these\nreliability functions to optimize automated fault and performance\ndiagnostics and reconfigurations for quick failure recovery.\n\nNetwork failures can either occur within the physical layer\ninfrastructure, or as errors within the higher protocol\nlayers, e.g., in the classical data link (L2), network (L3), of\ntransport (L4) layers.\nIn the context of SDONs, physical\nlayer failures present important future research opportunities.\nPhysical layer devices need to be carefully monitored by sending\nfeedback from the devices to the controller.\nThe research and development on communication between\nthe SDN controller and the network devices has mainly focused on sending\nflow rules to the network devices while feedback communicated from the devices\nto the controller has received relatively little attention.\nFor example, there are three types of OpenFlow messages, namely Packet-In,\nPacket-Out, and Flow-Mod. The Packet-In messages are\nsent from the OpenFlow switches to the controller,\nthe Packet-Out message is sent from\nthe controller to the device, and the Flow-Mod message is used to\nmodify and monitor\nthe flow rules in the flow table. Future research should examine extensions of\nthe Packet-In message to send specific status updates in support of\nnetwork and device failure monitoring to the controller.\nThese status messages could be monitored\nby a dedicated failure monitoring service.\nThe status update messages could be broadly defined to\ncover a wide range of network management aspects, including\nsystem health monitoring and network failure protection.\n\nA related future research direction is to secure configuration and\noperation of SDONs through trusted encryption and key management\nsystems~\\cite{ahm2015sec}. Moreover, mechanisms to\nensure the privacy of the communication should be explored.\nThe security and privacy mechanisms should strive to exploit the\nnatural immunity of optical transmission segments to electro-magnetic\ninterferences.\n\nIn summary, security and privacy of SDON communication are largely open\nresearch areas.\nThe optical physical layer infrastructure has traditionally not\nbeen controlled remotely, which in general reduces the occurrences\nof security breaches.\nHowever, centralized SDN management and control increase the risk of\nsecurity breaches, requiring extensive research on SDON security, so\nas to reap the benefits of centralized SDN management and control in a\nsecure manner.\n\n\\subsection{Scalability} \\label{scalability:fut}\nOptical networks are expensive and used for high-bandwidth\nservices, such as long-distance network access and data center interconnections.\nOptical network infrastructures either\nspan long distances between multiple geographically distributed locations,\nor could be short-distance incremental additions (interconnects)\nof computing devices. Scalability in multiple dimensions\nis therefore an important aspect for future SDON research.\nFor example, a\nmyriad of tiny end devices need to be provided with network access in\nthe emerging Internet of Things (IoT) paradigm~\\cite{wan2015nov}.\nThe IoT requires access network architectures and\nprotocols to scale vertically (across protocol layers and technologies)\nand horizontally (across network domains).\nAt the same time, the ongoing growth of\nmultimedia services requires data centers to scale up optical network bandwidths\nto maintain the quality of experience of the multimedia services.\nBroadly speaking, scalability includes in the vertical\ndimension the support for multiple network devices and technologies.\nScalability in\nthe horizontal direction includes the communication between\na large number of different domains as well as support for existing non-SDON\ninfrastructures.\n\nA specific scalability challenge arising with SDN infrastructure is that the\nscalability of the control plane (OpenFlow protocol signalling)\ncommunication\nand the scalability of the data plane communication which transports\nthe data plane flows\nneed to be jointly considered. For example, the Openflow protocol~1.4\ncurrently supports 34 Flow-Mod messages~\\cite{OF2016}, which can\ncommunicate between the network devices and the controller. This\nnumber limits the functionality of the SBI communication. Recent studies\nhave explored a protocol-agnostic\napproach \\cite{HuLi2015, bosshart2014p4}, which is a data plane\nprotocol that extends the use of multiple protocols for\ncommunication between the control plane and data plane.\nThe protocol-agnostic approach resolves\nthe challenges faced by OpenFlow and, in general, any particular protocol.\nExploring this novel protocol-agnostic approach presents many new SDON\nresearch directions.\n\nScalability would also require SDN technology to overlay and scale\nover existing non-SDN infrastructures. Vendors provide support\nfor known non-SDN devices, but this area is still a challenge. There\nare no known protocols that could modify the flow tables of existing\npopularly described ``non-OpenFlow'' switches. In the case of optical\nnetworks, as SDN is still being incrementally deployed, the overlaying\nwith non-SDN infrastructure still requires significant attention.\nIdeally, the overlay mechanisms should ensure seamless integration and should\nscale with the growing deployment of SDN technologies while incurring only\nlow costs.\nOverall, scalability poses highly important future SDON research directions\nthat require economical solutions.\n\n\n\\subsection{Standardization}   \\label{std:sec}\nNetworking protocols have traditionally followed a uniform\nstandard system for all the\ncommunication across multiple domains. Standardization has helped vendors\nto provide products that work in and across different network infrastructures.\nIn order to ensure the compatible inter-operation of SDON components (both hardware and software) from a various vendors,\nkey aspects of the inter-operation protocols need to be standardized.\nTowards the standardization goal,\ncommunities, such as Open Networking Foundation (ONF), have created boards\nand committees to standardize protocols, such as OpenFlow.\nStandardization should ensure that SDON infrastructures can be flexibly\nconfigured and operated with components from various vendors.\nThe use of open-source\nsoftware can further facilitate the inter-operation.\nProprietary hardware and software components generally create vendor lock-in,\nwhich restricts the flexibility of network operation and reduces\nthe innovation of network and service providers.\n\nAs groundwork for standardization, it may be necessary to develop\nand optimize a common (or a small set) of SDON architectures\nand network protocol configurations that can serve as a basis\nfor standardization efforts.\nThe standardization process may involve a common platform that\nis built thorough the cooperation of multiple manufacturers.\nAnother thrust of standardization groundwork could be the development of\nopen-source software that supports SDON architectures.\nFor example, Openstack is a cloud based management framework that has been\nadopted and supported by multiple networking vendors. Such efforts should\nbe extended to SDONs in future work.\n\n\\subsection{Multilayer Networking} \\label{multilayer:fut}\nAs discussed in Section~\\ref{orch:sec}, multilayer networking\ninvolves vertical multilayer networking\nacross the vertical layers as well as horizontal multilayer (multidomain)\nnetworking across multiple domains.\nWe proceed to outline open challenges and\nfuture research directions for vertical multilayer networking in the context of\nSDON, which includes an optical physical layer, in this subsection.\nHorizontal multilayer (multidomain) networking is considered in\nSection~\\ref{multidomain:fut}.\n\nFor the vertical multilayer networking in a single domain,\nthe optical physical layer is the key distinguishing feature of\nSDONs compared to conventional SDN architectures for general IP networks.\nMost of the higher layers in SDONs have similar multilayer\nnetworking challenges as general IP networks.\nHowever, the optical physical layer\nrequires the provisioning of specific optical\ntransmission parameters, such as wavelengths and signal strengths.\nThese parameters are managed by optical devices, such as the OLT in PON\nnetworks. For SDON networks, so-called \\textit{optical orchestrators},\nwhich are commercially available, e.g., from ADVA Optical Networking,\nprovide a single interface to provision the optical layer parameters.\nWe illustrate this optical orchestrator layer in the context of an\nSDON multilayer network in the rightmost branch of Fig.~\\ref{fig_control_orch}.\nThe optical orchestrator resides\nabove the optical devices and below the SDN controller.\nThe optical orchestrator uses common SDN SBI interface protocols,\nsuch as OpenFlow,\nto communicate with the optical devices in the south-bound direction\nand with the controller in the north-bound direction.\n\nThe SDN controller in the control plane is responsible for the management of the\nSDN-enabled switches, potentially via an optical orchestrator.\nCommunicating over the SBI using different\nprotocols can be challenging for the controller.\nThis challenge can\nbe addressed by using south-bound renderers. South-bound renderers are\nAPIs that reside within the\ncontroller and provide a communication channel to any desired\nSBI protocol.  Most SDN controllers currently have an\nOpenFlow renderer to be able to communicate to Openflow network switches.\nBut there are also SNMP and NETCONF-based renderers, which\ncommunicate with traditional non-OpenFlow switches. This enables the\nexistence of hybrid networks with already existing switches.\nThe effective support of such hybrid networks, in conjunction with\nappropriate south-bound renderers and optical orchestrators, is an important\ndirection for future research.\n\n\\subsection{Multidomain Networks} \\label{multidomain:fut}\nA network domain usually belongs to a single organization that\nowns (i.e., financially supports and uses) the network domain.\nThe management of multidomain networking involves the\nimportant aspects of configuring the access control as well as the\nauthentication, authorization, and accounting. Efficient\nSDN control mechanisms for configuring these multidomain networking\naspects is an important direction for future research and development.\n\nMultidomain SDONs may also need novel routing\nalgorithm that enhance the capabilities of the currently used BGP\nprotocol. Multidomain research \\cite{phe2014dis} has now taken\ninterest in the Intent-Based Networking (NBI) paradigm for\nSDN control, where Intent-APIs can solve the\nproblems of spanning across multiple domains. For instance, the intent of an\napplication to transfer information across multiple domains is\ntranslated into service instances that access configurations between\ndomains that have been pre-configured based on contracts.\nCurrently, costly manual configurations between\ndomains are required for such applications.\nFuture research needs to develop concrete models for\nNBI based multidomain networking in SDONs.\n\n\\subsection{Fiber-Wireless (FiWi) Networking} \\label{wireless:fut}\nThe optical (fiber) and wireless network domains have\nmany differences.\nAt the physical layer, wireless networks are characterized\nby varying channel qualities, potentially high losses, and generally lower\ntransmission bit rates than optical fiber.\nWireless end nodes are typically mobile and may connect dynamically\nto wireless network domains.\nThe mobile wireless nodes are generally the end-nodes in a\nFiWi network and connect via intermediate optical nodes to\nthe Internet.\nDue to these different characteristics, the management of\nwireless networks with mobile end nodes is very different from the\nmanagement of optical network nodes.\nFor example, wireless access points should maintain their own\nrouting table to accommodate access to dynamically connected mobile\ndevices. Combining the control of both wireless and\noptical networks in a single SDN controller requires\nconcrete APIs that handle the respective control functions of\nwireless and optical networks.\nCurrently, service providers maintain separate physical management services\nwithout a unified logical control and management plane for\nFiWi networks.\nDeveloping integrated controls for FiWi networks\ncan be viewed as a special case of multilayer networking and integration.\n\nDeveloping specialized multilayer networking strategies for\nFiWi networks is an important future research directions as many aspects of\nwireless networks have dramatically advanced in recent\nyears. For instance, the cell structure of wireless cellular\nnetworks~\\cite{Ohlen2016} has advanced to femtocell\nnetworks~\\cite{cha2008fem} as well as heterogeneous and multitier\ncellular structures~\\cite{els2013sto,lou2011tow}. At the same time,\nmachine-to-machine communication~\\cite{has2013ran,lay2014ran} and\nenergy savings~\\cite{ana2015opt,has2011gre} have drawn research attention.\n\n\\subsection{QoS and Energy Efficiency}  \\label{futworkene:sec}\nDifferent types of applications have vastly different\ntraffic bit rate characteristics and QoS requirements.\nFor instance, streaming high-definition video requires high bit rates,\nbut can tolerate\nsome delays with appropriate playout buffering. On the other hand,\nVoIP (packet voice) or video conference applications have\ntypically low to moderate bit rates, but require low latencies.\nAchieving these application-dependent QoS levels in an energy-efficient\nmanner~\\cite{has2011gre,ShZL14,wan2015ene}\nis an important future research direction.\nA related future research direction is to\nexploit SDN control for QoS adaptations of real-time media and\nbroadcasting services.\nBroadcasting services involve typically data rates ranging from\n3--48~Gb\/s to deliver video at various resolutions to\nthe users within a reasonable time limit.\nIn addition to managing the QoS, the network\nhas to manage the multicast groups for efficient\nrouting of traffic to the users.\nRecent studies \\cite{Butler2015, Ellerton2015} discuss\nthe potential of SDN, NFV, and optical technologies\nto achieve the growing demands of broadcasters and media.\nMoreover, automated\nprovisioning strategies of QoS and the incorporation of quality of\nprotection and security with traditional QoS are important direction for\nfuture QoS research in SDONs.\n\n\\subsection{Performance Evaluation}\nComprehensive performance evaluation methodologies and metrics need to\nbe developed to assess the SDON designs addressing the\npreceding future research directions ranging from simplicity and\nefficiency (Section~\\ref{simpl_fut:sec}) to optical-wireless networks\n(Section~\\ref{wireless:fut}).\nThe performance evaluations need to encompass the data plane,\nthe control plane, as well as the overall data and control plane interactions\nwith the SDN interfaces and need to take virtualization\nand orchestration mechanisms into\nconsideration. In the case of the SDON infrastructure, the performance\nevaluations will need include the optical\nphysical layer~\\cite{Azodolmolky2014}.\nWhile there have been some efforts to develop evaluation frameworks for\ngeneral SDN switches~\\cite{OFTest,rot2014ope}, such evaluation frameworks\nneed to be adapted to the specific characteristics of SDON architectures.\nSimilarly, some evaluation frameworks for general SDN controllers have\nbeen explored~\\cite{jar2012fle,jar2014ofc}; these need to be extended\nto the specific SDON control mechanisms.\n\nGenerally, performance metrics obtained with SDN and virtualization\nmechanisms should be benchmarked against the corresponding\nconventional network without any SDN or virtualization components.\nThus, the performance tradeoffs and costs of the flexibility\ngained through SDN and virtualization mechanism can be quantified.\nThis quantified data would then need to be assessed and compared\nin the context of business needs. To identify some of the important aspects of\nperformance we analyze the sample architecture in Fig.~\\ref{fig_app_i2rs}.\nThe SDN controller in the SDON architecture in Fig.~\\ref{fig_app_i2rs}\nspans across multiple elements, such as ONUs, OLTs,\nrouters\/switches in the metro-section, as well as PCEs in the core section.\nA meaningful performance evaluation of such a network\nrequires comprehensive analysis of data plane performance aspects and\nrelated metrics,\nincluding noise spectral analysis, bandwidth and link rate monitoring,\nas well as evaluation of failure resilience.\nPerformance evaluation mechanisms need to be\ndeveloped to enable the SDON controller to obtain and analyze these\nperformance data. In addition, mechanisms for control layer\nperformance analysis are needed.\nThe control plane performance evaluation should, for instance\nassess the controller efficiency and performance characteristics,\nsuch as the OpenFlow message rates and the rates and delays of flow table\nmanagement actions.\n\n\\section{\\MakeUppercase{Conclusion}}  \\label{sec:conclusion}\nWe have presented a comprehensive survey of software defined\noptical networking (SDON) studies to date.\nWe have mainly organized our survey according to the SDN\ninfrastructure, control, and application layer structure. In addition,\nwe have dedicated sections to SDON virtualization and orchestration\nstudies.\nOur survey has found that SDON infrastructure studies\nhave examined optical (photonic) transmission and switching components\nthat are suitable for flexible SDN controlled operation.\nMoreover, flexible SDN controlled switching paradigms and optical\nperformance monitoring frameworks have been investigated.\n\nSDON control studies have developed and evaluated SDN control\nframeworks for the wide range of optical network transmission\napproaches and network structures. Virtualization allows for\nflexible operation of multiple Virtual Optical Networks (VONs) over a given\ninstalled physical optical network infrastructure.\nThe surveyed SDON virtualization studies have examined the provisioning\nof VONs for access networks, exploiting the\nspecific physical and Medium Access Control (MAC) layer characteristics of\naccess networks. The virtualization studies have also examined\nthe provisioning of VONs in metro and\nbackbone networks, examining algorithms for embedding the VON topologies\non the physical network topology under consideration of the\noptical transmission characteristics.\n\nSDON application layer studies have developed mechanisms for achieving\nQuality of Service (QoS), access control and security, as well as\nenergy efficiency and failure recovery.\nSDON orchestration studies have examined coordination mechanisms\nacross multiple layers (in the vertical dimension of the network protocol\nlayer stack) as well as across multiple network domains (that may belong\nto different organizations).\n\nWhile the SDON studies to date have established basic principles\nfor incorporating and exploiting SDN control in optical networks,\nthere remain many open research challenges. We have\noutlined open research challenges for each individual category of\nstudies as well as cross-cutting research challenges.\n\n\\chapter{\\MakeUppercase{SDN Based Smart Gateways (Sm-GWs) for Multi-Operator Small Cell\n\tNetwork Management}}\n\n\\section{\\MakeUppercase{Introduction}}\n\\subsection{Motivation: Small Cells}\nRecent wireless communications research has examined the\nbenefits of splitting the conventional cells in wireless cellular\ncommunications into small cells for supporting the growing\nwireless network traffic.\nSmall cells can coexist with\nneighboring small cells while sharing the same spectrum resources \\cite{khan2016cognitive}, and are thus\nan important potential strategy for accommodating wireless network traffic\ngrowth~\\cite{jaf2015sma}.\nSmall cells are also sometimes referred to as ``femto'' cells in\nthe context of the Third Generation Partnership Project (3GPP)\nLong Term Evolution (LTE) wireless standard; \nwe use the general terminology ``small'' cells throughout.\nHowever, small cells pose new challenges, including\ninterference coordination~\\cite{Cong2014},\nbackhaul complexity \\cite{Wei2013,Siddique2015},\nand increased network infrastructure cost~\\cite{Mugume2015}.\nIn this article~\\cite{thyagaturu2016sdn} we propose a solution to\nreduce the infrastructure cost and complexity of backhaul access\nnetworks supporting small cells.\n\nSmall cell networks are expected to be\nprivatively owned~\\cite{Haider2016}.\nTherefore it is important\nto enable usage flexibility and the freedom of investment in the new\nnetwork entities (e.g., gateways and servers) and the network infrastructures\n(e.g., switches and optical fiber)\nby the private owners of small cells.\nWhile a plethora of studies has examined advanced enhanced Node~B (eNB)\nresource\nmanagement, e.g.,~\\cite{chen2015virtual,liu2015qos,samdanis2016td},\nthe implications of small cell deployments for backhaul\ngateways have largely remained unexplored~\\cite{chuang2015resource}.\nGenerally, backhaul access networks that interconnect\n  small cell deployments with LTE gateways\n   can employ a wide variety of link layer (L2) technologies, including\nSONET\/SDH, native Ethernet, \nand Ethernet over MPLS~\\cite{Briggs2010,CiscoDeployment,men2013hie}.\nIn order to accommodate these heterogeneous L2 technologies, \ncellular LTE network interfaces, such as S1 and X2 interfaces,\nare purposefully made independent of the L2 technology between\nsmall cell deployments and gateways. Due to the independent nature \nof L2 technologies, a dedicated link with prescribed \nQoS, which can support the fundamental operations of cellular protocols,\nmust be established for each interface connection~\\cite{Ghebretensae2010}. \nStatistical multiplexing is then limited \nby the aggregate of the prescribed QoS \\cite{guck2016function, fitzek2002providing} requirements\nand only long-term re-configurations, e.g.,\nin response to deployment changes, can optimize the\nbackhaul transmissions~\\cite{Mathew2015}.\nPresent wireless network deployments based on the 3GPP LTE standard do \nnot provide feedback from the eNBs to a\ncentral decision entity, e.g., an SDN orchestrator, \nwhich could flexibly allocate network resources based on eNB traffic demands.\nThus, present wireless backhaul architectures are characterized by\n$(i)$ essentially static network\nresource allocations between eNBs and operator gateways,\ne.g., LTE Servicing\/Packet Gateways (S\/P-GWs), and\n$(ii)$ lack of coordination between the eNBs and the operator gateways\nin allocating these network resources,\nresulting in under-utilization of the backhaul transmission resources.\nAdditionally, exhaustion of available ports at the operator gateways can\nlimit the eNB deployment in practice.\n\nThe static resource allocations and lack of eNB-gateway cooperation\nare highly problematic since the aggregate uplink transmission bitrate of the\nsmall cells within a small geographic area, e.g., in a building,\nis typically much higher than the\nuplink transmission bitrate available from the cellular operators.\nThus, small cell deployments create a bottleneck between the eNBs and the\noperator gateways.  For instance, consider the deployment of\n100 small cells in a building, whereby each small cell supports\n1~Gbps uplink transmission bitrate.\nEither each small cell can be allocated only one hundredth of the\noperator bitrate for this building or the operator would need to\ninstall 100~Gbps uplink transmission bitrate for this\nsingle building, which would require cost-prohibitive operator gateway\ninstallations for an organization with several buildings in a small\ngeographical area.\nHowever, the uplink transmissions from the widespread data communication\napplications consist typically of short high-bitrate bursts, e.g.,\n100 Mbps bursts.\nIf typically no more than ten small cells burst simultaneously, then\nthe eNBs can dynamically share a 1~Gbps operator uplink\ntransmission bitrate.\nAn additional problem is that with the typically limited port counts on\noperator gateways, connections to many new small cells may require\nnew operator gateway installations.\nAn intermediate Sm-GW can aggregate the small cell connections\nand thus keep the required port count at operator gateways low.\n\n\\subsection{Overview of Network Management with SDN-based Sm-GW}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=4.5in]{finman_smgw\/fig1.pdf}\n    \\caption{The proposed smart gateway (Sm-GW).} \n    \\label{fig_enterprise}\n\\end{figure}\nWe present a new backhaul network framework for supporting small cell\ndeployments based on a new network entity, the Smart GateWay (Sm-GW).\nConsider an exemplary small cell deployment throughout multiple\nbuildings of a university.\nEach building has hundreds of small cells that are flexibly\nconnected to an Sm-GW, as illustrated in Fig.~\\ref{fig_enterprise}.\nMultiple Sm-GWs are then connected to core networks, i.e., the S-GWs\nand P-GWs, of multiple cellular operators via\nphysical links (e.g., optical or microwave links)\n\\cite{mcgarry2012investigation, mcgarry2008ethernet, aurzada2008delay, mcgarry2006wdm, mcgarry2004ethernet, reisslein6metropolitan, mercian2013offline, aurzada2014delay},\nas illustrated for a single Sm-GW in Fig.~\\ref{fig_SMGW_overall}.\nAn SDN orchestrator owned by the university\nmanages the cellular infrastructure of the entire university.\nThe SDN orchestrator coordinates the resource allocations from\nthe operators to the Sm-GWs.\n\nThe main original contributions of this article are:\n\\begin{enumerate}\n\\item A novel comprehensive Smart Gateway (Sm-GW) architecture\n  and protocol framework that accommodates a flexible number of eNBs\n  while reducing the requirements at the operator's core, e.g.,\n  at LTE S-GW and MME. The Sm-GW physically and logically aggregates\n the eNB connections so that a set of eNBs appears as a single virtual\n eNB to the operator gateways, see Section~\\ref{arch:sec}.\n    \\item A Sm-GW scheduling framework to flexibly share the limited\n     uplink transmission bitrate among all the small cell eNBs connected\n      to an Sm-GW,  see Section~\\ref{gwsch:sec}.\n    \\item An adaptive SDN-based multi-operator management framework that\n      dynamically shares the uplink transmission bitrates of multiple\n      operators among the Sm-GWs. An SDN orchestrator dynamically coordinates\n      the sharing among the Sm-GWs, the transport network connecting the\n      Sm-GWs to the operator gateways, and the operator gateways,\n  see Section~\\ref{MOM:sec}.\n\\end{enumerate}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{finman_smgw\/fig2.pdf}\n    \\caption{The Smart Gateway (Sm-GW) architecture.} \n    \\label{fig_SMGW_overall}\n\\end{figure}\n\n\\subsection{Related Work}\nRecently proposed SDN based backhaul architectures, such as\nCROWD~\\cite{AliAhmad,seb2015dyn}, iJOIN~\\cite{Dongyao},\nU-WN~\\cite{Shengli}, Xhaul~\\cite{oli2015xha},\nthe multi-tiered SDN based backhaul architecture~\\cite{elg2015thr},\nand similar\narchitectures~\\cite{cos2015sof,cos2015,dra2015dyn,hur2015sdn,\n  Jungnickel,liu2013cas},\nare revolutionary designs proposing new cellular infrastructure\n  installations.\nIn contrast, our proposed SDN-based Sm-GW enables the softwarization of\n\\textit{existing}\ncellular infrastructures consisting of eNBs and conventional operator gateways,\nsuch as the S\/P-GW in LTE core networks.\nThe proposed Sm-GW is inserted in the existing backhaul\ninfrastructure to inter-network and co-exist with the existing\nLTE network core entities, such as the S\/P-GW.\n\nSDN based backhaul architectures with centralized\ninterference coordination have been proposed\nin~\\cite{akh2016syn, gop2016gen, nie2015wir}.\nThe SDN controller in these architectures maintains a global database\nof spectrum resources \\cite{akhtar2016white} and dynamically assigns the resources to base stations so\nas to minimize the mutual interference among base stations in dense\ndeployments.\nWe note that wireless interference is a localized phenomenon.\nTherefore, a base station is most affected by its neighboring base stations\nin dense deployments.\nThe centralized interference coordination techniques\nin~\\cite{akh2016syn, gop2016gen, nie2015wir} are complementary\nto our proposed Sm-GW architecture in that they can be\nimplemented at the Sm-GW instead of the SDN controller\/orchestrator.\n\nSchedulers at the eNB allow multiple user equipment (UE) devices to\nshare the wireless resources at the eNB.  For example, the LTE standard medium\naccess control (MAC) protocol~\\cite{3gppLTEMAC} coordinates the\nscheduling of wireless resources between an eNB and multiple UEs.\nGenerally, most wireless resource scheduling studies to date have focused on the\nsharing of the wireless resources at a given single eNB.\nFor instance, quality of service\n(QoS) aware uplink scheduling and\nresource allocation at a given single small cell eNB in an LTE network have been\nexamined in~\\cite{Chaudhuri2015}.\nIn contrast, we propose a novel\nscheduling framework at the Sm-GW based on uplink transmission bitrate requests\nfrom \\textit{multiple eNBs}, i.e., we propose the sharing of\nthe backhaul network resources among multiple eNBs.\n\nA similar sharing of network resources among small cell base\nstations has been studied in~\\cite{lak2013h}.\nSpecifically, the H-infinity scheduler for limited capacity\nbackhaul links~\\cite{lak2013h} schedules the traffic\nin the downlink. The centralized H-infinity scheduler\nfocused on buffer size requirements at\nthe base stations in the small cell networks.\nIn contrast, we focus on the \\textit{uplink} traffic from the eNBs to the Sm-GW.\nTo the best of our knowledge, we propose the first\nnetwork protocol framework for the uplink transmissions\nfrom multiple eNBs to the operator gateways in the context of LTE small cells.\nWe note that our Sm-GW framework is complementary to several recently studied\nresource allocation mechanisms in cellular networks.\nFor instance, D2D resource allocation through traffic\noffloading to small cell networks has been studied in~\\cite{sem2015con}; this\nD2D approach can be readily supported by our proposed Sm-GW.\nCoordinated scheduling \\cite{mer2016ups, wei2014dyc, che2015sim, das2016gro} in the context of small cells\nwith dynamic cell muting to mitigate the interference has\nbeen discussed in~\\cite{wan2015dyn}. The cell muting technique can be\nfurther extended based on our approach of traffic scheduling to eNBs.\nA flexible wireless resource allocation mechanism\nbased on the SDN programmability of traffic flows\nfrom a single UE device to multiple\nbase stations in dense small cell networks\nhas been examined in~\\cite{gas2015pro}.\nThe offloading of UE traffic for efficient traffic management in\n  small cell networks has been examined in~\\cite{tal2015eff}.\nIn contrast, we propose an SDN-based multi-operator resource allocation\nmechanism that allocates limited backhaul link capacities to\nmultiple Sm-GWs (which in turn can flexibly allocate the capacities\nto multiple eNBs).\nThe UE to eNB communication approach from~\\cite{gas2015pro}\nand UE traffic offloading~\\cite{tal2015eff} are thus\ncomplementary to our eNB to Sm-GW and Sm-GW to S\/P GW\nnetwork management approaches.\n\n\\section{\\MakeUppercase{Background: Conventional LTE Small Cell Backhaul}}\nIn this section we describe the conventional architectural model for\nHome-eNodeB (HeNB) access  networks~\\cite{HeNB3GPP} and the network\nsharing mechanism in 3GPP LTE.\nHeNBs are the small cell base stations of the LTE standard.\n  We use the general terminology ``eNB'' to denote all types of\nsmall cell base stations.\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=4.5in]{finman_smgw\/fig3.pdf}\n    \\caption{HeNB architectural models in 3GPP LTE.}\n    \\label{fig_HeNB}\n\\end{figure}\n\n\\subsection{HeNB Architectural Models in 3GPP LTE}\nIn Figure \\ref{fig_HeNB}\nwe show the 3GPP HeNB architectural models:\n1) with dedicated HeNB-GateWay (HeNB-GW), 2) without HeNB-GW,\nand 3) with HeNB-GW for the control plane.\n\n\\subsubsection{With Dedicated HeNB-GW}\nWith a dedicated HeNB-GW, communication between\nthe HeNB and the HeNB-GW is secured by a mandatory security gateway\n(Se-GW) network function. The HeNB-GW aggregates the\ncontrol plane connections (S1-MME) and user plane connections (S1-U) of all\nHeNBs connected to the HeNB-GW to a single control and user plane\nconnection. The HeNB gateway appears as a single eNB to the outside\nentities, such as S-GW and MME.\nIn a similar way, the HeNB-GW appears as both an\nS-GW and an MME to the eNBs connected to the HeNB-GW.  The numbers of\nports required at the MME and S-GW are reduced through the\naggregation at the HeNB-GW.\nOur proposed Sm-GW architecture is similar to the\n  dedicated HeNB-GW architecture, in that the Sm-GW aggregates the eNBs\n  connections both physically and logically.\nIn addition, our Sm-GW flexibly allocates uplink transmission\nbitrates to small cell eNBs (see Section~\\ref{gwsch:sec})\nand allows for the adaptive allocation of\noperator uplink transmission bitrates to the Sm-GW by the SDN orchestrator\n(see Section~\\ref{MOM:sec}).\n\n\\subsubsection{Without HeNB-GW}\nDeployments of HeNBs without the HeNB-GWs increase the requirements on\nthe S-GW and MME to support large numbers of connections.  Large\ndeployments of small cells without gateway aggregation at the HeNBs\nwould greatly increase the total network infrastructure cost.\n\n\\subsubsection{With HeNB-GW for the Control Plane}\nHeNB control plane connections are terminated at the HeNB-GW and a\nsingle control plane connection is established from the\nHeNB gateway to the MME. Although\nthe number of connections required at the MME is reduced due to the\ncontrol plane aggregation at the HeNB-GW, data plane connections are still\nterminated directly at the S-GW, increasing requirements at the\nS-GW. The Se-GW typically secures the communication to and from the\nHeNB. In contrast, our proposed Sm-GW terminates all\nthe control and data connections from HeNBs.\n\n\\subsection{3GPP Network Sharing} \nNetwork sharing was introduced by 3GPP in Technical Specification\n  TS~23.951~\\cite{3GPPNetSharing}\n  with the main motivation to share expensive radio spectrum\n  resources among multiple operators.\n For instance, an operator without available spectrum \nin a particular geographic area can offer cellular\nservices in the area through sharing the spectrum of another operator. \nIn addition to spectrum sharing, 3GPP specifies\ncore network sharing among multiple operators\nthrough a gateway core network (GWCN) configuration~\\cite{3GPPNetSharing}.\nGWCN configurations are statically pre-configured at deployment\nfor fixed pre-planned core network sharing.\nThus, GWCN sharing can achieve only limited statistical\nmultiplexing gain as the sharing is based on the pre-configured \nQoS requirements of the eNB interface connections and not on the \nvarying eNB traffic demands. \nAlso, the GWCN configuration lacks a central entity for \noptimization of the resource allocations with global knowledge of the \neNB traffic demands.\nIn contrast, our Sm-GW framework includes a central SDN orchestrator for\noptimized allocations of backhaul transmission resources  \naccording to the varying eNB traffic demands (see Section~\\ref{MOM:sec}).\n \n\\section{\\MakeUppercase{Proposed Smart Gateway (Sm-GW)}} \\label{arch:sec}\nIn this section we introduce the proposed Smart Gateway (Sm-GW) network\narchitecture for existing LTE deployments.\nWe describe the fundamental protocol mechanisms and interfaces\nthat integrate the proposed Sm-GW into the conventional LTE\nprotocols.\n\n\\subsection{LTE Protocol Modifications}  \\label{prot:sec}\nFig.~\\ref{fig_protocol} illustrates the proposed protocol\nmechanisms between a set of $N_s$ eNBs and a given Sm-GW $s$.\n\\begin{figure*}[!t]\n    \\centering\n    \\includegraphics[width=6in]{finman_smgw\/fig4.pdf}\n    \\caption{Illustration of proposed protocol mechanisms at eNB,\nSm-GW, and S-GW.} \n    \\label{fig_protocol}\n\\end{figure*}\n\n\\subsubsection{eNB} \\label{prot:sec:eNB}\nAt the eNB, we newly introduce the eNB-to-Sm-GW\nreporting protocol, which\noperates on top of the GPRS tunneling protocol (GTP)~\\cite{GTP}\nand stream control transmission protocol (SCTP).\nThe reporting protocol $(i)$ evaluates the required uplink transmission\nbitrate, and $(ii)$ sends the bitrate request messages to the Sm-GW.\nThe reporting protocol formulates the operator specific uplink\ntransmission bitrate requests\nbased on the requests of the UEs that are connected\nvia the eNB to multiple operators $o,\\ o = 1, 2, \\ldots, O$.\n\nThe eNB wireless resource scheduler is responsible for the sharing\nof wireless resources between the eNB and the UEs.\nThe eNB wireless resource scheduler ensures\nthat only the resources available at the eNB are granted to the UEs.\nUEs periodically send buffer status reports\n(BSRs) to the eNB which they are connected to.\nTherefore, the eNB-to-Sm-GW reporting protocol can estimate the UE traffic\nrequirements by interacting with the wireless resource scheduler.\n\n\\subsubsection{Smart Gateway (Sm-GW)} \\label{prot:sec:Sm-GW}\nThe protocol stack at the Sm-GW is similar to the HeNB-GW protocol stack.\nHowever, in the Sm-GW, an additional eNB coordination protocol,\na scheduler for the dynamic resource\nallocation, and SDN capabilities are introduced.\n\nThe eNB coordination protocol collects request messages from eNBs.\nThe eNB uplink transmission grants are sized based on the\neNB requests and the available Sm-GW resources\naccording to the Sm-GW scheduling described in Section~\\ref{gwsch:sec}.\nThe eNB coordination protocol sends grant messages\nto all eNBs within a reasonable processing delay.\n\nS1 based handovers for the downlink transmissions are typically\nanchored at the S-GW. (For the uplink transmissions, an anchoring,\nor buffering of packets, at a network entity, e.g., eNBs or S-GW, is\nnot required.) We emphasize that the\nSm-GW will be transparent to all the downlink packets from the S-GW\nand hence not be limited by the network protocol scheduler.\nThis ensures that the S1 based handover mechanisms\nat the S-GW and eNBs continue to function normally.\n\n\\subsubsection{SDN Operations of Sm-GW}\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[width=3.5in]{finman_smgw\/fig5.pdf}\n\n        \\caption{Illustration of Sm-GW embedding in the SDN ecosystem.}\n\n\t\\label{fig_Sm-GW_SDN}\n\\end{figure}\n\n\\paragraph{SDN Infrastructure}\nThe Sm-GW SDN capabilities can be provided by an\nOpenFlow (OF) agent and\/or\na configuration manager at each Sm-GW, as illustrated\nin Fig.~\\ref{fig_Sm-GW_SDN}.\nOpenFlow is a popular protocol for the southbound\ninterface (SBI) and can be employed on the SBI\nbetween the Sm-GW SDN controller and Sm-GW.\nThe OpenFlow agent supports OpenFlow SDN functionalities at the Sm-GW,\nmaking the Sm-GW configurable through the OpenFlow protocol.\nThe Sm-GW configuration manager can be controlled by the Sm-GW SDN controller,\ne.g., through the NETCONF (or OpenFlow) SBI,\nto dynamically reconfigure the Sm-GW.\n\nThe Sm-GW SDN controller configures the Sm-GWs to enable the\ninternal LTE X2 tunnel interfaces among all connected small cell eNBs,\nas elaborated in Section~\\ref{X2int:sec}. Also, the Sm-GW SDN\ncontroller manages the external LTE X2 and S1 interfaces at the Sm-GW through\ntunnel establishments to the external LTE network core entities,\ni.e., MMEs and S\/P-GWs.\n\nWhereas the conventional LTE transport network between the eNBs and\nS\/P-GWs is configured with static MPLS\/IP paths~\\cite{Ghebretensae2010},\nthe flexible Sm-GW operation requires a flexible transport\nnetwork, that is controlled by a transport SDN controller, as illustrated in\nFig.~\\ref{fig_Sm-GW_SDN}.\nThe flexible transport network can, for instance, be implemented through a\nSoftware Defined Elastic Optical Network \n(SD-EON)~\\cite{thy2016sof,Yoshida2015}.\n\n\\paragraph{Sm-GW Virtualization}   \\label{smgwvirt:sec}\nThe SM-GW can support a variety of virtualization strategies, e.g., to\nprovide independent virtual networks for different operators.\nOne example virtualization strategy could let the\nSm-GWs abstract the connected eNBs. Sm-GWs could then be abstracted by\na hypervisor~\\cite{ble2015sur,mij2016ne,nak2015wif,rig2015pro}\nthat intercepts the SBI, as illustrated in Fig.~\\ref{fig_Sm-GW_SDN}.\nBased on the operator configurations that are sent via the SDN orchestrator\nto the Sm-GW SDN controller, resources at the Sm-GWs and the small cell eNBs\n(which are privately owned by an organization~\\cite{Haider2016}) can\nbe sliced to form operator-specific virtual networks of Sm-GWs and\neNBs. The configuration manger at each Sm-GW can allocate resources to each of\nthese virtual networks.\n\nFrom the operator perspective,\n  the Sm-GW virtualization \\cite{ble2016con} essentially allows multiple operators to share the\nphysical small cell infrastructure of Sm-GWs and eNBs.\nThus, simultaneous services can be enabled to large UE populations\nthat belong to multiple operators, i.e., that have contracts with multiple\noperators, while using the same physical small cell infrastructure.\nExisting conventional cellular deployment structures do not support\nthe infrastructure sharing among multiple operators.\n\n\\paragraph{SDN Orchestration}\nThe SDN orchestrator coordinates the network management across multiple\ndomains of Sm-GWs (whereby each Sm-GW domain is controlled by its own\nSm-GW SDN controller), transport networks, and core networks.\nThe SDN orchestrator implements the multi-operator management\nintroduced in Section~\\ref{MOM:sec} and configures the\nSm-GWs and transport networks based on\nglobal multi-operator network optimizations.\nFor example, the SDN orchestrator\ncommunicates with the path computation element (PCE) SDN application\non the transport SDN controller.\nThe PCE dynamically evaluates the label switched paths,\nsuch as MPLS\/IP paths, so as to flexibly enable and reconfigure\nthe transport network~\\cite{thy2016sof,Yoshida2015}.\n\n\\subsubsection{LTE X2 Interfaces of eNBs with Sm-GW}  \\label{X2int:sec}\nX2 interfaces enable critical functionalities in LTE small cells,\nsuch as X2-based handover as well as interference coordination and mitigation.\nTypically, each eNB connected to a given Sm-GW\npertaining to an operator shares the same\nMME; thus, each eNB needs an X2 interface to all other eNBs\nwithin the same MME coverage area, the so-called tracking area.\nHence, eNBs connected to an Sm-GW must be\ninterconnected with X2 interfaces.\n\n\\paragraph{To External Macro-eNBs}\nX2 traffic flows destined to eNBs located outside the scope of an\nSm-GW (typically to a macro-eNB) are not be limited by the scheduler\nat the Sm-GW. X2 packets flow out of Sm-GW into the backhaul (i.e., to an\nS-GW) as they originate at the eNBs. The Sm-GW appears as an\nexternal router (or gateway) to the X2 external interfaces.\n\n\\paragraph{To Internal Small-eNBs} \\label{prot:X2:conn}\nThe Sm-GW appears as a simple bridge or a router to the internal X2\ninterfaces, routing the internal X2 packets within.\nTherefore, the scheduler at the Sm-GW does not limit any X2 packets.\nFor small cell deployments, an eNB can have multiple neighboring eNBs\nin the tracking area;\nthese neighboring eNBs need to be interconnected with each other\nwith X2 connections.\nOn the order of $O(N(N-1))$ dedicated links would be required\n  to interconnect the X2 interfaces of $N$ eNBs in the tracking area in a full\n  mesh topology.\nIn contrast, a star\ntopology with the Sm-GW at the center requires only $O(N)$\nconnections to connect each eNB to the Sm-GW.\nIn summary, in our Sm-GW architecture, the Sm-GW manages the X2 interfaces\nof all the internal small cell eNBs, thus eliminating\nthe distributed management of X2 interfaces at each eNB.\n\n\\subsubsection{Authentication of Sm-GW with EPC Core}\nTypically HeNBs use IPSec tunneling for their\nsecurity and encryption, which creates overhead.\nIf Sm-GWs are authenticated, the HeNBs would no longer need IPsec tunneling.\nSpecifically, upon boot-up, the Sm-GW is authenticated with\nan LTE Evolved Packet Core (EPC)\nso as to eliminate the need for a network security-gateway\n(Se-GW) function or IPsec tunneling between the eNBs and the P-GWs.\nCritical cellular network functions, such as\nsecurity, authentication, and reliability,\nrequire additional effort to be enabled in WiFi networks.\nWiFi Passpoint~\\cite{Passpoint} (Hotspot 2.0)\naims at providing an experience similar to\ncellular connectivity in\nWiFi networks by providing the cellular authentication mechanisms.\nWith the authentication of Sm-GWs,\nthe simplicity of WiFi networks\ncan be achieved by the small cell cellular networks.\n\n\\subsection{Downlink vs.~Uplink Communication} \\label{updown:sec}\n\\subsubsection{Downlink Packets at the Sm-GW}\nTraffic flows in the conventional downlink path from\nan S\/P-GW to an eNB are typically sent at rates that do not\nexceed the wireless transmission rates from the eNB to the UE devices.\nThus, as long as the link rates from the S\/P-GW to the\ninserted Sm-GW and from the Sm-GW to the eNB are at least as high\nas the conventional S\/P-GW to eNB links, the Sm-GW can be\ntransparent to the downlink packets from the S\/P-GW.\n\n\\subsubsection{Uplink Packets at Sm-GW}\nIn contrast to the downlink data traffic, the uplink data\ntraffic from the eNBs to an Sm-GW needs to be\nregulated as the traffic flows from all the eNBs terminating at the\nSm-GW can overwhelm the outgoing link towards the operator S-GW.\nEnforcing QoS strategies and fairness among eNBs requires scheduling\nof the uplink packet traffic arriving from the eNBs at an Sm-GW.\nTherefore, our focus is on frameworks for the uplink transmission scheduling of\nthe communication $(i)$ from eNBs to an Sm-GW (Section~\\ref{gwsch:sec}),\nand $(ii)$ from Sm-GWs to S-GWs (Section~\\ref{MOM:sec}).\n\n\\section{\\MakeUppercase{Proposed Sm-GW Scheduling Framework}} \\label{gwsch:sec}\n\\subsection{Purpose}\nThe main purpose of the Sm-GW scheduling framework\nis to maximize the utilization of\nthe network resources, and to ensure fair uplink transmission service for\n all eNBs connected to an Sm-GW.\nWithout scheduling, highly\nloaded eNBs can impair the service for lightly loaded\neNBs connected to the same Sm-GW.\nWhen many eNBs are\nflexibly connected to an Sm-GW, traffic bursts from heavily loaded eNBs can\noverwhelm the queue of an Sm-GW, resulting in excessive packet\ndrops and high delays, even for lightly loaded eNBs.\nOn the other hand, with scheduling, a large number of\neNBs can be flexibly connected to the Sm-GW while ensuring\nprescribed QoS and fairness levels.\nEach eNB can possibly have a different service level agreement.\nThe Sm-GW allows for the flexible deployment of a wide variety of\nscheduling algorithms.\nWe outline two classes of Sm-GW scheduling algorithms, and illustrate\nan elementary algorithm for each class.\n\n\n\\begin{table}[t]\n\\caption{Summary of Notation of Sm-GW Network Management}\n     \t\\centering\n\\begin{tabular}{|p{0.4cm} p{7.2cm}|}\n\\hline\n\\multicolumn{2}{|c|}{\\textbf{\\rule{0pt}{1\\normalbaselineskip}\n    Sm-GW Sched. Framework (Sm-GW $\\leftrightarrow$ eNBs),\n           Sec.~\\ref{gwsch:sec}}} \\\\   [.5ex]\n$N_s$ &  Number of small cell eNBs at Sm-GW $s$ \\\\\n$G_{so}$ &  Available uplink transm. bitrate [bit\/s] from Sm-GW $s$ to operator $o$  \\\\\n$W$ &  Duration [s] of scheduling cycle   \\\\\n$\\Gamma_{so}$  &  $= G_{so} W \/ N_s$, Max. eNB uplink transm. data amount [bit] per cycle with equal sharing \\\\\n$\\rho_{son}$ & Data amount [bit] that eNB $n$ at Sm-GW $s$ wants to\n  transmit to operator $o$ in a cycle, i.e., request by eNB $n$ \\\\\n$\\gamma_{son}$ & Data amount [bit] that eNB $n$ at Sm-GW $s$ is allowed\nto transmit to operator $o$ in a cycle, i.e., grant by Sm-GW $s$ \\\\%[0.5ex]\n\\hline\n\\multicolumn{2}{|c|}{\\textbf{\\rule{0pt}{1\\normalbaselineskip}\n   SDN Based Multi-Operator Managm. Framework, Sec.~\\ref{MOM:sec}}} \\\\\n            \\multicolumn{2}{|c|}{\\textbf{\n    (Sm-GWs $\\leftrightarrow$ Operator Gateways)}} \\\\[.5ex]\n$o$  &  Index of operators, $o = 1, 2, \\ldots, O$\\\\\n$s$   & Index of Sm-GWs, $s = 1, 2, \\ldots, S$ \\\\\n$R_{so}$   &  Smoothed uplink transmission bitrate [bit\/s] request\n     from Sm-GW $s$ to operator $o$ \\\\\n$K_o$ &  Max. available uplink transm. bitrate through operator $o$\\\\ \n$G_{so}$ & Granted uplink transm. bitrate from Sm-GW $s$ to operator $o$\\\\\n$X_{so}$ & Actual uplink traffic bitrate from Sm-GW $s$ to operator $o$\\\\ [0.5ex]\n\t\t\\hline\n\t\t\\end{tabular}\n\t\t\\end{table}\n\n\\subsection{Configuration Adaptive Scheduling} \\label{gwsch:sec:equal}\nConfiguration adaptive scheduling adapts the scheduling,\ni.e., the allocation of uplink transmission bitrates, according to\nthe number of eNBs connected to a given Sm-GW.\nThe Sm-GW tracks the number of connected eNBs and sends\na configuration message to all eNBs in the event of a change in\nconnectivity at the Sm-GW, i.e., addition of new eNB or\ndisconnection of existing eNB.\nMore specifically, consider $N_s$ eNBs at a given Sm-GW $s$ that has been\nallocated the uplink transmission bitrate $G_{so}$ [bit\/s] toward a\ngiven operator $o$ (through the coordination techniques in\nSection~\\ref{MOM:sec}).\n\nAn elementary equal share scheduling shares the available uplink\ntransmission bitrate at the Sm-GW toward a given operator $o$\nequally among all eNBs connected to the Sm-GW.\nEach eNB $n,\\ n = 1, 2, \\ldots, N_s$, can then transmit at most\n$\\Gamma_{so} =  G_{so} W \/ N_s$ [Byte] of traffic during a cycle\nof duration $W$ [seconds].\nThe traffic amount limit $\\Gamma_{so}$ and cycle duration $W$\nare sent to the eNBs as a part of the initial configuration message.\nEach eNB schedules the uplink transmissions\nsuch that no more than $\\Gamma_{so}$ [Byte] of traffic are\nsend in a cycle of duration $W$ [seconds].\n\nThe simple equal share scheduler can flexibly accommodate\nlarge numbers $N_s$ of eNBs.\nHowever, the equal bandwidth assignments by the elementary\nequal share scheduler to\nthe eNBs under-utilize the network resources when some eNBs have very\nlittle traffic while other eNBs have high traffic loads.\n\n\\subsection{Traffic Adaptive Scheduling} \\label{gwsch:sec:excess}\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=3.5in]{finman_smgw\/fig6.pdf}\n\\caption{Illustration of traffic adaptive Smart Gateway (Sm-GW)\n    scheduling.} \n\\label{fig_ExcessTiming}\n\\end{figure}\nWith traffic adaptive scheduling, the Sm-GW collects uplink transmission\nrequests from the eNBs. The Sm-GW then adaptively allocates portions of\nthe uplink transmission bitrate $G_{so}$ to the individual eNBs\naccording to their requests.\nTraffic adaptive scheduling operates with a\nrequest-allocate-transmit cycle of duration $W$ [seconds]\nillustrated in Fig.~\\ref{fig_ExcessTiming}. At the start of the\ncycle, each eNB $n,\\ n = 1, 2, \\ldots, N_s$, sends an\nuplink transmission bitrate request to Sm-GW~$s$.\nWe let $\\rho_{son}$ denote the amount of traffic [in Byte] that\neNB $n$ wants to transmit to operator $o$ over the next cycle of duration $W$.\nOnce all requests have been received, i.e., following the\nprinciples of the offline scheduling\nframework~\\cite{zhe2009sur}, portions of $G_{so}$ can be\nallocated to the eNBs according to some scheduling policy.\n\nAn elementary excess share scheduling\npolicy~\\cite{bai2006fai}\nallocates the eNB grants as follows. Lightly loaded eNBs with\n$\\rho_{son} < \\Gamma_{so}$ are granted their full request, i.e.,\nreceive the grant size $\\gamma_{son} = \\rho_{son}$, while their\nunused (excess) portion of the equal share allocation is accumulated\nin an excess pool:\n\\begin{equation}\n\\xi = \\sum_{\\forall \\rho_{son} \\leq \\Gamma_{so}} \\Gamma_{so} - \\rho_{son}.\n\\label{ExcessAccum}\n\\end{equation}\nFollowing the principles of controlled equitable excess\nallocation~\\cite{bai2006fai}, highly loaded eNBs are allocated\nan equal share of the excess up to their request. That is,\nwith $|\\mathcal{H}|$ highly loaded eNBs, the grants are\n\\begin{equation}\n\\gamma_{son} = \\min \\left(\\rho_{son},\\; \\Gamma_{so} +\n       \\frac{\\xi}{|\\mathcal{H}|}\\right).\n\\label{HighGrant}\n\\end{equation}\n\n\\subsection{Scheduling Fairness} \\label{sec:fariness}\nWithin the context of our proposed Sm-GW scheduling framework, fairness is\nthe measure of network accessibility of all $N_s$ eNBs\nconnected to an Sm-GW $s$ based on individual eNB uplink throughput level\nrequirements.\nWe denote $T_{son}$ for the long-run average throughput level [bit\/s] of\nuplink traffic generated at eNB $n,\\ n = 1, 2, \\ldots, N_s$,\nat Sm-GW $s$ for operator $o$.\nThe throughput level $T_{son}$ can for instance be obtained through\nsmoothing of the requests $\\rho_{son}$ over successive cycles $w$.\nIn order to avoid clutter, we omit the subscripts $s$ and $o$ in the remainder\nof this fairness evaluation.\nWe define the following fair target throughput levels $\\Omega_n$ [bit\/s]:\nLightly loaded eNBs $l \\in \\mathcal{L}$\nwith throughput levels $T_l < \\Gamma\/W$,\nshould be able to transmit their full traffic load, i.e., $\\Omega_l = T_l$.\nNext, consider highly loaded eNBs $h \\in \\mathcal{H}$ with throughput\nlevels $T_h > \\Gamma\/W$.\nIf the total throughput requirement of all eNBs\n$\\sum_{l \\in \\mathcal{L}} T_{l} +\\sum_{h \\in \\mathcal{H}}T_h$\nis less than or equal to the\nuplink transmission bitrate $G$, then the highly loaded\neNBs should be able to transmit their full traffic load, i.e.,\n$\\Omega_h = T_h$.\nOn the other hand, if the total traffic load exceeds the uplink transmission\nbitrate, i.e., if $\\sum_{l \\in \\mathcal{L}} T_{l} +\\sum_{h \\in \\mathcal{H}}T_h > G$, then\nthe highly loaded eNBs should be able to transmit traffic up to\nan equitable share of the uplink\ntransmission bitrate not used by the lightly loaded eNBs.\nThus, overall:\n$\\Omega_h = \\min \\{T_h,\\ (G - \\sum\\limits_{l \\in \\mathcal{L}} T_{l} )  \/\n\t|\\mathcal{H}| \\}$. We define the normalized distance $\\mathcal{E}_n$\nof the actually achieved (observed) throughput $\\tau_n$ and the\ntarget throughput $\\Omega_n$, i.e.,\n$\\mathcal{E}_n = \\tau_n - \\Omega_n$.\n\nBased on the preceding target throughput definitions, we obtain\nthe normalized distance throughput fairness index~\\cite{JainIndex}\n\\begin{equation}\n\\mathcal{F}_T =\n\\frac{ \\sqrt{ \\sum_{n = 1}^N \\mathcal{E}^2_n}}\n{\\sqrt{\\sum_{n =1}^N\\Omega^2_n}},\n\\label{FairIndex}\n\\end{equation}\nwhereby $\\mathcal{F}_T$ close to zero indicates\nfair Sm-GW scheduling.\n\n\\subsection{Sm-GW Scheduling Overhead}\nIn configuration adaptive Sm-GW scheduling, a reconfiguration event,\ni.e., an eNB connect or disconnect event, triggers the re-evaluation\nof the grant size limit $\\Gamma_{so}$, see Section \\ref{gwsch:sec:equal}.\nThe new $\\Gamma_{so}$ value is sent to all eNBs.\nSince reconfiguration events occur typically only rarely, e.g., on\nthe time scale of minutes or hours, the overhead for\nconfiguration adaptive scheduling is negligible.\n\nTraffic adaptive Sm-GW scheduling requires each eNB $n$ to send a request\nevery cycle of duration $W$ seconds.\nUpon reception of the requests from all $N_{s}$ eNBs,\nthe Sm-GW evaluates and sends the grants to the respective eNBs,\nas illustrated in Fig.~\\ref{fig_ExcessTiming}(a).\nThe requests and grants can typically be sent simultaneously, i.e., in\nparallel, over the individual eNB-to-Sm-GW links.\nThus, within a cycle duration $W$,\nthe overhead amounts to the transmission delays of the request and\ngrant messages, the maximum round-trip propagation delay between eNBs and Sm-GW,\nand the schedule processing delay at the Sm-GW.\nFor typical parameter settings, such as\n70~Byte messages transmitted at 1~Gbps, up to 500~m eNB-to-Sm-GW\npropagation distance, $W=1$~ms cycle duration, and schedule processing\ndelay on the order of microseconds, the overhead is less than half a\npercent.\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\centering\n\t\\includegraphics[width=3.5in]{finman_smgw\/fig7a.pdf}\\\\\n\n\t\\footnotesize (a) Observed avg. throughput\n\tof lightly loaded eNBs  \\\\ \\footnotesize\n\t\\includegraphics[width=3.5in]{finman_smgw\/fig7b.pdf} \\\\\n\t(b) Observed avg. throughput of highly loaded eNBs\\\\\n\t\\label{fig_SmGW_sch2}\n\\end{figure*}\n\n\\begin{figure*}[t]\n      \\centering\n      \\centering\n      \\includegraphics[width=3.5in]{finman_smgw\/fig7c.pdf}  \\\\\n      \\footnotesize (c) Avg. delay of lightly loaded $T_L$ eNBs,\n      when $T_H = 80$ Mbps \\\\\n      \\includegraphics[width=3.5in]{finman_smgw\/fig7d.pdf} \\\\\n      \\footnotesize (d) Fairness Index $\\mathcal{F}_T$,\n      when $T_H$ = 200 Mbps.\n      \\caption{Simulation results for Sm-GW scheduling.}\n      \\label{fig_SmGW_sch}\n\\end{figure*}\n\n\\subsection{Evaluation of Sm-GW Scheduling}\n\\subsubsection{Simulation Setup}\nWe evaluate the performance of Sm-GW scheduling\nwith the discrete event simulator OMNET++.\nWe consider a given Sm-GW $s$ with an uplink\ntransmission bitrate to a given operator $o$ of\n$G_{so} = 1$~Gbps. We omit the subscripts $s$ and $o$ in the remainder\nof this evaluation section to avoid notational clutter.\nThe LTE access network typically requires the\npacket delay to be less than 50~ms~\\cite{3GPPQoS}.\nTherefore, we set the Sm-GW queue size to\n20~MBytes, which is equivalent to a maximum queuing delay of 20~ms\nover the $G = 1$~Gbps link. Without any specific scheduling,\nthe Sm-GW operates in first-come-first-served mode with taildrop.\n\nWe simulate the typical bursty eNB traffic\ngeneration pattern, with two eNB traffic rate states: low and heavy.\nThe sojourn time in\na given traffic rate state is randomly drawn from a\nuniform distribution over 1~ms to 4~ms. At the end of the sojourn time,\na switch to another state occurs with a probability of 70~\\%\nin the low traffic state\nand 30~\\% in the heavy traffic state. The traffic bitrate\nratio between the heavy and low traffic states is $4:1$.\nWithin a given traffic rate state, data packets are randomly\ngenerated according to independent Poisson processes.\n\nWe consider $|\\mathcal{L}| = 10$ lightly\nloaded eNBs and $|\\mathcal{H}| = 10$ highly loaded eNBs connected to\nthe considered Sm-GW.\nEach eNB, irrespective of whether it is lightly or highly loaded, generates\ntraffic according to the two traffic rate state (low and heavy) model.\nThe low and heavy traffic rates are set such that the long-run\naverage generated traffic rate corresponds to a prescribed required\nthroughput (load) level\n$T_L < G\/N = 50$~Mbps for a lightly loaded eNB and a prescribed required\nthroughput (load)\nlevel $T_H > G\/N$ for a highly loaded eNB.\nFor all simulations, the 95~\\% confidence intervals are less\nthan 5~\\% of the corresponding sample mean.\n\n\\subsubsection{Simulation Results}\n\\paragraph{Without Sm-GW Scheduling}\nIn Fig.~\\ref{fig_SmGW_sch}, we show representative evaluation results\ncomparing configuration adaptive equal-share Sm-GW scheduling\nand traffic adaptive excess-share Sm-GW scheduling with the\nconventional backhaul without Sm-GW scheduling.\nFigs.~\\ref{fig_SmGW_sch}(a) and (b) show the actual (achieved, observed)\nthroughput\n$\\tau$ of lightly loaded and highly loaded eNBs, respectively, as a function\nof the generated lightly loaded ($T_L$) and highly loaded ($T_H$)\nthroughput levels.\nWe observe from Figs.~\\ref{fig_SmGW_sch}(a) that without scheduling,\nthe lightly loaded eNB suffer reductions in the achieved throughput, that\nare especially pronounced (over 30~\\%) for the high $T_H = 200$~Mbps\nload of the highly loaded eNBs.\nAt the same time, we observe from Figure~\\ref{fig_SmGW_sch}(b)\nthat without scheduling, the highly loaded eNBs achieve more than\ntheir fair throughput share.\nFor instance, for the highly loaded eNB throughput requirement (load)\n$T_H = 140$~Mbps, and $T_L = 30$~Mbps, the observed throughout of the highly\nloaded eNBs is $\\tau_H = 76$~Mbps, which is significantly higher than\nthe fair share of $(G - |\\mathcal{L}| T_L) \/|\\mathcal{H}| = 70$~Mbps.\nThe unfairness arising without scheduling is further illustrated in\nFig.~\\ref{fig_SmGW_sch}(c), where we observe a sharp delay increase\nat $T_L = 20$~Mbps, when the total traffic load\n$|\\mathcal{L}| T_L + |\\mathcal{H}| T_H$ approaches the uplink transmission\nbitrate $G$.\nMoreover, from Fig.~\\ref{fig_SmGW_sch}(d), we observe an increasing\nfairness index $\\mathcal{F}_T$ as the lightly loaded eNBs generate more\ntraffic, i.e., as $T_L$ increases.\nThat is, as the lightly loaded eNBs try to transmit more traffic,\ntheir achieved throughput falls more and more below their\nfair share [see growing divergence between\nthe no scheduling curves and straight lines for\nscheduling in Fig.~\\ref{fig_SmGW_sch}(a)], leading to increasingly\nunfair treatment of the lightly loaded eNBs.\n\n\\paragraph{Equal-share Sm-GW Scheduling}\nWe observe from Fig.~\\ref{fig_SmGW_sch}(a) and (c) that\nlightly loaded eNBs benefit from equal-share scheduling in that they\nget the full share of their fair target throughput and experience\nlow delay.\nHowever, we observe from Fig.~\\ref{fig_SmGW_sch}(b) that\nhighly loaded eNBs achieve only a throughput of\n$G\/(|\\mathcal{L}| + |\\mathcal{H}|) = 50$~Mbps as\nequal-share Sm-GW scheduling assigns a configuration adaptive allocation of\nequal shares of the limited uplink transmission bitrate $G$ to all\neNBs irrespective of their traffic generation rates.\nCorrespondingly, we observe from Fig.~\\ref{fig_SmGW_sch}(d), a high\nfairness index $\\mathcal{F}_T$ for low traffic loads of the\nlightly loaded eNBs, as the highly loaded eNBs receive only\nunfairly small shares of the uplink transmission bitrate $G$.\n\n\\paragraph{Excess-share Sm-GW Scheduling}\nWe observe from Fig.~\\ref{fig_SmGW_sch}(a) and (b)\nthat with excess-share Sm-GW scheduling, both\nlightly loaded eNBs and highly loaded eNBs\nachieve their fair target throughput.\nWe further observe from Figs.~\\ref{fig_SmGW_sch}(c) and (d)\nthat excess-share Sm-GW scheduling gives also\nfavorable delay and fairness index performance.\n\n\\paragraph{Summary}\nWe conclude that scheduling\nof the Sm-GW uplink transmission bitrate $G$ is necessary to prevent backhaul\nbandwidth starvation of lightly loaded eNBs due to the overwhelming\ntraffic rates of highly loaded eNBs. On the other hand,\nsimple configuration adaptive\nallocation of equal uplink transmission bitrate\nshares to each eNB wastes bandwidth.\nFlexible traffic adaptive scheduling according to the\ntraffic loads of the eNBs, e.g., through excess-share\nscheduling, can ensure fairness while efficiently utilizing the\nuplink transmission bitrate.\n\n\\section{\\MakeUppercase{SDN Based Multi-Operator Management}}  \\label{MOM:sec}\n\\subsection{Overview}\nIn this section we introduce a novel SDN based network management\nframework for flexible sharing of the backhaul resources of\nmultiple operators.\nIn particular, the framework\nintroduced in Sections~\\ref{rap:sec}--\\ref{optex:sec}\nallows a set of Sm-GWs to flexibly share the\nuplink transmission bitrate of a given single operator.\nThe inter-operator sharing introduced in Section~\\ref{roam:sec}\nallows the Sm-GWs to flexibly share the uplink transmission bitrates\nof multiple operators.\nOur proposed multi-operator management framework accommodates\ndynamic changes of the traffic requirements of the small cells, such as\nchanges of the generated uplink traffic bitrates, as well as dynamic changes of\nthe operator characteristics, such as changes of the available\nuplink traffic bitrates.\nIn the proposed multi-operator management framework,\nan SDN orchestrator dynamically\nconfigures the Sm-GWs and the transport network connecting\nthe Sm-GWs to the operator gateways to flexibly adapt to changes in\nsmall cell traffic loads and the operator characteristics.\n\n\\subsection{Request and Allocation Procedures}  \\label{rap:sec}\nIn a small cell deployment environment, such as a large organization,\nmultiple Sm-GWs can serve multiple buildings.\nFor example, in a university setting,\na library can be equipped with an Sm-GW\nand the administration building can be equipped with another Sm-GW.\nThe throughput requirements and priorities\nof these buildings typically vary widely over time.\nFor instance, the administration building experiences a large\nvisitor influx during graduation and student admission periods, while\nmany students visit the library during exam week.\nMoreover, services from multiple operators may need to be\nshared among the buildings in a given organization,\ni.e., among multiple Sm-GWs.\nHence, there is a need for highly flexible traffic management\nwithin the large organization based on time-varying priorities\nand throughput requirements.\n\n\\begin{figure*}[t!]\n    \\centering\n    \\includegraphics[width=1\\textwidth]{finman_smgw\/fig8.pdf}  \n    \\caption{Illustration of SDN based multiple-operator management\nserving multiple Smart Gateways (Sm-GWs).}\n    \\label{fig_OPmanagement}\n\\end{figure*}\n\nSuppose, the Sm-GWs $s,\\ s = 1, 2, \\ldots, S$, and\noperators $o,\\ o = 1, 2, \\ldots, O$, are interconnected\nin a full mesh transport network, as illustrated in Fig.~\\ref{fig_OPmanagement}.\nAs described in Section~\\ref{gwsch:sec:excess}, with traffic adaptive\nSm-GW scheduling, each eNB $n$ sends its operator $o$ specific uplink\ntransmission bitrate request to Sm-GW $s$ in every cycle.\nThe requested uplink transmission data amounts $\\rho_{son}$\nwill typically vary over time scales that are long enough to\nreasonably permit adaptations of the Sm-GW configurations.\nFor instance, the requests will typically change\n on the time scales of several seconds or minutes, or possibly even\nlonger, such as the seasonal variations in the visitor volume to\nuniversity buildings.\nIn order to obtain the variational characteristics of the\neNB requirements, the operator specific requests at the Sm-GWs can be\naggregated over the eNBs and appropriately smoothed, i.e., averaged over time,\nto obtain an aggregate smoothed uplink transmission bitrate request\n$R_{so}$ [bit\/s] from Sm-GW $s$ to operator $o$.\n\nIdeally, the backhaul network should adapt to varying requirements at\nthe Sm-GWs to maximize the network utilization.\nWe exploit the centralized control property of SDN to\nadaptively configure the network for variable requirements.\nMore specifically, the SDN orchestrator in Fig.~\\ref{fig_OPmanagement}\noptimizes the allocations $G_{so}$ of operator $o$\nuplink transmission bitrate [bit\/s] to the individual Sm-GWs $s$.\nThe SDN orchestrator thus ensures that the grants to\nthe eNBs are globally maximized (subject to the operators' constraints\nand requirements).  When the optimized allocations $G_{so}$ are used at the\nSm-GW scheduler (Section~\\ref{gwsch:sec}),\nthe maximum allowed traffic flow is sent from the Sm-GWs\nto each operator core.\n\n\\subsection{Optimization Decision Variables and Constraints for Multi-Operator Management with Sm-GWs}\nIn this section,\n  we define a general optimization model for the multi-operator\nmanagement framework. Specifically, we define the constraints and\ndecision variables for optimizing the multi-operator management.\nThe defined decision variables and constraints are employed\nfor the operation of the SDN orchestrator,\nas detailed in Section~\\ref{sdnorchpro:sec}.\nThe SDN orchestrator can employ arbitrary objective functions and\nconstraint allocation strategies for the optimization, as illustrated\nfor an elementary example in Section~\\ref{optex:sec}.\n\n\\subsubsection{Constraints}\nRequests for the uplink transmission of $\\rho_{son}$ [bits] from\neNBs $n,\\ n = 1, 2, \\ldots, N_s$, arrive at Sm-GW $s,\\ s = 1, 2, \\ldots, S$,\nevery cycle of duration $W$ seconds, i.e., on the order of milliseconds,\nrequesting uplink transmission bitrates from operator $o,\\ o = 1, 2, \\ldots, O$.\nThe Sm-GW aggregates the requests over the eNBs $n$ and smoothes the\naggregated requests to obtain the smoothed aggregated requests $R_{so}$.\nDenoting $w$ for the cycle index, an elementary\nweighted sampling smoothing computes\n\\begin{equation}\n  R_{so}(w) = \\alpha\\; \\left( \\frac{1}{N_s}\n  \\sum_{n=1}^{N_s}  \\frac{\\rho_{son}(w)}{W}  \\right)\n   + (1-\\alpha)\\; R_{so}(w-1),\n\\label{filteredReports}\n\\end{equation}\nwhere $\\alpha$ denotes the weight for the most recent request sample.\nA wide variety of other smoothing mechanism can be employed and optimized\naccording to the specific deployment settings.\nThe smoothed requests $R_{so}$ are periodically (with a period typically\nmuch longer than the eNB reporting window) sent to the\nSDN orchestrator.\nIn particular, each Sm-GW $s$, sends a vector of smoothed requests\n$\\overrightarrow{R_{s}} = [R_{s1} \\; R_{s2} \\; \\cdots \\;R_{sO}]$\ncontaining the aggregated and smoothed requests for each operator $o$\nto the SDN orchestrator.\nThe SDN orchestrator combines the request vectors $\\overrightarrow{R_{s}}$ to\nform the request matrix\n\\begin{equation}  \\label{reportMatrix}\n\\mathbf{R} = [ R_{so} ],\\ s = 1, 2, \\ldots, S;\\ o = 1, 2, \\ldots, O.\n\\end{equation}\n\nEach operator $o,\\ o = 1, 2, \\ldots, O$, can enforce\na set of constraints $K_{oc},\\ c = 1, 2, \\ldots, C$,\nrepresented by a constraint vector\n $\\overrightarrow{K_{o}} = [K_{o1} \\; K_{o2} \\; \\cdots \\;K_{oC}]$\nthat is sent to the SDN orchestrator.\nEach constraint $c$ may be associated with a particular specification from\noperator $o$, e.g., for traffic shaping of the flows or for the aggregate\nmaximum bitrate.\nIn order to avoid clutter and not to obscure the main ideas of our\noverall multi-operator management framework, we consider in this study\na single constraint for each operator $o$.\nThat is, in place of the constraint vector $\\overrightarrow{K_{o}}$\nwe consider a single (scalar) constraint $K_o$.\nThe SDN orchestrator combines the scalar constraints from the\nvarious operators $o$ to form the constraint vector\n\\vspace{0cm}\n\\begin{equation}\n\\mathbf{K} = [K_{1} \\; K_{2} \\; \\cdots \\;K_{O}].\n\\label{constrantsVec}\n\\end{equation}\n\n\\subsubsection{Decision Variables}\nThe Sm-GW $s$ scheduler uses the operator $o$ specific grant size limits\n$\\Gamma_{so}$ to schedule\/assign uplink transmission grants to eNBs\n(see Sections~\\ref{gwsch:sec:equal} and~\\ref{gwsch:sec:excess}). By\ncontrolling the variable $\\Gamma_{so}$ specific to operator $o$ we\ncan control the flow of traffic outward from the Sm-GW, i.e.,\ntowards the respective operator $o$. The long-term average traffic\nflow rates $X_{so}$ [bit\/s] from the Sm-GW $s,\\ s = 1, 2, \\ldots, S$, to the\noperators $o,\\ o = 1, 2, \\ldots, O$, can be expressed as matrix\n\\begin{equation}   \\label{flowMatrix}\n\\mathbf{X} = [X_{so}],\\ s = 1, 2, \\ldots, S; o = 1, 2, \\ldots, O.\n\\end{equation}\n\nThe operator $o$ specific uplink transmission bitrates\n$G_{so}$ granted to the Sm-GWs are evaluated at the\nSDN orchestrator, based on the\nrequest matrix $\\mathbf{R}$ and the constraint vector $\\mathbf{K}$.\nThe orchestrator\nresponds to the request vector $\\overrightarrow{R_s}$ from each\nSm-GW $s$ with a grant vector $\\overrightarrow{G_s}$.\nAt the SDN orchestrator, the grant vectors $\\overrightarrow{G_{s}}$\ncan be combined to form the orchestrator grant matrix\n\\begin{equation}   \\label{GmaxMatrix}\n\\mathbf{G} = [G_{so}],\\ s = 1, 2, \\ldots, S; o = 1, 2, \\ldots, O.\n\\end{equation}\n$\\mathbf{G}$ is a positive (non-negative) matrix, since the\nmatrix elements $G_{so},\\ G_{so} \\geq 0$, correspond granted\nuplink transmission bitrates.\n\nOur objective is to maximize the traffic flow rates $X_{so}$ from the\nSm-GWs $s$ to the operators $o$ subject to the operator\nconstraints $\\mathbf{K}$.\nIn particular, the aggregated traffic\nsent from the Sm-GWs $s,\\ s = 1, 2, \\ldots, S$,\nto the operator $o$ core should satisfy the operator\nconstraint $K_o$, i.e.,\n\\begin{equation}\n\\sum_{s = 1}^S X_{so} \\leq K_o,\\ \\ \\ \\forall o,\\ o = 1, 2, \\ldots, O.\n\\label{Eq:OpCnstGen}\n\\end{equation}\n\nUsing the grant vector $\\overrightarrow{G_{s}}$ at Sm-GW $s$\nto assign, i.e., to schedule, uplink traffic grants to the eNBs\n(see Section~\\ref{gwsch:sec}) ensures that the\ntraffic flow rates $X_{so}$ from Sm-GW $s$ to operator $o$ are bounded\nby $G_{so}$, i.e., \\begin{equation}\nX_{so} \\leq G_{so},\\ \\ \\forall (s,o).\n\\label{eq:geqx}\n\\end{equation}\nThus, in order to ensure that the traffic flows\n$X_{so}$ satisfy the operator constraints $\\mathbf{K}$,\nthe grants $G_{so}$ must satisfy the operator constraints, i.e.,\n\\begin{equation}\n\\sum_{s = 1}^S G_{so} \\leq K_o, \\;\\;\\;\\;\\; \\forall o,\\ o = 1, 2, \\ldots, O.\n\\label{Eq:OpCnstGenGrant}\n\\end{equation}\nIn order to maximize the traffic flows $X_{so}$ to each operator $o$,\nthe SDN orchestrator needs to grant each Sm-GW $s$\nthe maximum permissible uplink transmission bitrate $G_{so}$.\n\n\n\\begin{algorithm}[t!]\n    \\caption{SDN Orchestrator procedure}\n    \\label{algo:SDN}\n    \\SetKwInOut{Input}{input}\n    \\SetKwInOut{Output}{output}\n    \\nonl {\\bfseries{1. Sm-GWs}} \\newline\n    \\nl (a) Evaluate aggregate smoothed requests $R_{so}$ from\n    eNB requests $\\rho_{son}$,  Eqn~(\\ref{filteredReports}). \\newline\n    (b) Periodically send request vector $\\overrightarrow{R_s}$\n     to SDN orchestrator\n\\newline\n    \\nonl \\If{Grant vector $\\overrightarrow{G_s}$ is received}{\n     \\nonl   Update SM-GW (to eNBs) grant size limits $\\Gamma_{so}$,\n    \\nonl }\n\n    \\nonl   {\\bfseries{2. Operators}} \\newline\n    (a) Send constraint $K_o$ to SDN orchestrator\n\n    \\nonl   {\\bfseries{3. SDN Orchestrator}}   \\newline\n    \\nonl     \\If{Request vector $\\overrightarrow{R_s}$ is received\n   \\textbf{OR} constraint $K_o$ is received}{\n     \\nonl   Re-optimize orchestrator (to Sm-GW) grants $\\mathbf{G}$ \\;\n      \\nonl  Send grant vector $\\overrightarrow{G_s}$ to Sm-GW $s$;\n    \\nonl }\n\n\\end{algorithm}\n\\subsection{SDN Orchestrator Operation}  \\label{sdnorchpro:sec}\nThe operational procedures for evaluating the\nSDN orchestrator grant matrix $\\mathbf{G}$ (\\ref{GmaxMatrix})\nare executed in parallel in the Sm-GWs, operators, and the SDN orchestrator,\nas summarized in Algorithm~\\ref{algo:SDN}.\nThe Sm-GWs aggregate and smooth the eNB requests and periodically\n  send the request vector $\\overrightarrow{R_s}$ to the SDN orchestrator.\nThe SDN orchestrator optimizes the grant matrix $\\mathbf{G}$\nupon the arrival of a new Sm-GW request vector $\\overrightarrow{R_s}$ or\n a change in an operator constraint $K_o$.\nThe orchestrator updates the Sm-GWs with the newly\nevaluated orchestrator grant vectors $\\overrightarrow{G_s}$,\nwhich update their grant size limits $\\Gamma_{so}$.\n\nOur SDN based multi-operator management framework\n  allows for a wide variety of\nresource (uplink transmission bitrate) allocations from the multiple operators\nto the Sm-GWs.\nIn order to illustrate the introduced framework, we consider\nnext an elementary specific optimization problem formulation with\na linear objective function and a proportional constraint allocation\nstrategy that allocates the uplink transmission bitrate constraints\nproportional to the requests.\nMore complex objective functions and allocation strategies,\ne.g., objective functions\nthat prioritize specific grants are an interesting direction for\nfuture research.\nWe note that this illustrative example does not exploit inter-operator\nsharing, which is examined in Section~\\ref{roam:sec}.\n\n\\subsection{Illustrative Optimization Example with Linear Objective\n  Function and Request-Proportional Constraint Allocations}  \\label{optex:sec}\nSince the grants $G_{so}$ are non-negative, an elementary objective\nfunction can linearly sum the grants $G_{so}$, i.e., as\n$\\sum_{s = 1}^S \\sum_{o = 1}^O G_{so}$.\nFor the constraint allocation,\nwe consider the aggregate over all Sm-GWs $s$ of the\naggregated smoothed requests $R_{so}$\nfor a specific operator $o$, i.e., we consider\nthe unit norm of the request vector\n$\\| \\overrightarrow{R_o} \\|_1 = \\sum_{s = 1}^S R_{so}$.\nIf $\\| \\overrightarrow{R_o} \\|_1$ is less than the operator constraint $K_o$,\nthen the corresponding grants $G_{so}$ are set to the requests,\ni.e., $G_{so} = R_{so}$.\nOn the other hand, if $\\| \\overrightarrow{R_o} \\|_1 > K_o$,\nthen we proportionally assign the operator $o$ backhaul bandwidth $K_o$, i.e.,\nwe assign the proportion $R_{so} \/ \\| \\overrightarrow{R_o} \\|_1$\nof the constraint $K_o$. Thus,\n\\begin{equation}\nG_{so} = \\min \\left( R_{so},\\\n    \\frac{R_{so}}{\\|\\overrightarrow{R_o}\\|_1} K_o\\right).\n\\label{Eq:propGrantThresh}\n\\end{equation}\nThe resulting elementary optimization problem can be summarized as:\n\\vspace{0cm}\n\\begin{equation}\n\\begin{split}\n\\text{Maximize} \\;\\;\\; \\sum_{s = 1}^S \\sum_{o = 1}^O G_{so}\\;\\;\\;\\;\n\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\\\\n\\text{Subject to:} \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\:\\:\\;\\;\\;\\;\\;\\;\\;\\;\n\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\:\\:\\;\\;\\;\\;\\\\\n\\forall s \\;\\in\\; \\{ 1, 2,\\ldots,S\\}\\;\\;\\; \\text{and}\n\\;\\;\\;\\forall o \\; \\in \\; \\{ 1, 2,\\ldots,O\\},\\\\\n-G_{so} \\leq 0,\\;\\;\\;\\;\\;\\;\\;\\:\\;\\;\\;\\;\\;\\;\\;\\;\n\\;\\;\\;\\:\\;\\;\\;\\;\\;\\;\\;\\;\\;\\\\\n\\;\\;\\; \\;G_{so} \\leq R_{so}, \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\n\\:\\:\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\\\\nG_{so} \\leq K_{o} \\frac{R_{so}}{\\| \\overrightarrow{R_o}\\|_1 }.\n\\;\\;\\;\\;\\;\\;\\;\\;\\;\\:\\;\\;\\;\\;\\;\\:\n\\end{split}\n\\label{optimization}  \\end{equation}\n\n\\subsection{Inter-Operator Sharing}\n\\label{roam:sec}\nWhen the aggregate backhaul bandwidth $\\|R_o\\|_1$ requested from\nan operator $o$ exceeds its constraint $K_o$, inter-operator sharing\ncan be employed to route the additional traffic through the\n network managed by another operator.\nOur proposed Sm-GW multi-operator management\nprovides a distinctive advantage in\nmaintaining active connections with multiple operators to\neasily route the excess traffic to a sharing operator.\nWe denote $o = m$ for the operator\nthat accepts the sharing traffic from an other operator $o = e$ whose\ntraffic constraint has been exceeded.\nIn this study, we focus on one operator accepting sharing traffic and\none operation with excess traffic.\nThe extension to sets of multiple operators accepting\nsharing traffic and multiple operators with excess traffic\nis left for future research. An operator in sharing $m$ should have\nlow incoming traffic as compared to the constraints $K_m$ in order to accept\nthe traffic from the operator in excess $e$. Therefore, for\nthe sharing ($o = m$) and excess ($o = e$) operators the requests\n$R_{so}$ need to satisfy,\n\\begin{equation}\n\\sum_{s = 1}^S R_{sm} < K_m,\\ \\ \\mbox{and}\\ \\\n\\sum_{s = 1}^S R_{se} > K_e.    \\label{Eq:op:class}\n\\end{equation}\nThe traffic rate from excess operator~$e$ that\ncan be carried by sharing operator $m$ depends on the\nunutilized slack uplink transmission bitrate of operator~$m$:\n\\begin{equation}\n\\begin{split}\n\\zeta = K_m - \\sum_{s = 1}^S R_{sm}.\n\\end{split}\n\\label{eq:op:ex}\n\\end{equation}\nIf $\\zeta > 0$, the last constraint in optimization problem\n(\\ref{optimization}) for the excess operator $e$ is replaced by the constraint\n\\begin{equation}\nG_{se} \\leq \\left(K_{e} + \\zeta\\right)\\frac{R_{se}}\n   {\\| \\overrightarrow{R_e}\\|_1 }\\ \\ \\ \\forall s.\n\\end{equation}\n\n\\begin{figure}[t!]  \\centering\n    \\includegraphics[width=3.2in]{finman_smgw\/fig9.pdf}  \n    \\caption{Traffic rates simulation.}\n    \\label{fig_opti1}  \\end{figure}\n\n\\begin{figure*}[t!]\n\t\\centering\n    \\includegraphics[width=0.9\\textwidth]{finman_smgw\/fig10.pdf}\n\t\\caption{Inter-operator sharing evaluation.}\n\t\\label{fig_roam2}\n\\end{figure*}\n\n\\subsection{Evaluation of Multi-Operator Management} \\label{OMres:sec}\nIn order to showcase the effectiveness of the SDN based\nmulti-operator management framework,\nwe conducted simulations for the elementary optimization with linear\nobjective function and proportional constraint sharing\n(see Section~\\ref{optex:sec}).\nWe consider $S=2$ Sm-GWs and $O=2$ operators.\nAs comparison benchmark, we consider a static equal allocation of\noperator uplink transmission bitrate $K_o$ to the $S$ Sm-GWs, i.e.,\neach Sm-GW $s,\\ s = 1, 2$, is allocated $K_o \/ S$ of the operator $o$\nuplink transmission bitrate.\n\n\\subsubsection{Without Inter-Operator Sharing}\nIn Fig.~\\ref{fig_opti1} we plot the Sm-GW $s$ to operator $o$\ntraffic flow rates $X_{so}$ resulting from the optimized\nSDN orchestrator grants $G_{so}$ as a function of\nthe uplink transmission bitrate requested by Sm-GWs $s = 1$ and $s = 2$\nfrom operator $o = 1$. Specifically, Sm-GW $s = 1$ requests bitrate\n$R_{11} = 2R$ and Sm-GW $s = 2$ requests bitrate $R_{21} = R$ from operator\n$o = 1$. The bitrate requests from operator $o = 2$ are fixed at 50~Mbps.\nEach operator $o,\\ o = 1, 2$,\nhas uplink transmission bitrate constraint $K_o = 100$~Mbps.\n\nWe observe from Fig.~\\ref{fig_opti1} that for requests for\n operator $o = 1$ bitrate up to $R = 25$~Mbps,\nthe traffic rates $X_{11}$ and $X_{21}$ are equal to the requests, irrespective\nof whether SDN orchestrated optimization is employed for not.\nIn contrast, as the requested bitrate increases above $R = 25$~Mbps,\ni.e., the bitrate $R_{11} = 2R$ requested by Sm-GW $s = 1$ from operator\n$o = 1$ increases above $K_1 \/ S = 50$~Mbps, the granted bitrate $G_{11}$\nwith SDN orchestration and the corresponding traffic flow $X_{11}$\ncontinue to increase. On the other hand, the granted bitrate $G_{11}$ and traffic\nflow $X_{11}$ without SDN orchestration stay limited at the static\nequal share $X_{11} = G_{11} = K_1 \/ S = 50$~Mbps.\n\nAs the requested bitrate $R$ increases above 33.3~Mbps, i.e., a total of\n$3R = 100$~Mbps requested from operator $o = 1$,\nwe observe from Fig.~\\ref{fig_opti1} that without orchestration,\nthe traffic flow $X_{21} $ from Sm-GW $s = 2$ to operator $o = 1$\ngrows to and then remains at the static equal share $K_1 \/ S = 50$~Mbps.\nThat is, the conventional static uplink transmission bitrate allocation\nresults in unfair disproportional backhaul service.\nIn contrast, our dynamic multi-operator management with\nSDN orchestrated optimization based on proportional sharing adaptively\nassigns the operator $o = 1$ bitrate to Sm-GWs $s=1$ and $s = 2$\nproportional to their requests.\n\n\\subsubsection{With Inter-Operator Sharing} \\label{sec:withroaming}\nIn Fig.~\\ref{fig_roam2}, we plot the Sm-GW $s$ to operator $o$\ntraffic flow rates $X_{so}$\nas a function of the uplink transmission bitrate $R_{11} = R$ requested by\nSm-GW $s = 1$ from operator $o = 1$ when inter-operator sharing is employed.\nSm-GW\n$s = 1$ requests bitrate $R_{12} = 100 - R$~Mbps from operator $o = 2$.\nAlso, Sm-GW $s = 2$ requests fixed bitrates\n$R_{21} = R_{22} = 20$~Mbps from each operator. Each operator $o$ has a fixed\nuplink transmission bitrate constraint of $K_o = 50$~Mbps. Note that operator\n$o = (m) = 1$ has slack uplink transmission bitrate\nwhen $R \\leq 30$~Mbps and can thus serve\nas roaming operator for the excess traffic to operator $e = 2$.\nAs $R$ increases and starts to exceed 70~Mbps, the roles are reversed, so\nthat operator $e = 1$ can send some of its excess traffic to roaming\noperator $m = 2$.\n\nFocusing initially on the case $R = 100$~Mbps, i.e., the right edge of\nFig.~\\ref{fig_roam2}, we observe that without SDN orchestrated optimization,\nSm-GW $s = 1$ can only transmit at its fixed static allocation of\n$X_{11} = K_1 \/ S = 25$~Mbps to operator $o = 1$, even though\nSm-GW $s = 1$ has a traffic load demanding $R_{11} = R = 100$~Mbps.\nAt the same time, Sm-GW $s = 2$ transmits at its requested rate\n$X_{21} = R_{21}  = 20~\\mbox{Mbps} < K_1\/S$. Thus, the operator $o = 1$\nuplink transmission bitrate $K_1$ is underutilized,\neven though Sm-GW $s = 1$ has\nmore traffic to send, but cannot due to the inflexible static\nuplink transmission bitrate allocations.\n\nWith SDN orchestrated optimization with proportional sharing,\nthe overloaded uplink transmission bitrate\n$K_1 = 50$~Mbps of operator $o = 1$ is shared between\nthe two Sm-GWs, allowing Sm-GW $s = 1$ to transmit\n$X_{11} = R_{11} \/ (R_{11} + R_{21}) = 41.7$~Mbps, while Sm-GW $s = 2$\ntransmits $X_{21} = R_{21} \/ (R_{11} + R_{21}) = 8.3$~Mbps.\nHowever, the uplink transmission bitrate\n$K_2$ of operator $o = 2$ is underutilized\nwith only Sm-GW $s = 2$ transmitting $X_{22} = 20$~Mbps.\n\nWith inter-operator sharing, the unutilized uplink transmission bitrate\n$\\zeta = K_2 - X_{22} = 30$~Mbps\nof operator $o = 2$, is used to carry excess traffic from\noperator $o = 1$.\nIn particular, the aggregate of the regular operator $o = 1$ uplink transmission\nbitrate $K_1$ and the uplink transmission bitrate available to operator $o = 1$ through\ntraffic sharing to operator $o = 2$ ($\\zeta = 30$~Mbps),\ni.e., $K_1 + \\zeta = 80$~Mbps is available to operator $o = 1$.\nWith proportional sharing, Sm-GW $s = 1$ can transmit\n$X_{11} = (K_1 + \\zeta) R_{11} \/ (R_{11} + R_{21}) = 66.7$~Mbps, while Sm-GW $s = 2$\ncan correspondingly transmit $X_{21} = 13.3$~Mbps, fully utilizing\nthe backhaul capacities of both operators.\n\nOverall, we observe from Fig.~\\ref{fig_roam2} that across the\nentire range of traffic loads $R$ from Sm-GW $s = 1$ for operator $o = 1$,\nour SDN based multi-operator orchestration with sharing is able to fully\nutilize the uplink transmission bitrates of both operators. Note in\nparticular, that depending on the traffic load, the roles\nof the two operators (excess or sharing) are dynamically adapted.\n\n\\section{\\MakeUppercase{Conclusions}}\nWe have introduced a new backhaul architecture by inserting a novel\nSmart Gateway (Sm-GW) between the wireless base stations (eNBs) and\nthe conventional operator gateways, e.g., LTE S\/P-GWs. The Sm-GW enables\nflexible support for large numbers of small cell base stations. In\nparticular, the Sm-GW adaptively schedules uplink\nbackhaul transmission grants to the individual eNBs on a\nfast (typically millisecond) timescale. In addition, an SDN\norchestrator adapts the allocation of the uplink transmission bitrate of the\nconventional gateways of multiple operators to the Sm-GWs on a slow\n(typically minutes or hours) time scale. Simulation results have\ndemonstrated that the scheduling of eNB grants by the Sm-GW can\ngreatly improve the backhaul service over conventional\nstatic backhaul uplink transmission bitrate allocations. Moreover, the\nSDN orchestrator substantially improves the utilization of the\nbackhaul bandwidth, especially when inter-operator sharing is permitted.\n\nThere are several important directions for future research on the\nSm-GW architecture and small cell backhaul in general.\nOne direction is to examine a variety of scheduling algorithms\nin the context of the Sm-GW.\nAnother direction\nis to examine different specific optimization objective functions\nwithin the general SDN orchestrator optimization introduced in this\narticle.\nMoreover, it is of interest to investigate\nQoS strategies for different traffic types, such as\ndata, voice, and video traffic \\cite{li2009ene, see2015wvs, pul2013tra,van2008tra, rei2002tra}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{MOSSCO concepts}\n\nThe modularity and coupling concepts proposed in this paper\nelaborate the design of a novel software system that emphasizes the needs of\nresearchers who want to  make maximum use of their existing knowledge in a  specific\nfield (e.g., geomorphology or marine ecology) but wish to conduct integrative research\nin a wider and flexible context. In strengthening modularity \\emph{sensu} independence\nof specific physical drivers, the new concept should, in addition to addressing the\nproblems listed above, support\n\\begin{inparaenum}[(1)]\n\\item liaisons  between traditionally separated modelling communities (e.g., coastal\nengineers, physical oceanographers and biologists),\n\\item inter-comparison  studies of, e.g., physical, geological, and biological modules, and\n\\item up-scaling studies where models developed at the  laboratory scale in a\nnon-dimensional context are applied to regional, global and Earth System scales.\n\\end{inparaenum}\n\nThe design of MOSSCO is application-oriented and driven by the demands for enabling and improving integrated\nregional coastal  modelling. It is targeted towards building coupled systems\nthat support decision making for local policies implementing the European Union Water\nFramework Directive (WFD) and Marine Strategic Planning Directive (MSPD).  From a design \npoint of view we envisioned a system that is foremost flexible and equitable.\n\n\\begin{description}\n\\item[Flexibility] means that the system itself is able to deal on the one hand \nwith a diverse small or large  constellation of coupled model components and on\nthe other hand with different orders of magnitude of spatial and temporal\nresolutions; it is able to deal equally well with zero-, one-, two- and three-dimensional\nrepresentations of the coastal system. Flexibility implies the capability to encapsulate\nalso existing legacy models to create one or more different ``ecosystems'' of models.\nThis feature should allow seamless replacement of individual model components, which\nis an important procedure in the continual development of integrated systems. Flexibly\nreplacing components finally creates a test-bed for model intercomparison studies.\n\n\\item[Equitability] means that all models in the coupled framework are treated as \nequally important, and that none is more important than any other.  This principle\ndissolves the primacy of the hydrodynamic or atmospheric models as the\ncentral hub in a coupled system.   Also, data components are as important as process \ncomponents or model output; any de facto difference in model importance should be\ngrounded on the research question, and not on technological legacy.  As complexity\ngrows by coupling\nmore and more models, this equitability also demands that experts in one particular\nmodel can rely on the functionality of other components in the system without having\nto be an expert in those models, as well. \n\\end{description}\n\nThe equitability design extends to participation:  contributions to the development\nof  components or the coupling framework itself is allowed and\nencouraged.  Anyone can use and modify the coupled framework or parts of both\nin a legal sense by open source licensing, and in an accessibility sense through\ntemplate codes and extensive documentation.\n\n\\subsection{Wrapping legacy models -- first steps in  \\texttt{PARSE}}\n\nAs MOSSCO is built around the ESMF hierarchy of components, any existing code\nthat can be wrapped in an ESMF component can be a component in MOSSCO, too.  \nThe ESMF user guide \\citep{ESMF2013usr} suggests a best \npractice method \\texttt{PARSE} to achieve this componentization of a legacy code.  \n\n\\begin{description}\n\\item[\\texttt{P}]repare the user code by splitting it into three phases that initialize, run and finalize a model;\n\\item[\\texttt{A}]dapt the model's data structures by wrapping them in ESMF infrastructure like states and fields;\n\\item[\\texttt{R}]egister the user's initialize, run, and finalize routines through ESMF;\n\\item[\\texttt{S}]chedule data exchange between components;\n\\item[\\texttt{E}]xecute a user application by calling it from an ESMF driver.\n\\end{description}\n\nThis \\texttt{PARSE} concept allows a smooth transition from a legacy model to an ESMF component. In this concept, the first \nthree steps have to be performed on the model side, and the latter two on the framework side and have been taken care \nof by the MOSSCO coupling layer. The \\emph{preparation} of the code is independent of the use of ESMF and provides \nthe basic couplability of the model; many existing models already implement this separation into initialize, run, and finalize phases, \neither structurally or more formally by implementing a Basic Model Interface \\citep[BMI,][]{Peckham2013}.  For the run \nphase, it is mandatory that this phase refers to a single model timestep and not to the entire run loop.\n\nThe \\emph{adaption} of a model's internal structures to ESMF consists of technically wrapping data into ESMF \ncommunication objects, and in providing sufficient metadata for communication.  Among these are grid definition and \ndecomposition, units and semantics of data, optimally following a metadata scheme like the widespread Climate and \nForecast \\citep[CF,][]{Eaton2011} or the more bottom-up Community Surface Dynamics Modeling System \n(CSDMS, following a scheme like  \nobject + [ operation ] + quantity, \\citealt{Peckham2014iemss}).  Both are currently being included in the emerging \nGeoscience Standard Names Ontology (GSN, \\href{http:\/\/geoscienceontology.org}{geoscienceontology.org}).  \n\nESMF provides the interfaces for models written in either the Fortran or C~programming languages; data \narrays are bundled together with related metadata in ESMF field objects.  All field objects from components are then \nbundled into exported and imported ESMF state objects to be passed between components.  As a third step, the\nESMF \\emph{registration} facility needs to be added to a user model; this step is achieved by using template code from \nany one of the examples or tutorials provided with ESMF.  The second and third step (\\emph{adapt} and \n\\emph{register}) are  typical tasks of what  \\citet[][]{Peckham2013} refers to as a component model interface (CMI);\nit  is very similar between models (and thus easily accessible from template code) and targets the interface of a \nspecific coupling framework.  \n\nMOSSCO contains CMIs for ESMF in all of its provided components (Fig.~\\ref{fig:bmicmi}).  The \ncurrent naming scheme  follows the CF convention for standard names except for quantities that are not defined by CF; \nthese names (often from biological processes) are modeled onto existing CF standard names as much as possible.  \nMOSSCO also allows the specification within other naming schemes and includes a name matching \nalgorithm to mediate between different schemes. For future development, adoption of the GSN ontology is \nforeseen.  \n\n\\subsection{Scheduling in a coupled system -- the ``S'' in  \\texttt{PARSE}}\n\nMOSSCO adds onto ESMF a scheduling system (corresponding to the fourth step in\n\\texttt{PARSE}) that  calls  the different phases of participating coupled models.  The\ncoupling time step duration of this new scheduler relies on the ESMF concept of alarms\nand a user specification of pairwise coupling intervals between models. The scheduler\nminimizes calls to participating models by flexibly adjusting time step duration to the\ngreatest common denominator of coupling intervals pertinent to each coupled model.\nUpon reading the user's coupling specification, \n\\begin{inparaenum}[(i)]\n\\item models are initialized in random order but with consideration of special initialization dependencies set by the user;\n\\item a list of alarm clocks is generated that considers all pairwise couplings a model is involved in;\n\\item special couplers associated with a pairwise coupling are executed;\n\\item the scheduler then tells each model to run until that model reaches its next alarm time;\n\\item  advancing of the scheduler to the minimum next alarm time repeats until the end of the simulation.\n\\end{inparaenum}\n\nThe MOSSCO scheduler allows for both sequential and for concurrent coupling of model components, or a hybrid \ncoupling mode.  In the concurrent mode, components run at the same time on different compute elements; in the \nsequential mode, components are executed one after another.  Recently,  \\citet{Balaji2016} demonstrated how a hybrid \ncoupling mode and fine granularity could be used to increase the performance of a system that consists of both highly \nscalable and less scalable components: In their system, they ran an ocean component concurrently with the radiation \ncode of the atmosphere sequentially to all other atmospheric process components.  \n\nFor both concurrent and sequential modes, the MOSSCO scheduler runs the connectors and mediator components\nthat exchange the data before the components are run, i.e. computations are performed on data with the same timestamp.  For sequential mode, the \ncoupling configuration allows also a scheme where consecutive components rely always on the most recently \ncalculated data from all other components (Fig.~\\ref{fig:scheduling}, see also Sect.~\\ref{sec:linkcopy}).\n\n\\subsection{Deployment of the coupled system -- the ``E'' in  \\texttt{PARSE}}\n\nMOSSCO provides a Python-based generator that dynamically creates an ESMF driver component\nin a star topology that then acts as the scheduler for the coupled system.  This generator reads a text-based\nspecification of pairwise couplings (including the coupling interval, dependencies and instantiation under\ndifferent names) and generates a Fortran source file that represents the scheduler component.  The\ngenerator takes care of compilation dependencies of the coupled models, and of coupling dependencies, such\nas grid inheritance; in addition to the basic init--run--finalize BMI scheme, it also honors multi-phase\ninitialization (as in the National Unified Operational  Prediction Capability, NUOPC, ESMF extension) and a restart phase.\n\nA MOSSCO command line utility provides a user-friendly interface to generating the scheduler,\n(re-)compiling all source codes into an executable and submitting the executable to a multi-processor system,\nincluding different high-performance computing (HPC) queueing implementations; this is the fifth step in \n\\texttt{PARSE}.  MOSSCO has been successfully deployed at several\nnational HPC centers, such as the Norddeutsche Verbund f\\\"ur Hoch- und H\\\"ochstleistungsrechnen (HLRN), \nthe  German Climate Computing Center (DKRZ), or the J\\\"ulich Supercomputing Centre  (JSC); equally,\nMOSSCO is currently functioning on a multitude of Linux and macOS laptops, desktops and multiprocessor\nworkstations using the same MOSSCO (bash-based) command line utility on all platforms.  \n\nThe MOSSCO coupling layer is coded in Fortran while most of the supporting structure is coded in\nPython and partially in Bash shell syntax.  The system requirements are a Fortran 2003 compliant\ncompiler, the CMake build system, the Git distributed version control system, Python with YAML\nsupport (version~2.6 or greater), a Network Common Data Form \\citep[NetCDF,][]{Rew1990}  library, and\nESMF (version~7 or greater).   For parallel applications, a Message Passing  library (e.g., OpenMPI) \nis  required. Many HPC centers have toolchains available that already meet all of\nthese requirements.  For an individual user installation, all requirements can be taken care of with\none of the package managers distributed with the operating system, except for the installation of\nESMF, which needs to be manually installed; MOSSCO provides a semiautomated tool for helping\nin this installation of ESMF. \n\n\\section{MOSSCO components and utilities}\n\nDriven by user needs, MOSSCO currently entails utilities for I\/O, an extensive model library, and\ncoupling functionalities (Fig.~\\ref{fig:functionalities} and Table~\\ref{tab:components}).   As a utility\nlayer on top of ESMF, MOSSCO also extends the  Application Programming Interface (API) of\nESMF by providing convenience methods to facilitate the handling of time, metadata (attributes),\nconfiguration, and to unify the provisioning and transfer of scientific data across the coupling\nframework.  The use of this utility layer is not mandatory;  any ESMF based component can be\ncoupled to the MOSSCO provided components without using this utility layer.\n\nOne of the major design principles of MOSSCO is  seamless scaling from zero-dimensional to three-dimensional spatial\nrepresentations, while maintaining the coupling configuration to the maximum extent possible.\nThis design principle builds on the dimensional-independency concept of FABM achieved by local description of processes\n(often referred to as a box model),  where the dimensionality is defined by the hydrodynamic model to which FABM is \ncoupled; MOSSCO generalizes this concept to enable especially the developers of new biological and chemical models to scale up from a\nbox-model (zero-dimensional) to a water-column (one-dimensional), sediment plate or a vertically resolved transect\n(two-dimensional), and a full atmosphere or ocean (three-dimensional) setup.\nAs a concrete example, upscaling of the novel Model for Adaptive Ecosystems in Coastal Seas \\citep[MAECS,][]\n{Wirtz2016,Kerimoglu2017} has been developed in this way going forth and back between the lab\nscale and the regional coastal ocean scale. \n\nAll utility functions and components, especially the generic I\/O facilities from MOSSCO, are able to\nhandle data of any spatial dimension.  MOSSCO communicates different dimensional information\nby grid inheritance: components that implement this are able to obtain the spatial information from another model in the coupled context.  \nUsually (but not necessarily) biological and chemical models inherit the spatial configuration from a hydrodynamic model; \nequally well, this information can be obtained from data in standardized grid description formats like Gridspec \n\\citep{Balaji2007} or the Spherical Coordinate Remapping and Interpolation Package  \\citep[SCRIP,][]{Jones1999mwr}.\n\n\\subsection{Model library: Basic Model Interfaces for scientific model components}\n\nThe model library (right branch in Fig.~\\ref{fig:functionalities}) includes new models (e.g., for filter\nfeeders and  surface waves) and wrappers to legacy models and frameworks such as FABM or\nGETM.  Some of these wrappers are  under development (e.g., Hamburg Shelf Ocean Model\n\\citep[HamSOM,][]{Harms1997} and a Lagrangian particle tracer model). Here, we briefly document\nthe model collection in particular with respect to their preparation and functioning within the new\ncoupling context. \n\n\\subsubsection{Pelagic ecosystem component}\\label{sec:fabm_pelagic}\n\nThe pelagic ecosystem component (\\texttt{fabm\\_pelagic\\_component}) collects (mostly biological) \nprocess models for aquatic systems. This component makes use of the Framework for Aquatic\nBiogeochemical Models  \\citep[][]{Bruggeman2014}.  FABM is a coupling layer to a multitude\nof biogeochemical models which provide the source-minus-sink terms for variables, their vertical\nlocal movement (e.g., due to sinking or active mobility), and diagnostic data. Each model variable \nis equipped with meta data, which is transferred by the ecosystem component into\nESMF field names and attributes. Similarly, the forcing required by the biogeochemical models\nis communicated within the framework and linked to FABM. The pelagic ecosystem component\nincludes a numerical integrator for the boundary fluxes and local state variable dynamics. \nAdvective and diffusive transport are not part of this component but are left to the hydrodynamical\nmodel through the \\texttt{transport\\_connector} (\\refsec{transport}). The close connection between\ntransport  and the pelagic ecosystem requires grid inheritance by the pelagic ecosystem\nmodel from the hydrodynamic model component.\n\nMany well known biogeochemical process models have been coded in the FABM standard by\nvarious institutes, such as the European Regional Seas Ecosystem Model\n\\citep[ERSEM,][]{Butenschoen2016}, ERGOM \\citep{Hinners2015}, PCLake \\citep{Hu2016} and\nthe Bottom RedOx Model \\citep[BROM,][]{Yakushev2017}.  All biogeochemical models complying\nwith the FABM standard can equally be used in MOSSCO, while retaining their functionality.\n\n\\subsubsection{Sediment\/soil component}\\label{sec:fabm_sediment}\n\nThe sediment component \\texttt{fabm\\_sediment\\_component}  hosts (mostly biogeochemical)\nprocess descriptions for aquatic soils. To allow efficient coupling to a pelagic ecosystem, the sediment \ncomponent inherits a horizontal grid or mesh from the coupled system and adds its own vertical coordinate, a number \nof layers of horizontally equal height for the upper soil (typical domain heights range from 10--50\\,\\unit{cm}). State \nvariables within the sediment are defined through the FABM framework within the 3D grid or mesh in the sediment. \nAs in the pelagic ecosystem component, the state variables, meta data, diagnostics and forcings are \ncommunicated via the ESMF framework to the coupled system. The sediment component is the numerical integrator for \nthe tracer dynamics within each sediment column in the horizontal grid or mesh, including diffusive transport, \ndriven by molecular diffusion of the nutrients or bioturbative mixing.\nAdditionally, a 3D variable porosity defines the fraction of pore water as part of the bulk \nsediment, while all state variables are measured per volume pore water in each cell.  FABM's infrastructure of state \nvariable properties is used to label the new Boolean property \\texttt{particulate} in FABM models to define whether a state \nvariable belongs to the solid phase within the domain.\nA typical model used in applications of the sediment component is the biogeochemical model of the \nOcean Margin Exchange Experiment OMEXDIA \\citep[][]{Soetaert1996}; a version of this model\nwith added phosphorous cycle is contained  in the FABM model library as OMEXDIA\\_P \\citep{Hofmeister2014iche}. \n\n\\subsubsection{1D Hydrodynamics: General Ocean Turbulence Model (GOTM))}\\label{sec:gotm}\n\nThe  General Ocean Turbulence Model \\citep[GOTM,][]{Burchard1999,Burchard2006jms} is\na one-dimensional  water column model for hydrodynamic and thermodynamic processes\nrelated to vertical mixing. MOSSCO provides a component for GOTM and a component\nhierarchy that considers a coupled GOTM with internally coupled FABM within one\ncomponent (\\texttt{gotm\\_fabm\\_component}), as many existing available model setups rely on the direct\ncoupling of FABM to GOTM.  This way, the modularization -- taking a coupled GOTM\/FABM apart and\nrecoupling it through the MOSSCO infrastructure, can be verified; the encapsulation of GOTM is implemented\nin the \\texttt{gotm\\_component}.\n\n\\subsubsection{3D Hydrodynamics: General Estuarine Transport Model (GETM)}\\label{sec:getm}\nMOSSCO provides an interface to the 3D coastal ocean model GETM \\citep{Burchard2002ec}.\nGETM solves the Navier-Stokes Equations under Boussinesq approximation, optionally\nincluding the nonhydrostatic pressure contribution \\citep{Klingbeil2013}. A direct interface to\nGOTM (see Section~\\ref{sec:gotm}) provides state-of-the-art turbulence closure in the vertical.\nGETM supports horizontally curvilinear and vertically adaptive meshes\n\\citep{Hofmeister2010,Graewe2015}; the different mesh topologies need to be made accessible\nas ESMF grid objects. Typically,  the GETM component exports its grid and subdomain decomposition to the\ncoupled system where the spatial and parallelization information is inherited by other \ncomponents. The interface to GETM is provided by the \n\\texttt{getm\\_component};  any model coupled to GETM via the transport component can\nhave its state variables conservatively transported by GETM (see Section~\\ref{sec:transport}).\n\n\\subsubsection{Model components for erosion, sedimentation, and their biological alteration}\\label{sec:erosed}\n\nThe erosion\/sedimentation routines of the Deltares Delft3D model \\citep[EROSED,][]\n{VanRijn2007} were encapsulated in a MOSSCO component.   EROSED uses a Partheniades--Krone equation \n\\citep{Partheniades1965} for calculating the net sediment flux of cohesive sediment at the water--sediment interface for \nmultiple SPM size classes.  MOSSCO's \\texttt{erosed\\_component} always uses the current version of \nEROSED maintained by Deltares, and with the help of subsidiary infrastructure isolates the EROSED code from the deeply intertwined \ndependencies of the original implementation.  The functionality of this erosion and sedimentation component is described in more \ndetail by \\citet{Nasermoaddeli2014ich}.\n\nFlow and sediment transport can be affected by the presence of benthic organisms in many ways. Protrusion of \nbenthic animals and macrophytes in the boundary layer changes the bed roughness and thus the bed shear stress and \nconsequently the sediment transport. The erodibility of sediment can be modified by the mucus produced by benthic \norganisms; the \nerodibility of the upper bed sediment can be altered by bioturbation generated by macrofauna \\citep{DeDeckere2001}.  \nIn the \\texttt{benthos\\_component}, these biological effects of microphytobenthos and of benthic macrofauna on \nsediment erodibility and critical bed shear stress are parameterized. The benthos effect model is described in detail by \n\\citeauthor{Nasermoaddeli2017} (submitted to Estuarine, Coastal, and Shelf Science).\n\n\\subsubsection{Filter feeding model}\\label{sec:filtration}\n\nThe generic filtration component describes the instantaneous filtration by filter feeders within the water column. The biological \nfiltration model follows \\citet{Bayne1993} and describes the filtration rate as a function of food supply; it  can be adapted \nto different species of filter feeders and was recently applied to describing the ecosystem effect of blue mussels on \noffshore wind farms as the \\texttt{filtration\\_component} of MOSSCO (\\citeauthor{Slavik2017}, to be submitted to Hydrobiologia).\nThe filtration model uses an arbitrary chemical species or compound, say phytoplankton carbon as the ``currency'' for \nprocessing.  The amount of ambient phytoplankton carbon concentration is sensed by the model organisms and it is \nfiltered along with the other nutrients (in stoichiometric proportion) out of the environment, creating a sink term for \nsubsequent numerical integration in the pelagic ecological model.  \n\n\\subsubsection{Wind waves}\\label{sec:waves}\n\nA simple wind wave model is part of the MOSSCO suite. Based on the parameterization by \\citealt{Breugem2007},\nsignificant wave height and peak wave period are estimated in terms of local water depth, wind speed and fetch length.\nThis wave data enable the inclusion\nof wave effects especially for idealized 1D water column studies, e.g. the consideration of erosion processes\ndue to wave-induced bottom stresses.\nCoupling to 3D ocean models and the calculation of additional wave-induced momentum forces there, following either the Radiation stress or\nVortex Force formulation \\citep{Moghimi2013a}, is possible as well.\nFor the inclusion of wave--wave or wave--current interaction in realistic 3D applications,\nthe coupling to a more advanced third generation wind wave model like SWAN, WaveWatch~III or Wave Atmospheric Model (WAM)\nwould be necessary. \n\n\\subsection{Input\/Output utilities} \nThe  I\/O utilities include generic coupling functionalities that deal with boundary conditions, provide a\nrestart facility, add surface, lateral and point source fluxes (lower left branch in Fig.~\\ref{fig:functionalities}).  \n\n\\subsubsection{Generic output}\\label{sec:output}\n\nThis utility component of MOSSCO provides a generic output facility \\texttt{netcdf\\_component} for any data that is \ncommunicated in the coupling framework. The component writes one- to three-dimensional time sliced data into a \nNetCDF \\citep[][]{Rew1990} file and adds metadata on the simulation to this output.  \nMultiple instances of this component can be used within a simulation, such that output of different variables, differently \nprocessed data, and output at various output time steps can be recorded. The output component is fully parallellized\nwith a grid decomposition inherited from one of the coupled science \ncomponents.\nIn order to reduce interprocess communication during runtime, each write process considers only the part of the data that resides within its compute domain. This comes \nat a cost to the user, who has to postprocess the output tiles to combine for later analysis;  a python script is provided \nwith MOSSCO that takes care of joining tiled files. \n \nThe generic output also adds meta data that is collected from the system and the user environment when the output is \nwritten to disk.   Diagnostics about the processing element and run time between output steps are recorded.   The \nstructure of the NetCDF output  follows the Climate and Forecast \\citep[CF,][]{Eaton2011} convention for physical \nvariables, geolocation, units, dimensions and methods modifying variables.  When (mostly biological) terms are not \navailable in the controlled vocabulary of CF,  names are built to resemble those contained in the standard.  \n\n\\subsubsection{Generic input}\\label{sec:input}\n\nThe generic  \\texttt{netcdf\\_input\\_component} of MOSSCO reads from NetCDF files and provides the file content \nwrapped in ESMF data structures (fields) to the coupling framework.  It is parallelized identical to the generic output \ncomponent, and inherits its decomposition from other components in the coupled system.  Data can be read for the \nentire domain or for all decomposed compute elements separately.  Upon reading of data, fields can be renamed and \nfiltered before they are passed on to the coupled system. \n\nThe input component is typically used to initialize other components, for restarting, to provide boundary conditions, and \nfor assimilating data into the coupled system.  The generic input facility supports interpolation of data in time upon \nreading the data, with nearest, most recent, and linear interpolation.   It also supports reading  climatological data and \ntranslates the climatological timestamp to a simulation present time stamp in the coupling framework.   \n\n\\subsection{MOSSCO connectors and mediators}\nInformation in the form of ESMF states that contain the output fields of every component are communicated to the\nESMF driver; requests for data by every component are also communicated to the ESMF driver component.\nMOSSCO connectors are separate components that link output and requested fields between pairwise coupled\ncomponents.   MOSSCO informally distinguishes between connector components that do not manipulate the field\ndata on transfer at all (or only slightly), and mediator components that extract and compute new data out of the input data.\n\n\\subsubsection{Link, copy and nudge connectors}\\label{sec:linkcopy}\n\nThe simplest and default connecting action between components is to share a reference (i.e. a link)\nto a single field that resides in memory and can be manipulated by each component; in contrast, the\n\\texttt{copy\\_connector} duplicates a field at coupling time.  The consideration of a link or copy connector is \nimportant for managing the data flow sequence in a coupled system: the copy mechanism ensures that two coupled  \ncomponents work on the same lagged state of data, whereas the link mechanism ensures that each\ncomponent works on the most recent data available.\n\nThe  \\texttt{nudge\\_connector} is used to consolidate output from two components by weighted averaging of the\nconnected fields.  It is typically used as a simple assimiliation tool to drive model states towards observed\nstates, or to impose boundary conditions.\n\nThese connectors can only be applied between components that run on the same grid (but maybe with a different subdomain decomposition).\nThe \\texttt{link\\_connector} can only be applied between components with an identical subdomain decomposition\nso that the components have access to the same memory.\nComponents on different grids require regridding, which is currently under development in MOSSCO.\n\n\\subsubsection{Transport connector}\\label{sec:transport}\n\nA model component qualifies as a transport component when it  offers to\ntransport  an arbitrary number of tracers in its numerical grid; this facility is\npresent, for example, in the current \\texttt{gotm\\_component} and\n\\texttt{getm\\_component}. The \\texttt{transport\\_connector} provides generic\ninfrastructure that communicates the tracer fields to be transported to the transporting\ncomponent based on the availability of both, the tracer concentrations as well as\ntheir rate of vertical movement independent of the water currents. This connector\nis usually called only once per coupled pair of components during the initialization phase.\n\n\\subsubsection{Mediators for soil--pelagic coupling}\\label{sec:bpcoupler}\n\nOne aspect of the generalized coupling infrastructure in MOSSCO is the use of\nconnecting components that mediate between technically or \nscientifically incompatible data field collections. The soil--pelagic coupling of biogeochemical\nmodel components with a variety of different state variables raises the need for these  mediators.\nThe use of mediators leaves\nthe level of data aggregation and dis-aggregation, and unit conversion to the \ncoupling routine, instead of requiring specific output from a model component depending\non its coupling partner component.\n\nFor soil--pelagic (or benthic--pelagic) coupling, the  \\texttt{soil\\_pelagic\\_connector}\nmediates the soil biogeochemistry output towards the pelagic ecosystem input and\nthe  \\texttt{pelagic\\_soil\\_connector} mediates the pelagic ecosystem output towards\nthe soil biogeochemistry input, e.g.:\n\\begin{inparaenum}[(i)]\n\\item dis-aggregation of dissolved inorganic nitrogen to dissolved ammonium and dissolved nitrate\n\\item filling missing pelagic state fields for phosphate using the Redfield-equivalent for dissolved inorganic nitrogen\n\\item calculation of the vertical flux of particulate organic matter (POM) from the water column into the sediment depending \non POM concentrations in the near-bottom water, its sinking velocity and a sedimentation efficiency depending on the \nnear-bottom turbulence. The effective vertical flux is communicated into the pelagic ecosystem component to budget the \nrespective loss, and is communicated to the soil biogeochemistry component to account for the respective new mass of \nPOM. The mediator also handles\n\\item dis-aggregation of a single oxygen concentration (allowing positive and negative values) into dissolved oxygen \nconcentration, if positive, and dissolved reduced substances, if negative.\n\\item aggregation of pelagic POM composition (variable nitrogen to carbon ratio) into fixed stoichiometry POM pools in \nthe soil biogeochemistry.\n\\end{inparaenum}\n\n\n\\section{Selected applications as feasibility tests and use cases}\\label{sec:apps}\n\nMOSSCO was designed for enhancing flexibility and equitability in environmental data and model coupling. These design goals \nhave been helpful in generating new integrated models for coastal research ranging from one-dimensional water-\ncolumn to three-dimensional, with applications at different marine stations, transects, and sea domains.  Below, we \ndescribe from a user perspective the added value and success of the design goals  obtained from using MOSSCO in \nselected applications; here, the focus is not on the scientific outcome of the application (these are described elsewhere \nby, e.g., \\citeauthor{Nasermoaddeli2017}, submitted, \\citeauthor{Slavik2017}, to be submitted, \\citealt{Wirtz2016}, \\citealt{Kerimoglu2017}). All setups\ndescribed in the use cases are available as open source (with limited forcing data due to space and bandwidth constraints).\n\n\\subsection{Helgoland station}\\label{sec:helgoland}\n\nThe seasonal dynamics of nutrients and turbidity emerges from\nthe interaction of physical, ecological and biogeochemical processes in the water column and the \nunderlying sea floor.  We resolve these dynamics in a coupled application for a \n1D~vertical water column for a station near the German offshore  island Helgoland.  Average water\ndepth around the island is 25\\,m; tidal currents are affected by\nthe M2 and S2~tides with a characteristic spring--neap cycle, with current velocity not \nexceeding 1 m\\,s$^{-1}$. \n\nThe Helgoland 1D application is realized by a coupled system consisting of GOTM hydrodynamics,\nthe pelagic FABM component with a nutrient--phytoplankton--zooplankton--detritus (NPZD) ecosystem model  \\citep{Burchard2005} and two SPM size classes,\ninteracting with the erosion and sedimentation module,  the sediment component with the OMEXDIA\\_P \nearly diagenesis submodel, and coupler components for soil--pelagic, pelagic--soil and \ntracer transport. This system and setup is described in more detail by \\citet{Hofmeister2014iche}.\n\nSimulations with this application show a prevailing seasonal cycle in the model states (Fig.~\\ref{fig:helgoland}). \nDissolved nutrients  (referred as dissolved inorganic nitrogen) are taken up by phytoplankton, which fills\nthe pool of particulate organic nitrogen during the spring bloom (Fig.~\\ref{fig:helgoland}d). The particulate organic matter sinks\n into the sediments, where it is remineralized along axis, sub-oxic and anoxic pathways;  denitrification, for example, shows a peak in late summer\n (Fig.~\\ref{fig:helgoland}b). At the end of a year, nutrient concentrations are high in the sediment and diffuse back into the\n water column up to winter values of 20--25 mmol\\,m$^{-3}$.  The seasonal variation of turbidity is a result of higher\n erosion in winter and reduced vertical transport in summer  (Fig.~\\ref{fig:helgoland}c).\n\n\\subsection{Idealized coastal 2D transect}\\label{sec:transect}\n\t\t\nThe coastal nitrogen cycle is resolved in an idealized coupled system\nfor a tidal shallow sea. This two-dimensional setup represents a\nvertically resolved cross-shore transect of 60\\,km length and 5--20\\,m\nwater depth and has  been used  by \\citet{Hofmeister2017}\nto simulate  sustained horizontal nutrient gradients by particulate matter\ntransport towards the coast.  Within the MOSSCO coupling framework,\nthe 2D~transect scenario additionally provides insights into horizontal variability of\nerosion\/sedimentation and benthic biogeochemistry. \nIts coupling configuration builds on the one used for the 1D~station\nHelgoland  (Sect.~\\ref{sec:helgoland}); the water-column hydrodynamic model\nGOTM, however, is replaced by the 3D~model GETM; a local wave component and\ndata components for open boundaries and restart has been added.\n\nFigure \\ref{fig:transect} shows exchange fluxes between the water column\nand the sediment for one year of simulation.  The simulation of turbidity, as a\nresult of pelagic SPM transport and resuspension by currents  and wave stress,\nprovides the light climate for the pelagic ecosystem. The flux of particulate\norganic carbon (POC) into the sediment reflects bloom \nevents in summer during calm weather conditions. Macrobenthic activity in the sea floor brings fresh organic matter into \nthe deeper suboxic layers of the sediment, where denitrification removes nitrogen from the pool of dissolved nutrients. \nThe coupled simulation reveals decoupled signals of benthic respiration, denitrification and nutrient reflux into the water \ncolumn, which is not resolved in monolithically coded regional ecosystem models of the North Sea \n\\citep{Lorkowski2012,Daewel2013}.\n\n\\subsection{Southern North Sea bivalve ecosystem applications}\n\nA Southern North Sea (SNS) domain was used in two  studies concerning the\neffects of bivalves on the pelagic ecosystem.  \\citeauthor{Slavik2017} (to be\nsubmitted) investigated  how the accumulation of epifauna on wind turbine\nstructures (Fig.~\\ref{fig:bivalves}d) impacts pelagic primary production and\necosystem functioning in the SNS at larger spatial scales. This study is the\nfirst of its kind that extrapolates ecosystem impacts of anthropogenic offshore\nwind farm structures from a local to a regional sea scale.  The authors use  a \nMOSSCO coupled system consisting of the hydrodynamic model GETM, the\necosystem model MAECS as described by \\citet{Kerimoglu2017}, the transport\nconnector, the filter feeder component, and several input and output components\n(Fig.~\\ref{fig:bivalves}e).   They assess the impact of anthropogenically\nenhanced filtration from  blue mussel (\\emph{Mytilus edulis}) settlement on\noffshore wind farms that are planned to meet the 40-fold increase in offshore\nwind electricity in the European Union until~2030; they find a small but \nnon-negligible large-scale effect in both phytoplankton stock and primary production, \nwhich possibly contributes to better water clarity (Fig.~\\ref{fig:bivalves}f).\n\nBiological activities of macrofauna on the sea floor mediate suspended sediment\ndynamics, at least locally. In the study  by \\citeauthor{Nasermoaddeli2017}\n(submitted), the large-scale biological contribution of benthic\nmacrofauna, represented by the key species \\emph{Fabulina fabula}\n(Fig.~\\ref{fig:bivalves}a), to suspension of sediment was investigated. Simulation\nresults for a typical winter month revealed that SPM is increased not only locally \nbut beyond the mussel inhabited zones. This effect is not limited to the near-bed\nwater layers but can be observed throughout the entire water column, especially\nduring storm events (Fig.~\\ref{fig:bivalves}c).  In this coupled application, the\nhydrodynamic model GETM, the pelagic ecosystem component with three SPM\nsize classes, the erosion--sedimentation and benthic mediation components,\nseveral input and one output components, and the transport connector were\nused (Fig.~\\ref{fig:bivalves}d).   \n\n\\section{Discussion and Outlook}\nIn merging existing frameworks that address distinct types of modularity and\nby developing a super-structure for making the multi-level coupling approach\napplicable in coastal research, the MOSSCO system largely  meets the design\ngoals \\emph{flexibility}  and \\emph{equitability}. In doing so, structural deficiencies of\nlegacy models and the need for practical compromises became very\napparent.   \n\nFor legacy reasons, \\emph{equitability} is the harder to achieve \ndesign goal.  Both the distribution of compute resources as well as the \nspatial grid definition can be in principle determined by any one of the participating\ncomponents; de facto, in marine or aquatic research, they are prescribed by\nthe hydrodynamic models \nthat have so far not been enabled to inherit a grid specification or a resource\ndistribution from a coupler or coupled system.  With the ongoing development and\ndiversification of hydrodynamic models, and no immediate benefit for the different physical\nmodels to outsource grid\/resource allocation, this situation is not likely to\nchange.  MOSSCO compromises here by its flexible grid inheritance scheme and\nwith the grid provisioning component that delivers this information to the \ncoupled system whenever a hydrodynamic component is not used.\n\nBeyond grid\/resource allocation, however, the \\emph{equitability} concept is successfully driving\nindependent developments of submodules.  We found that indeed experts\nin one particular model, e.g. the erosion module, could rely on the \nfunctionality of the other parts of the system without having to be an expert\nthemselves in all of the constituent models in the coupled application. \nThe limitations to this black-box approach became evident in the scientific\napplication and evaluation of the coupled model system, which was only\npossible when a collaboration with experts in these other model systems was\nsought.  By taking away the inaccessibility barrier and by enforcing clear separation\nof tasks the modular system stimulated a successful collaboration.  \nSustained granularity also helped to align with ongoing development\nin external packages. These can be integrated fast into the coupled system, which\ndoes not rely on specific versions of the externally provided software\nunless structural changes occur.\nLong-term supported interfaces on the external model side facilitate MOSSCO being\nup-to-date with e.g. the fast evolving GETM and FABM code bases.  \n\nWhen legacy codes were equipped with a framework-agnostic  interface\nwe encountered  four major difficulties:\n\\begin{enumerate}\n\\item For organizing the data flow between the components, MOSSCO\nuses standard names and units compatible with the infrastructure and \nlibrary of standard names and units provided in the pelagic component \nfor the FABM framework (mostly modelled on CF).  Other components, \nsuch as the BMIs of wrapped legacy models, do not provide such a \nstandard name in their own implementation, and in particular, often do\nnot adhere to a naming standard. We found ambiguity arising, e.g.,\nwith temperature to be represented as  \\texttt{temperature} vs.\\\n\\texttt{sea\\_water\\_temperature} vs.\\ \\texttt{temperature\\_in\\_water}.\nWhile this can be resolved\nbased on CF for temperature, most ecological and biogeochemical\nquantities currently lack a consistent naming scheme.  The forthcoming\nGSN ontology \\citep[building on CSDMS names,][]{Peckham2014iemss}\ncould adequately address this coupling challenge. \n\n\\item Deep subroutine hierarchies of existing models made it difficult\nto isolate desired functionality from the structural external overhead.\nIn one example, where a single functional module was taken out of the \ncontext of an existing third-party coupled system, the module depended on\nmany routines dispersed throughout that third-party system repository.\n\n\\item Components based on standalone models are developed and tested\nwith their own I\/O infrastructure and typically supply a BMI implementation\nonly for part of their state and input data fields. A new, coupled application or\ndata provisioning\/request within a coupled system can therefore easily require\na change in the model?s BMI. The implementation potential input and\noutput for all quantities, including replacement of the entire model-specific I\/O\nin the BMI is therefore desirable for new developments and refactoring.\n\n\\item Two-way coupling of mass and energy fluxes between components has\nto be integrated numerically in a conservative way, despite different time\ndiscretization schemes used within the different components. We relied on\nconservative integration of the transferred fluxes within pairwise coupled\ncomponents for thir respective coupling timestep, which is most flexible for\nasynchronous scheduling. The coupled system  itself, however, cannot ensure\nthe conservative integration of mass and energy fluxes between components. \nHere, the user needs to take care of a correct coupling configuration.\n\\end{enumerate}\n\nEfforts in making legacy models couplable, either for MOSSCO or \nsimilar frameworks, however, leads to additional benefits besides the \nimmediate applicability in an integrated context. Couplability strictly demands\nfor communication of sufficient metadata, which stimulates the quality and \nquantity of documentation and of scientific and technical reproducibility of\nlegacy models. Indeed, transparency has been greatly increased by wrapping legacy\nmodels in the MOSSO context.   All participating components  performed\nintrospection and leveraging of a collection of metadata at assembly time of the coupled\napplication and during output.  Transparency is\nexpected to be continuously increasing by new coupling demands and more\ngenerous metadata provisioning from wrapped science models.  MOSSCO \nis moving towards adopting the Common Information Model~(CIM) that is also \nrequired by Climate Model Intercomparison Project~(CMIP) participating\ncoupled models \\citep{Eyring2016}.  \n\nWith a current small development base of twelve contributors, the\nopenness concept of MOSSCO in terms of including contributions\nfrom outside the core developer team has not yet been tested; in the\ncategorization by \\citet{DeLaat2007} internal governance with \nsimple structure is sufficient at this size.  Formally, external contributions\ncan be included in MOSSCO by way of contributor license agreements.\nThe openness concept has been useful  in instigating\ndiscussions about the need for explicit (and preferably open)\nlicensing of related scientific software and data as demanded in current\nopen science strategies \\citep[e.g.,][]{Scheliga2016}.\n\nSo far, scalability in current MOSSCO applications is excellent: \nStrong scaling experiments with a coupled application using GETM,\nFABM with MAECS ($\\approx$20~additional transported 3D tracers), and FABM \nwith OMEXDIA\\_P on Jureca \\citep{Krause2016} show linear (perfect) speedup\nfrom 100 to 1000~cores, and a small leveling-off (to~85\\% of perfect scaling)\nat 3000~cores. We have not observed  loss of compute time  due to the\ninfrastructure and superstructure overhead of ESMF. \n\nProblems of multi-component systems \nneed to be solved in terms of acceptance by the research community. \nMulti-component systems are much harder to be implemented and maintained\nby individual groups, where researchers solve coastal ocean problems of a\nlarge range of complexity,  from purely hydrodynamic applications via coupled\nhydrodynamic--sediment dynamic applications to fully coupled systems. Many \nacademic problems  focus on specific mechanisms and thus do not require\nthe complete and fully coupled modular system, such that the application of the\nfull system might mean a large structural overhead and additional workload.\nMost researchers would agree on the potential necessity of following a holistic\napproach when tackling grand research questions in environmental science\nsuch as related to system responses to anthropogenic intervention. Yet, it is\nnot clear whether the up-scaling approach inherent to the addition of many\nmodular components can lead to meaningful results.\n\nAs evident form the test cases (\\refsec{apps}), MOSSCO also encourages\ncoupled applications that are far from a complete system level description.  \nWith few coupled components, the technical threshold to getting an application\nrunning on an arbitrary system is relatively low.  The user can reach a fast first success.\nMOSSCO provides a full documentation, step by step recipes, and a public bug tracker;\nit adopts abundant error reporting from ESMF and a fail fast design that\nstops a coupled applications as soon as a technical error is detected \\citep{Shore2004}.\nUsability is especially high due to an available master script that compiles,\ndeploys, and schedules a coupled application. To address a wide range of\nusers, the system is designed to run on a single processor or on a user's laptop\nequally well as on a high-performance computer using several\nthousand compute nodes.\n\nAn obvious advantage of modular coupling is the opportunity to bridge the gap\nbetween different scientific disciplines. It allows in principle to combine, e.g.,\nhydrodynamic models from oceanography with sediment transport models\nfrom coastal engineering. Thus different experts can work on their individual\nmodels but benefit from each others' progress. This seeming advantage, however,\nposes also a drawback for modular coupling approaches: An initial effort which is necessary\nfor individual models to meet the requirements of a modular modelling framework\nhas to be invested. This will only happen if there is either an urgent pressure\nto include specific model capabilities, which will be difficult to include\notherwise, or if convincing examples of possible benefits can be presented.\nIt cannot be expected that the coastal ocean modelling community\nwill agree about one coupler or one way of interfacing modules, such that it\nwill still require considerable implementation work to transfer a module from\none modular system to another. To solve this problem, coupling standards\nneed to become more general, but in turn this might even increase the\nstructural overhead in using these systems.\n\nOffsetting these concerns, the separate design of basic and component\ninterfaces (BMI\/CMI) ensures that the effort spent on wrapping an existing\nmodel, or on equipping a new model with a basic model interface is not tied\nto a particular coupling framework, or even a particular coupling framework\ntechnology.  A model that  follows BMI principles will be more easily\ninterfaced to other models no matter what coupler is used.  Wrapped legacy\nmodels from MOSSCO can thus be useful in non-ESMF contexts, as well; \nand models with an existing BMI can be integrated in MOSSCO more easily,\nin turn.\n\nOne demand for integrative modeling, which is likely best practised in open\nand flexible system approaches, arises from\ncurrent European Union legislation. The Water Framework Directive and the\nMarine Strategic Planning Directive require the description of marine\nenvironmental conditions and the development of action plans to achieve a\ngood environmental status. These objectives can initially be met by a monitoring\nprogram to determine present-day conditions but ultimately rely on numerical\nmodel studies to evaluate anthropogenic measures. This ecosystem based\napproach to management \\citep[e.g.,][]{Ruckelshaus2008} demands modelling\nsystems which are capable of taking into account hydrodynamics, biogeochemistry,\nsedimentology and their interactions to properly describe the\nenvironmental status. As further legal requirements can be expected for many\ncoastal seas worldwide, numerical\nmodelling systems applied for this task need to be flexible in terms of integrating\nadditional (e.g., site-specific) processes. In this ongoing process, the initial effort of creating a\nmodular system may be the only way forward that can take into account all relevant processes\nin the long run.\n\n\\subsection{Outlook}\n\nThe suite of components provided or encapsulated so far meets the demands\nthat were initially formulated by our users; they already allow for a wide range of\nnovel coupled applications to investigate the coastal sea.  To stimulate more\ncollaboration, however, and to bring forward a  general ``ecosystem'' of modular science\ncomponents, several legacy models could interface to MOSSCO components\nin the near future by building on complementary work at other institutions.  For example,\nthe Regional Earth\nSystem Model \\citep[RegESM,][]{Turuncoglu2013} provides ESMF interfaces for\nMITgcm, ROMS  and WAM, amongst others.\nConvergence of the development of MOSSCO and RegESM is feasible in the near term.\nAlso, the recently developed Icosahedral Non-Hydrostatic Atmospheric Model\n\\citep[ICON][]{Zaengli2015} is currently being equipped with an ESMF component model\ninterface.\n\nOnce ESMF interfaces have been developed for a legacy model, it is desirable that these\ndevelopments move out of the coupler system and are integrated into the development of\nthe legacy model itself.  This has been successfully  achieved with the ESMF interface\nfor the hydrodynamic model GETM, which is now distributed with the GETM code.\nMuch of the utility layer  developed in MOSSCO, or likewise in MAPL or\nin the ESMF extension of the WRF model, are expected to be propagated upstream into\nthe framework ESMF itself.   \n\nThe interoperability of current coupling standards will increase.   While currently there are\nthree flavors of ESMF: basic  ESMF as in MOSSCO, ESMF\/MAPL as in the GEOS-5 system,\nor ESMF\/NUOPC  as in the RegESM,  only a minor effort would be required to provide the basic\nESMF and ESMF\/MAPL implementations with a NUOPC cap and make them interoperable with\nthe entire ESMF ecosystem.    Even a coupling of ESMF based systems to OASIS\/MCT based\nsystems has been proposed; and investigation is ongoing on a \ncoupling of MOSSCO to the formal BMI for CSDMS.  \n\n\\conclusions\n\nWe problematized both the primacy of hydrodynamic models and the limited modularity in\ncoupled coastal modeling that can stand in the way of developing and applying novel and\ndiverse biogeochemical process descriptions. Such developmental potential is likely\nneeded to progress towards holistic regional coastal systems models.  We presented the\nnovel Modular System for Shelves and Coasts (MOSSCO), that is  built on coupling\nconcepts centered around equitability and flexibility to resolve the issue of obstructed modularity.\nThese concepts bring about also openness, usability, transparency and scalability. MOSSCO\nas an actual Fortran implementation of this concept includes the wrapped Framework\nfor Aquatic Biogeochemical Models (FABM) and a usability layer for the Earth System Modeling\nFramework (ESMF).\n\nMOSSCO's design principles emphasize basic couplability and rich meta information.  Basic\ncouplability requires that models communicate about flow control, compute resources, and\nabout exchanged data and metadata. We demonstrated that  the design principles flexibility\nand equitability enable the building of complex coupled models that adequately\nrepresent the complexity found in environmental modelling. In this first version, the MOSSCO\nsoftware wrapped several existing legacy models with basic model interfaces (BMI); \nwe added ESMF-specific component model interfaces (CMI) to these wrappers and other\nmodels and frameworks to build a suite of ESMF components that when coupled represent\na small part of a holistic coastal system.  These components deal with hydrodynamics,\nwaves, pelagic and sediment ecology and biogeochemistry, river loads, erosion,\nresuspension, biotic sediment modification and filter feeding. \n\nIn selected applications, each with a different research question, the applicability of the coupled\nsystem was successfully tested. MOSSCO  facilitates the development of new coupled\napplications for studying coastal processes that extend from the atmosphere through\nthe water column into the sea bed, and that range from laboratory studies to 3D\nsimulation studies of a regional sea. This system meets an infrastructural need that is\ndefined by experimenters and process modellers who develop biogeochemical, physical,\nsedimentological or ecological models at the lab scale first and who would like to\nseamlessly embed these models into the complex coupled three-dimensional coastal\nsystem. This upscaling procedure may ultimately support also the global Earth System community. \n\n\\codedataavailability{The MOSSCO software is licensed under the GNU General Public\nLicense~3.0, a copyleft open  source license that allows the  use, distribution and\nmodification of the software under the same terms.  All documentation for MOSSCO is\nlicensed under the Creative Commons Attribution Share-Alike~4.0 (CC-by-SA), a \ncopyleft open document license that allows use, distribution and modification of the\ndocumentation under the same terms. \n\nDevelopment code and documentation are currently  primarily hosted on Sourceforge\n(\\href{https:\/\/sf.net\/p\/mossco\/code}{https:\/\/sf.net\/p\/mossco\/code}) and mirrored on Github\n(\\href{https:\/\/github.com\/platipodium\/mossco-code}{https:\/\/github.com\/platipodium\/mossco-code}).\nThe release version 1.0.1 is permanently archived\non Zenodo and accessible  under the digital object identifier\n\\href{http:\/\/dx.doi.org\/10.5281\/zenodo.438922}{doi:10.5281\/zenodo.438922}.  \nAll wrapped legacy models are open source and freely available from the developing institutions;  \nfree registration is required  for accessing the Delft3D system at Deltares.\nSelected test cases are available from a separate Sourceforge repository\n\\href{https:\/\/sf.net\/p\/mossco\/setups}{https:\/\/sf.net\/p\/mossco\/setups}, where all of the data on\nwhich the presented use cases are based are freely available, with the \nexception of the meteorological forcing fields.  These are, for example, available at request online\nat \\href{http:\/\/www.coastdat.de}{http:\/\/www.coastdat.de}, from the coastDat model based\ndata base developed for the assessment of long-term changes by Helmholtz-Zentrum Geesthacht \\citep{Geyer2014}.\n}\\label{sec:availability}\n\n\\authorcontribution{C.L., R.H., K.K., H.N developed the MOSSCO components (CMI) and wrappers (BMI).  K.W, C.L., and K.K.\ndesigned the coupling philosophy, C.L. developed the user interface and the utility library. K.W., H.N., R.H., O.K. and C.L. \ncarried out and analysed simulations, based on contributions from all authors.  C.L., K.W. and R.H. wrote the manuscript with contributions from all other authors.}\n\n\\competinginterests{The authors declare that they have no conflict of interest.}\n\n\\begin{acknowledgements}\nMOSSCO is a project funded under the K\\\"ustenforschung Nordsee--Ostsee programme of the Forschung f\\\"ur \nNachhaltigkeit (FONA) agenda of the German Ministry of Education and Science (BMBF) under grant agreements \n03F0667A, 03F0667B, and 03FO668A.  This research contributes to the PACES~II programme of the Hermann von \nHelmholtz-Gemeinschaft Deutscher Forschungszentren.\nFurther financial support for K.K.\\ and H.B.\\ was provided by the Collaborative Research Centre TRR181\non Energy Transfers in Atmosphere and Ocean funded by the German Research Foundation (DFG).\nWe thank those MOSSCO developers that are not co-authors \nof this paper, amongst them Markus Kreus, Ulrich K\\\"orner and Niels Weiher, and acknowledge the support of \nWenyan Zhang in preparing the model setups.  This research is based on tremendous efforts by the open \nsource community, including but not limited to the developers of Delft3D, GETM, GOTM, FABM, ESMF, OpenMPI, \nPython, GCC and NetCDF who share their codes openly.\n\\end{acknowledgements}\n\n\\bibliographystyle{copernicus}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{SECT:intro}\n\nThe classical question in symbolic integration is whether the integral of\na given function can be written in \\lq\\lq closed form\\rq\\rq. In its most restricted form,\nthe question is whether for a given function~$f$ belonging to some domain $D$\nthere exists another function~$g$, also belonging to~$D$, such that $f=g'$. For\nexample, if $D$ is the field of rational functions, then for $f=1\/x^2$ we can\nfind $g=-1\/x$, while for $f=1\/x$ no suitable $g$ exists. When no $g$ exists\nin~$D$, there are several other questions we may ask. One possibility is to ask\nwhether there is some extension~$E$ of $D$ such that in $E$ there exists some\n$g$ with $g'=f$. For example, in the case of elementary functions, Liouville's\nprinciple restricts the possible extensions~$E$, and there are algorithms which\nconstruct such extensions whenever possible. Another possibility is\nto ask whether for some modification $\\tilde f\\in D$ of~$f$ there exists a $g\\in\nD$ such that $\\tilde f=g'$. Creative telescoping is a question of this\ntype. Here we are dealing with domains~$D$ containing functions in several\nvariables, say $x$ and~$t$, and the question is whether there is a linear\ndifferential operator~$P$, nonzero and free of~$x$, such that there exists a\n$g\\in D$ with $P\\cdot f=g'$, where $g'$ denotes the derivative of $g$ with\nrespect to~$x$. Typically, $g$~itself has the form $Q\\cdot f$ for some operator\n$Q$ (which may be zero and need not be free of~$x$). In this case, we call $P$\na telescoper for~$f$, and $Q$ a certificate for~$P$.\n\nCreative telescoping is the backbone of definite integration. Readers not\nfamiliar with this technique are referred to the literature~\\cite{PWZbook1996,Zeilberger1990c,Zeilberger1991,Zeilberger1990,Koepf1998}\nfor motivation, theory, algorithms, implementations, and applications. There are\nseveral ways to find telescopers for a given $f\\in D$. In recent years, an\napproach has become popular which has the feature that it can find a telescoper\nwithout also constructing the corresponding certificate. This is interesting\nbecause certificates tend to be much larger than telescopers, and in some\napplications only the telescoper is of interest. This approach was first\nformulated for rational functions $f\\in C(t,x)$ in~\\cite{BCCL2010} and later\ngeneralized to rational functions in several variables~\\cite{bostan13, lairez15}, to\nhyperexponential functions~\\cite{bostan13a} and, for the shift case, to hypergeometric\nterms~\\cite{chen15a} and binomial sums~\\cite{bostan15}. In the present paper, we will extend\nthe approach to algebraic functions.\n\nThe basic principle of the general approach is as follows. Assume that the\n$x$-constants $\\mathrm{Const}_x(D)=\\{\\,c\\in D:c'=0\\,\\}$ form a field and that $D$\nis a vector space over the field of $x$-constants. Assume further that there is\nsome $\\mathrm{Const}_x(D)$-linear map $[\\cdot]\\colon D\\to D$ such that for every\n$f\\in D$ there exists a $g\\in D$ with $f-[f]=g'$. Such a map is called a\n\\emph{reduction.} For example, in $D=C(t,x)$ Hermite reduction~\\cite{Hermite1872} produces for\nevery $f\\in D$ some $g\\in D$ such that $f-g'$ is either zero or a rational function\nwith a square-free denominator. In this case, we can take $[f]=f-g'$.\nIn order to find a telescoper, we can compute $[f]$, $[\\partial_t\\cdot f]$, $[\\partial_t^2\\cdot f]$, \\dots,\nuntil we find that they are linearly dependent over $\\mathrm{Const}_x(D)$.\nOnce we find a relation\n$p_0[f] + \\cdots + p_r[\\partial_t^r\\cdot f] = 0$,\nthen, by linearity,\n$[p_0 f + \\cdots + p_r \\partial_t^r\\cdot f] = 0$,\nand then, by definition of $[\\cdot]$, there exists a $g\\in D$ such that $(p_0+\\cdots + p_r\\partial_t^r)\\cdot f=g'$.\nIn other words, $P=p_0+\\cdots + p_r\\partial_t^r$ is a telescoper.\n\nThere are two ways to guarantee that this method terminates. The first requires that we already know for\nother reasons that a telescoper exists. The idea is then to show that the\nreduction $[\\cdot]$ has the property that when $f\\in D$ is such that there\nexists a $g\\in D$ with $g'=f$, then $[f]=0$. If this is the case and\n$P=p_0+\\cdots+p_r\\partial_t^r$ is a telescoper for~$f$, then $P\\cdot f$ is integrable\nin~$D$, so $[P\\cdot f]=0$, and by linearity $[f]$, \\dots, $[\\partial_t^r\\cdot f]$ are\nlinearly dependent over $\\mathrm{Const}_x(D)$. This means that the method won't\nmiss any telescoper. In particular, this argument has the nice feature that we\nare guaranteed to find a telescoper of smallest possible order~$r$. This\napproach was taken in~\\cite{chen15a}.\nThe second way consists in showing that $\\{\\,[f]:f\\in D\\,\\}$ is a finite-dimensional vector space over\n$\\mathrm{Const}_x(D)$. This approach was taken in~\\cite{BCCL2010,bostan13a}. It has the\nnice additional feature that every bound for the dimension of this vector space\ngives rise to a bound for the order of the telescoper. In particular, it implies\nthe existence of a telescoper.\n\nIn this paper, we show that Trager's Hermite reduction for algebraic\nfunctions directly gives rise to a reduction-based creative telescoping\nalgorithm via the first approach (Section~\\ref{SECT:CT-1}). We will combine Trager's Hermite reduction\nwith a second reduction, called polynomial reduction (Section~\\ref{sec:polynomial}), to obtain a reduction-based creative\ntelescoping algorithm for algebraic functions via the second approach (Section~\\ref{SECT:CT-2}).\nThis gives a new proof of a bound for the order of the telescopers, and in\nparticular an independent proof for their existence.\n\nA few years ago, Chen et al.~\\cite{chen12d} have already considered the problem of creative\ntelescoping for algebraic functions. They have pointed out that by canceling residues\nof the integrand, a given creative telescoping problem can be reduced to a creative\ntelescoping problem for a function with no residues, which may be much smaller than the\noriginal function. For this smaller function, however, they still need to construct a\ncertificate. The algorithms presented in the present paper are the first which can find\ntelescopers for algebraic functions without also constructing corresponding certificates.\nBy Theorem~6 of~\\cite{chen12d}, our results also translate into a certificate-free\ncreative telescoping algorithm for rational functions in three variables.\n\n\\section{Algebraic Functions}\n\nThroughout the paper, let $C$ be a field of characteristic zero, $K=C(t)$, and $\\bar K$ the algebraic closure of~$K$.\nWe consider algebraic functions over~$K$.  For some absolutely irreducible\npolynomial $m\\in K[x,y]$, we consider the field\n$A=K(x)[y]\/\\langle m\\rangle$. If $n=\\deg_ym$, then every element of $A$ can be written uniquely in the form\n$f=f_0+f_1y+\\cdots+f_{n-1}y^{n-1}$ for some $f_0,\\dots,f_{n-1}\\in K(x)$.\n\nThe element $y\\in A$ is a solution of the equation $m=0$,\nbecause in $A$ we have $m=0$ by construction. The polynomial $m$ also admits\n$n$ distinct solutions in the field\n\\[\n  \\bar K\\<x-a>:=\\bigcup_{r\\in\\set N\\setminus\\{0\\}} \\bar K(\\!(\\ (x-a)^{1\/r}\\ )\\!)\n\\]\nof formal Puiseux series around $a\\in\\bar K$. There are also $n$ distinct\nsolutions in the field\n\\[\n  \\bar K\\<x^{-1}>:=\\bigcup_{r\\in\\set N\\setminus\\{0\\}} \\bar K(\\!(x^{-1\/r})\\!)\n\\]\nof formal Puiseux series around~$\\infty$.\nSince $\\bar K\\<x^{-1}>$ and the $\\bar K\\<x-a>$ are fields, we can associate to every\n$f\\in A$ and every $a\\in\\bar K\\cup\\{\\infty\\}$ in a natural way $n$ distinct series\nobjects with fractional exponents, by plugging any of the $n$ distinct series solutions\nof $m$ into the representation $f=f_0+\\cdots+f_{n-1}y^{n-1}$.\nIn other words, for every $a\\in\\bar K\\cup\\{\\infty\\}$ there are $n$ distinct natural\nring homomorphisms from $A$ to $K\\<x-a>$ or $K\\<x^{-1}>$, respectively.\n\nIn the field $A$ as well as the fields $\\bar K\\<x-a>$ and $\\bar K\\<x^{-1}>$, we have\nnatural differentiations with respect to~$x$. For a series, differentiation is defined\ntermwise using the usual rules $\\bigl((x-a)^{\\nu+n}\\bigr)'=(\\nu+n)(x-a)^{\\nu+n-1}$ and\n$\\bigl((x^{-1})^{\\nu+n}\\bigr)'=-(\\nu+n)(x^{-1})^{\\nu+n+1}$. For the elements of~$A$, note\nfirst that $m(x,y)=0$ implies\n\\begin{equation}\\label{eq:yprime}\n  m(x,y)'=(\\frac d{dx}m)(x,y) + (\\frac d{dy}m)(x,y)y' = 0,\n\\end{equation}\nso $y'=-(\\frac d{dx}m)(x,y)\/(\\frac d{dy}m)(x,y)$. Regarding $m$ as element of $K(x)[y]$\nand observing that $0<\\deg_y\\frac d{dy}m<n$, we have $\\gcd(m,\\frac d{dy}m)=1$ in $K(x)[y]$,\nso $\\frac{d}{dy}m$ is invertible in $A=K(x)[y]\/\\langle m\\rangle$.\nNote that we have $x'=1$ and $c'=0$ for all $c\\in K=C(t)$, in particular also $t'=0$.\nThe derivative of an arbitrary\nelement $f\\in A$, say $f=p(x,y)$ for some $p\\in K(x)[y]$ of degree less than~$n$, is\n\\[f'=(\\frac{d}{dx}p)(x,y)+(\\frac d{dy}p)(x,y)y'.\\]\nThus we have an action of the algebra $K(x)[\\partial_x]$ of differential operators on~$A$.\n\nThe derivations on $A$ and on the series domains are compatible in the sense\nthat for every $f\\in A$, the series associated to $f'$ are precisely the\nderivatives of the series associated to~$f$.\n\nIn the context of creative telescoping, we will also need to differentiate with\nrespect to~$t$. The action of $K(x)[\\partial_x]$ on $A$ and on the series domains\nis extended to an action of $K(x)[\\partial_x,\\partial_t]$ on $A$ and on the series\ndomains. On $A$, the action of $\\partial_t$ is defined as the unique derivation with\n$\\partial_t\\cdot t=1$ and $\\partial_t\\cdot x=0$, analogously to the construction\nabove. For the series domains, $\\partial_t$ acts on the coefficients\n(which are elements of $\\bar K$) in the natural way, and does not affect~$x$.\nSince each particular element $c\\in\\bar K$ belongs to a finite algebraic extension of~$K$,\nthe result $\\partial_t\\cdot c$ is uniquely determined.\nThe actions of the larger operator algebra\n$K(x)[\\partial_x,\\partial_t]$ on $A$ and on the series domains are compatible to\neach other.\n\nIn this paper, the notation $f'$ will always refer to the derivative $\\partial_x\\cdot f$\nwith respect to~$x$, not with respect to~$t$.\n\n\\medskip\n\nTrager's Hermite reduction for algebraic functions rests on the notion of\nintegral bases. Let us recall the relevant definitions and properties.\nAlthough the elements of a Puiseux series ring $\\bar K\\<x-a>$ are formal\nobjects, the series notation suggests certain analogies with complex\nfunctions. Terms $(x-a)^\\alpha$ or $(\\tfrac1x)^\\alpha$ are\ncalled \\emph{integral} if $\\alpha\\geq0$. A series in\n$\\bar K\\<x-a>$ or $\\bar K\\<x^{-1}>$ is called integral if it only contains integral\nterms. A non-integral series is said to have a \\emph{pole} at the reference\npoint. Note that in this terminology also $1\/\\sqrt{x}$ has a pole\nat~$0$. Note also that the terminology only refers to $x$ but not to~$t$.\n\nIntegrality at $a\\in\\bar K$ is not preserved by differentiation,\nbut if $f$ is integral at~$a$, then so is $(x-a)f'$. Somewhat conversely,\nintegrality at infinity is preserved by differentiation, we even have the\nstronger property that when $f$ is integral at infinity, then not only $f'$ but also $xf'=(x^{-1})^{-1}f'$ is\nintegral at infinity.\n\nAn element $f\\in A=K(x)[y]\/\\langle m\\rangle$\nis called (locally) integral at $a\\in\\bar K\\cup\\{\\infty\\}$ if for every series\nassociated to $y$ the corresponding series for $f$ is integral.\nThe element $f$ is called (globally) integral if it is locally integral at every\n$a\\in\\bar K$ (``at all finite places'').\nThis is the case if and only if the minimal polynomial of $f$ in $K[x,y]$ is monic\nwith respect to~$y$.\nBecause of Chevalley's theorem~\\cite[page 9, Corollary 3]{Chevalley1951}, any\nnon-constant algebraic function has at least one pole. Equivalently, an element $f$ is\nintegral at all $a\\in\\bar K\\cup\\{\\infty\\}$ if and only if it is constant.\n\nFor an element $f\\in A$ to have a ``pole'' at $a\\in\\bar K\\cup\\{\\infty\\}$ means\nthat $f$ is not locally integral at~$a$; to have a ``double pole'' at $a$ means\nthat $(x-a)f$ (or $\\frac1xf$ if $a=\\infty$) is not integral; to have a ``double\nroot'' at $a$ means that $f\/(x-a)^2$ (or $f\/(\\frac1x)^2=x^2f$ if $a=\\infty$) is integral,\nand so on.\n\nThe set of all globally integral elements $f\\in A$ forms a $K[x]$-submodule of~$A$.\nA basis $\\{\\omega_1,\\dots,\\omega_n\\}$ of this module is called an \\emph{integral basis}\nfor~$A$. Such bases exist, and algorithms are known for computing them~\\cite{trager84,Rybowicz:1991:ACI:120694.120715,vanHoeij94}.\nFor a fixed $a\\in\\bar K$, let $\\bar K(x)_a$ be the ring of rational functions $p\/q$\nwith $q(a)\\neq0$, and write $\\bar K(x)_\\infty$ for the ring of all\nrational functions $p\/q$ with $\\deg_x(p)\\leq\\deg_x(q)$.\nThen the set of all $f\\in A$ which are locally integral at some\nfixed $a\\in\\bar K\\cup\\{\\infty\\}$ forms a $\\bar K(x)_a$-module. A basis of this module is\ncalled a \\emph{local integral basis} at $a$ for~$A$. Also local integral bases can\nbe computed.\n\nAn integral basis $\\{\\omega_1,\\dots,\\omega_n\\}$ is always also a $K(x)$-vector space\nbasis of~$A$. A key feature of integral bases is that they make poles explicit. Writing\nan element $f\\in A$ as a linear combination $f=\\sum_{i=1}^n f_i\\omega_i$ for some\n$f_i\\in K(x)$, we have that $f$ has a pole at $a\\in\\bar K$ if and only if at least one\nof the $f_i$ has a pole there.\n\n\\begin{lemma}\\label{lemma:1}\n  Let $\\{\\omega_1,\\dots,\\omega_n\\}$ be a local integral basis of $A$ at $a\\in\\bar K\\cup\\{\\infty\\}$.\n  Let $f\\in A$ and $f_1,\\dots,f_n\\in K(x)$ be such that $f=\\sum_{i=1}^nf_i\\omega_i$.\n  Then $f$ is integral at $a$ if and only if each $f_i\\omega_i$ is integral at~$a$.\n\\end{lemma}\n\\begin{proof}\n  The direction ``$\\Leftarrow$'' is obvious. To show ``$\\Rightarrow$'', suppose\n  that $f$ is integral at~$a$. Then there exist $w_1,\\dots,w_n\\in\\bar K(x)_a$ such that\n  $f=\\sum_{i=1}^nw_i\\omega_i$. Thus $\\sum_{i=1}^n(w_i-f_i)\\omega_i=0$, and then\n  $w_i=f_i$ for all $i$, because $\\omega_1,\\dots,\\omega_n$ is a vector space basis of~$A$.\n  As elements of $\\bar K(x)_a$, the $f_i$ are integral at~$a$, and hence also all the $f_i\\omega_i$\n  are integral at~$a$.\n\\end{proof}\n\nThe lemma says in particular that poles of the $f_i$ in a linear combination\n$\\sum_{i=1}^n f_i\\omega_i$ have no chance to cancel each other.\n\n\\begin{lemma}\\label{lemma:e}\n  Let $\\{\\omega_1,\\dots,\\omega_n\\}$ be an integral basis of~$A$.\n  Let $e\\in K[x]$ and\n  $M=((m_{i,j}))_{i,j=1}^n\\in K[x]^{n\\times n}$ be such that\n  \\[\n    e\\,\\omega_i'=\\sum_{j=1}^n m_{i,j}\\omega_j\n  \\]\n  for $i=1,\\dots,n$ and $\\gcd(e,m_{1,1},\\dots,m_{n,n})=1$.\n  Then $e$ is squarefree.\n\\end{lemma}\n\\begin{proof}\n  Let $a\\in\\bar K$ be a root of~$e$. We show that $a$ is not a multiple root.\n  Since $\\omega_i$ is integral, it is in particular locally integral at~$a$.\n  Therefore $(x-a)\\omega_i'$ is locally integral at~$a$.\n  Since $\\omega_1,\\dots,\\omega_n$ is an integral basis, it follows that\n  $(x-a)m_{i,j}\/e\\in\\bar K(x)_a$ for all~$i,j$.\n  Because of $\\gcd(e,m_{1,1},\\dots,m_{n,n})=1$, no factor $x-a$ of $e$\n  can be canceled by all the~$m_{i,j}$.\n  Therefore the factor $x-a$ can appear in $e$ only once.\n\\end{proof}\n\n\\begin{lemma} \\label{lemma:degM}\n  Let $\\{\\omega_1,\\dots,\\omega_n\\}$ be a local integral basis at infinity of~$A$.\n  Let $e\\in K[x]$ and $M=((m_{i,j}))_{i,j=1}^n\\in K[x]^{n\\times n}$\n  be defined as in Lemma~\\ref{lemma:e}. Then $\\deg_x(m_{i,j})<\\deg_x(e)$ for all $i,j$.\n\\end{lemma}\n\\begin{proof}\n  Since every $\\omega_i$ is locally integral at infinity, so is every $x\\,\\omega_i'$.\n  Since $\\omega_1,\\dots,\\omega_n$ is an integral basis at infinity, it follows that\n  $xm_{i,j}\/e\\in\\bar K(x)_\\infty$ for all~$i,j$. This means that $1+\\deg_x(m_{i,j})\\leq\\deg_x(e)$\n  for all~$i,j$, and therefore $\\deg_x(m_{i,j})<\\deg_x(e)$, as claimed.\n\\end{proof}\n\nA $K(x)$-vector space basis $\\{\\omega_1,\\dots,\\omega_n\\}$ of $A$ is\ncalled \\emph{normal} at $a\\in\\bar K\\cup\\{\\infty\\}$ if there exist $r_1,\\dots,r_n\\in\nK(x)$ such that $\\{r_1\\omega_1,\\dots,r_n\\omega_n\\}$ is a local integral basis\nat~$a$. Trager shows how to construct\nan integral basis which is normal at infinity from a given integral basis and\na given local integral basis at infinity~\\cite{trager84}.\n\nAlthough normality is a somewhat weaker condition on a basis than integrality,\nit also excludes the possibility that poles in the terms of a linear combination\nof basis elements can cancel:\n\n\\begin{lemma}\\label{lemma:3}\n  Let $\\{\\omega_1,\\dots,\\omega_n\\}$ be a basis of~$A$\n  which is normal at some $a\\in\\bar K\\cup\\{\\infty\\}$.\n  Let $f=\\sum_{i=1}^n f_i\\omega_i$ for some $f_1,\\dots,f_n\\in K(x)$.\n  Then $f$ has a pole at $a$ if and only if\n  there is some $i$ such that $f_i\\omega_i$ has a pole at~$a$.\n\\end{lemma}\n\\begin{proof}\n  Let $r_1,\\dots,r_n\\in K(x)$ be such that $r_1\\omega_1,\\dots,r_n\\omega_n$ is a\n  local integral basis at~$a$. By $f=\\sum_{i=1}^n\n  (f_ir_i^{-1})(r_i\\omega_i)$ and by Lemma~\\ref{lemma:1}, $f$~is integral at~$a$ iff all\n  $f_ir_i^{-1}r_i\\omega_i=f_i\\omega_i$ are integral at~$a$.\n\\end{proof}\n\n\\section{Hermite Reduction}\\label{sec:hermite}\n\nWe now recall the Hermite reduction for algebraic functions~\\cite{trager84,ACA1992,bronstein98}.\nLet $\\{\\omega_1,\\ldots,\\omega_n\\}$ be an integral basis for~$A$.\nFurther let $e, m_{i,j}\\in K[x]$ ($1\\leq i,j\\leq n$) be such that\n$e\\omega_i'=\\sum_{j=1}^n m_{i,j}\\omega_i$ and\n$\\gcd(e,m_{1,1},m_{1,2},\\ldots,m_{n,n})=1$.\nFor describing the Hermite reduction we fix an integrand $f\\in A$ and represent it in the\nintegral basis, i.e., $f=\\sum_{i=1}^n (f_i\/D)\\,\\omega_i$ with\n$D, f_1,\\ldots,f_n\\in K[x]$. The purpose is to find $g, h\\in A$ such that\n$f=g' + h$ and $h=\\sum_{i=1}^n(h_i\/D^\\ast)\\,\\omega_i$ with $h_1,\\ldots,h_n\\in K[x]$\nand $D^\\ast$ denoting the squarefree part of~$D$.\nAs differentiating the $\\omega_i$ can introduce\ndenominators, name\\-ly the factors of~$e$, it is convenient to consider those\ndenominators from the very beginning on, which means that we shall assume\n$e\\mid D$. Note that $\\gcd(D,f_1,\\ldots,f_n)$ can then be nontrivial.\nLet~$v\\in K[x]$ be a nontrivial squarefree factor of~$D$ of multiplicity~$\\mu>1$.\nThen~$D = uv^\\mu$ for some $u\\in K[x]$ with $\\gcd(u, v)=1$ and $\\gcd(v,v')=1$.\nOne step of the Hermite reduction is as follows:\n\\begin{equation}\\label{eq:hred}\n  \\sum_{i=1}^n \\frac{f_i}{uv^\\mu}\\omega_i =\n  \\biggl(\\sum_{i=1}^n\\frac{g_i}{v^{\\mu-1}}\\omega_i\\biggr)' +\n  \\sum_{i=1}^n \\frac{h_i}{uv^{\\mu-1}}\\omega_i,\n\\end{equation}\nwhere $g_i, h_i \\in K[x]$ and~$\\deg_x(g_i)< \\deg_x(v)$.\nThe existence of such~$g_i$'s and~$h_i$'s follows from the crucial fact that\nthe elements $s_i :=  uv^\\mu(v^{1-\\mu}\\omega_i)'$ with $i\\in \\{ 1, \\ldots, n\\}$\nform a local integral basis at each root of~$v$~\\cite[page 46]{trager84}.\nBy a repeated application of such reduction steps, one can decompose any $f\\in A$\nas $f=g' + h$ where the denominators of the coefficients of $h$ are squarefree\nand the coefficients of $g$ are proper rational functions (i.e., their numerators\nhave smaller degree than their denominators).\n\nIt was observed that Hermite reduction itself often takes less time than the construction\nof an integral basis. If Hermite reduction is applied to some other basis, for instance\nthe standard basis $\\{1,y,\\dots,y^{n-1}\\}$, it either succeeds or it runs into a division by zero.\nBronstein~\\cite{bronstein98a} noticed that when a division by zero occurs, then the basis can\nbe replaced by some other basis that is a little closer to an integral basis, just\nas much as is needed to avoid this particular division by zero. After finitely many\nsuch basis changes, the Hermite reduction will come to an end and produce a correct\noutput. This variant is known as lazy Hermite reduction.\n\n\\section{Telescoping via reductions: \\hskip0ptplus1fill\\break first approach} \\label{SECT:CT-1}\nRecall from the introduction that reduction-based creative telescoping requires\nsome $K$-linear map $[\\cdot]\\colon A\\to A$ with the property that\n$f-[f]$ is integrable in $A$ for every $f\\in A$. This is sufficient for the\ncorrectness of the method, but additional properties are needed in order to\nensure that the method terminates.\n\nAs also explained already in the introduction, one possibility consists in\nshowing that $[f]=0$ whenever $f$ is integrable. Trager showed that his\nHermite reduction has this property~\\cite[page 50, Theorem 1]{trager84}.\nFor the sake of completeness, we reproduce his proof here.\n\n\\begin{lemma}\\label{lemma:pole_at_inf}\nLet $W=\\{\\omega_1,\\dots,\\omega_n\\}$ be an integral basis for $A$ that is normal at\ninfinity. Let $g=\\sum_{i=1}^ng_i\\omega_i\\in A$ be such that all its\ncoefficients $g_i\\in K(x)$ are proper rational functions. If an integral\nelement $f\\in A$ has a pole at infinity, then also $f+g$ has a pole at\ninfinity.\n\\end{lemma}\n\\begin{proof}\nSince $f$ is assumed to be integral we can write it as\n$f=f_1\\omega_1+\\cdots+f_n\\omega_n$ with $f_i\\in K[x]$.\nIf $f$ has a pole at infinity, there is at least one index~$i$\nsuch that $f_i\\omega_i$ has a pole at infinity. There are two cases why this\ncan happen.\n\\renewcommand{\\labelenumi}{(\\alph{enumi})}\n\\begin{enumerate}\n\\item The polynomial~$f_i$ has positive degree.  This means that $f_i+g_i$ has a\n  pole at infinity, because the $g_i$ are proper rational functions.\n  Thus $(f_i+g_i)\\omega_i$ has a pole at infinity, because $\\omega_i$ has no poles\n  at finite places and therefore no root at infinity.\n\\item The integral basis element $\\omega_i$ is not constant and $f_i$ is not zero. Hence\n  $\\omega_i$ has a pole at infinity, and this also implies that $(f_i+g_i)\\omega_i$\n  has a pole at infinity, again employing the fact that $g_i$ is a proper rational function.\n\\end{enumerate}\nIn both cases, therefore, $f+g=\\sum_{i=1}^n(f_i+g_i)\\omega_i$ has a pole at\ninfinity by Lemma~\\ref{lemma:3}.\n\\end{proof}\n\n\\begin{theorem}\\label{thm:intiff0}\n  Suppose that $f\\in A$\n  has a double root at infinity (i.e., every series in $\\bar K\\<x^{-1}>$\n  associated to $f$ only contains monomials $(1\/x)^\\alpha$ with $\\alpha\\geq2$).\n  Let $W=\\{\\omega_1,\\dots,\\omega_n\\}$\n  be an integral basis for $A$ that is normal at infinity.\n  If $f=g'+h$ is the result of the Hermite reduction with respect to~$W$,\n  then $h=0$ if and only if $f$ is integrable in~$A$.\n\\end{theorem}\n\n\\begin{proof}\nThe direction ``$\\Rightarrow$'' is trivial. To show the implication\n``$\\Leftarrow$'' assume that $f$ is integrable in~$A$. From $f=g'+h$ it follows that\nthen also $h$ is integrable in~$A$; let $H\\in A$ be such that $H'=h$.  In order to show\nthat $h=0$, we show that $H$ is constant.  To this end, it suffices to show that\nit has neither finite poles nor a pole at infinity; the claim then follows from\nChevalley's theorem.\n\nIt is clear that $H$ has no finite poles because $h$ has at most simple poles\n(i.e., all series associated to $h$ have only exponents $\\geq-1$).\nThis follows from the facts that the $\\omega_i$ are integral and that\nthe coefficients of~$h$ have squarefree denominators.\n\nIf $H$ has a pole at infinity, then by Lemma~\\ref{lemma:pole_at_inf} also\n$g+H$ must have a pole at infinity, because Hermite reduction produces\n$g=\\sum_i g_i\\omega_i$ with proper rational functions~$g_i$.  On the other\nhand, since $f=g'+h=(g+H)'$ has at least a double root at infinity by\nassumption, $g+H$ must have at least a single root at infinity. This is\na contradiction.\n\\end{proof}\n\nNote that the condition in Theorem~\\ref{thm:intiff0} that $f$ has a double\nroot at infinity is not a restriction at all, as it can always be achieved by\na suitable change of variables. Let $a\\in C$ be a regular point; this means\nthat all series in $\\bar K\\<x-a>$ associated to $f$ are formal power series. By the substitution\n$x\\to a+1\/x$ the regular point~$a$ is moved to infinity. From\n\\[\n  \\int f(x) \\,\\mathrm{d}x = \\int f\\left(\\frac{1}{x}+a\\right)\\left(-\\frac{1}{x^2}\\right) \\mathrm{d}x\n\\]\nwe see that the new integrand has a double root at infinity.\n\nMoreover, since the action of $\\partial_t$ on series domains is defined coefficient-wise,\nit follows that when $f$ has at least a double root at infinity (with respect to~$x$),\nthen this is also true for $\\partial_t\\cdot f, \\partial_t^2\\cdot f, \\partial_t^3\\cdot f,\\dots$,\nand then also for every $K$-linear combination $p_0f+p_1\\partial_t\\cdot f+\\cdots+p_r\\partial_t^r\\cdot f$.\nThus Theorem~\\ref{thm:intiff0} implies that $p_0+p_1\\partial_t+\\cdots+p_r\\partial_t^r$ is a telescoper for $f$ if\n\\emph{and only if} $[p_0+p_1\\partial_t+\\cdots+p_r\\partial_t^r]=0$.\n\nWe already know for other reasons~\\cite{Zeilberger1990,chyzak00,chen12d} that\ntelescopers for algebraic functions exist, and therefore the re\\-duc\\-tion-based\ncreative telescoping procedure with Hermite reduction with respect to an\nintegral basis that is normal at infinity as reduction function succeeds when\napplied to an integrand $f\\in A$ that has a double root at infinity.\nIn particular, the method finds a telescoper of smallest possible order.\nAgain, if $f$ has no double root at infinity, we can produce one by a change of variables.\nNote that a change of variables $x\\to a+1\/x$ with $a\\in C$ has no effect on\nthe telescoper.\n\n\\begin{example}\\label{ex:ct}\nWe consider the algebraic function $f=y\/x^2$ where $y$ is a solution of the\nthird-degree polynomial equation $m(x,y) = y^3 + y + x + t = 0$. An integral\nbasis for $A=K(x)[y]\/\\langle m\\rangle$ that is normal at infinity is given by\n$\\omega_1=1$, $\\omega_2=y$, $\\omega_3=y^2$.  (This means that employing lazy\nHermite reduction avoids completely the computation of an integral basis in\nthis example.)\n\nBy solving Equation~\\eqref{eq:yprime} for $y'$ we obtain\n\\[\n  y' = \\frac{-6y^2 + 9(t+x)y - 4}{27x^2+54tx+27t^2+4}.\n\\]\nThen for the differentiation matrix~$\\frac1eM$, a simple calculation yields\n\\[\n  \\begin{pmatrix} \\omega_1' \\\\[1pt] \\omega_2' \\\\[1pt] \\omega_3' \\end{pmatrix} =\n  \\frac{1}{e} \\begin{pmatrix} 0 & 0 & 0 \\\\[1pt] -4 & 9 (t+x) & -6 \\\\[1pt] 12 (t+x) & 4 & 18 (t+x) \\\\ \\end{pmatrix}\n  \\begin{pmatrix} \\omega_1 \\\\[1pt] \\omega_2 \\\\[1pt] \\omega_3 \\end{pmatrix}\n\\]\nwith $e=27x^2+54xt+27t^2+4$. Thus we write $f=\\sum_{i=1}^3 (f_i\/D) \\omega_i$ with\n$f_1=f_3=0$, $f_2=e$, and $D=x^2e$.  After a single step the Hermite reduction\ndelivers the result\n\\[\n  f = \\biggl(\\, \\underbrace{\\vphantom{\\frac{1}{(x)}} -\\frac{y}{x}}_{\\textstyle=g_0} \\;\\biggr)' +\\>\n  \\underbrace{\\frac{-6y^2+9(x+t)y-4}{x(27x^2+54xt+27t^2+4)}}_{\\textstyle=h_0}.\n\\]\nAs the Hermite remainder~$h_0$ is nonzero, Theorem~\\ref{thm:intiff0} tells us that\n$f$ is not integrable in~$A$. Hence we continue by applying Hermite reduction to\n\\[\n  \\partial_t\\cdot f = \\frac{-6y^2+9(x+t)y-4}{x^2(27x^2+54xt+27t^2+4)}.\n\\]\nNote that we could as well take $\\partial_t\\cdot h_0$ instead of $\\partial_t\\cdot f$, which\nin general should result in a faster algorithm.\nAgain after a single reduction step, the decomposition $\\partial_t\\cdot f = g_1' + h_1$\nis obtained, where\n\\begin{align*}\n  g_1 &= \\frac{6y^2-9ty+4}{x(27t^2+4)} \\\\\n  h_1 &= \\frac{6\\bigl((9x+27t)y^2-(27xt+27t^2-2)y+6x+18t\\bigr)}{x(27t^2+4)(27x^2+54xt+27t^2+4)}.\n\\end{align*}\nSince $h_0$ and $h_1$ are linearly independent over $K=C(t)$, we continue with\n$\\partial_t^2\\cdot f$.\nThis time however, it is preferable to start the Hermite reduction\nwith $\\partial_t\\cdot h_1$, which is given by\n\\[\n  \\frac{1}{x(27t^2+4)^2(27x^2+54xt+27t^2+4)^2}.\n\\]\nSetting $v=27x^2+54xt+27t^2+4=e$ and doing one reduction step,\nthe Hermite remainder $h_2$ is found to be\n\\begin{multline*}\n \\bigl(6\\bigl((-729xt-1539t^2+96)y^2+(1215xt^2-144x+1215t^3-{}\\\\\n 306t)y-486xt-1026t^2+64\\bigr)\\bigr)\\mathrel{\\big\\slash}\\bigl(x(27t^2+4)^2e\\bigr).\n\\end{multline*}\nThe corresponding integrable part $g_2$ is not displayed here for space reasons.\n\nNow one can find a linear dependence between $h_0,h_1,h_2$ that gives rise to the telescoper\n$(27t^2+4)\\partial_t^2+81t\\partial_t+24$, which is indeed the minimal one for this example.\n\\end{example}\n\n\n\\section{Polynomial Reduction}\\label{sec:polynomial}\n\nRecall that instead of requesting that $[f]=0$ if and only if $f$ is integrable\n(first approach), we can also justify the termination of reduction-based\ncreative telescoping by showing that the $K$-vector space $\\{\\,[f]:f\\in A\\,\\}$\nhas finite dimension (second approach). If $[\\cdot]$ is just the Hermite\nreduction, we do not have this property. We therefore introduce below an\nadditional reduction, called \\emph{polynomial reduction,} which we apply after\nHermite reduction. We then show that the combined reduction (Hermite reduction\nfollowed by polynomial reduction) has the desired dimension property for the\nspace of remainders. As a result, we obtain a new bound on the order of the\ntelescoper, which is similar to those in~\\cite{chen12d,chen14a}.\n\nIn this approach, we use two integral bases. First we use a global integral basis (not\nnecessarily normal at infinity) in order to perform Hermite reduction. Then we write the\nremainder $h$ with respect to some local integral basis at infinity and perform the\npolynomial reduction on this representation.\n\nThroughout this section let $W=(\\omega_1,\\ldots,\\omega_n)^T\\in A^n$ be such\nthat $\\{\\omega_1, \\ldots, \\omega_n\\}$ is a global integral basis of~$A$, and\nlet $e\\in K[x]$ and $M=(m_{i,j})\\in K[x]^{n\\times n}$ be such that $eW'=MW$\nand $\\gcd(e, m_{1, 1}, m_{1, 2}, \\ldots, m_{n ,n})=1$. The Hermite reduction\ndescribed in Section~\\ref{sec:hermite} decomposes an input element $f\\in A$\ninto the form\n\\[\n  f = g' + h = g' + \\sum_{i=1}^n \\frac{h_i}{de} \\omega_i,\\qquad\n  g, h\\in A,\n\\]\nwith $h_i, d\\in K[x]$ such that $\\gcd(d, e)=\\gcd(h_i,de)=1$ and $d$ is squarefree.\n\\begin{lemma}\\label{LEM:d}\nIf $h$ is integrable in~$A$, then $d$ is in~$K$.\n\\end{lemma}\n\\begin{proof}\nSuppose that $h$ is integrable in~$A$, i.e., there exist $a, b_i\\in K[x]$\nsuch that $h = \\bigl(\\frac{1}{a}\\sum_{i=1}^n b_i \\omega_i\\bigr)'$. Then\n\\[\n  h = \\sum_{i=1}^n \\frac{h_i}{de}\\omega_i= \\sum_{i=1}^n \\biggl(\\Bigl(\\frac{b_i}{a}\\Bigr)' \\omega_i +\n  \\frac{b_i}{a e} \\sum_{j=1}^n  m_{i, j}\\omega_j\\biggr).\n\\]\nWe show that $a$ is constant. Otherwise, for any irreducible factor $p$ of~$a$, we would have that $h$ has a pole of\nmultiplicity greater than $1$ at the roots of~$p$. This contradicts\nthe fact that $d, e$ are squarefree. Thus, $d$ is a constant.\n\\end{proof}\n\nBy the extended Euclidean algorithm, we compute $u_i, v_i\\in K[x]$ such that\n$h_i = u_i d + v_i e$ and $\\deg_x(v_i) < \\deg_x(d)$. Then the Hermite remainder~$h$\ndecomposes as\n\\begin{equation}\\label{EQ:h}\n  \\sum_{i=1}^n \\frac{h_i}{de}\\omega_i =  \\sum_{i=1}^n \\frac{u_i}{e}\\omega_i + \\sum_{i=1}^n \\frac{v_i}{d}\\omega_i.\n\\end{equation}\n\nWe now introduce the \\emph{polynomial reduction} whose goal is to confine the $u_i$ to a finite-dimensional\nvector space over~$K$. Similar reductions have been introduced and used in creative telescoping\nfor hyperexponential functions~\\cite{bostan13a} and hypergeometric terms~\\cite{chen15a}.\nLet $V = (\\nu_1, \\ldots, \\nu_n)^T\\in A^n$ be such that its entries form a $K(x)$-basis of~$A$,\nand let $a\\in K[x]$ and $B = (b_{i, j})\\in K[x]^{n \\times n}$ be such that $aV'=BV$ and\n$\\gcd(a, b_{1, 1}, b_{1, 2}, \\ldots, b_{n ,n})=1$. Let $p = (p_1, \\ldots, p_n)\\in K[x]^n$. Then\n\\begin{equation} \\label{EQ:polyred}\n  (pV)' = \\sum_{i=1}^n (p_i \\nu_i)' = \\frac{ap' + pB}{a}\\, V.\n\\end{equation}\nThis motivates us to introduce the following definition.\n\\begin{defi}\nLet the map $\\phi_V\\colon K[x]^n \\rightarrow K[x]^n$\nbe defined by $\\phi_V(p) = ap' + pB$ for any $p\\in K[x]^n$.\nWe call $\\phi_V$ the \\emph{map for polynomial reduction} with respect to~$V$, and call\nthe subspace $\\im(\\phi_V) = \\{\\phi_V(p) \\mid p \\in K[x]^n\\}$\nthe \\emph{subspace for polynomial reduction} with respect to~$V$.\n\\end{defi}\n\nNote that, by construction and because of Lemma~\\ref{LEM:d}, $q\\in K[x]^n$ is in\n$\\im(\\phi_V)$ if and only if $\\frac{q}{a}V$ is integrable in~$A$.\n\nWe can always view an element of $K[x]^n$ (resp. $K[x]^{n\\times n}$) as a polynomial in~$x$\nwith coefficients in~$K^n$ (resp. $K^{n\\times n}$). In this sense we use the notation $\\lc(\\cdot)$\nfor the leading coefficient and $\\lt(\\cdot)$ for the leading term of a vector (resp. matrix).\nFor example, if $p\\in K[x]^n$ is of the form\n\\[\n  p = p^{(r)}x^r + \\dots + p^{(1)}x + p^{(0)},\\quad p^{(i)}\\in K^n,\n\\]\nthen $\\deg_x(p)=r$, $\\lc(p)=p^{(r)}$, and $\\lt(p)=p^{(r)}x^r$.\nLet $\\{e_1, \\ldots, e_n\\}$ be the standard basis of~$K^n$.\nThen the module $K[x]^n$ viewed as a $K$-vector space is generated by\n\\[\n  \\cS := \\bigl\\{e_ix^j \\mathrel{\\big|} 1\\leq i \\leq n,\\, j\\in \\bN\\bigr\\}.\n\\]\nWe define $K[x]_\\mu^n:=\\{p\\in K[x]^n \\mid \\deg_x(p) \\leq \\mu\\}$; as a $K$-vector\nspace it is generated by\n\\[\n  \\cS_\\mu := \\bigl\\{e_ix^j \\mathrel{\\big|} 1\\leq i \\leq n,\\, 0\\leq j\\leq \\mu\\bigr\\}.\n\\]\nAny element $p\\in K[x]_\\mu^n$ can be expressed in the\nbasis $\\cS_\\mu$ as a vector $\\vec{p}\\in K^{n(\\mu+1)}$ (in the following the\ndecoration~\\raisebox{-1pt}{$\\vec{}\\;$} always indicates such a typecast).\n\n\\begin{defi}\nLet $N_V$ be the $K$-subspace of $K[x]^n$ generated by\n\\[\n  \\bigl\\{t \\in \\cS \\mathrel{\\big|} t \\neq \\lt(p) \\ \\text{for all $p\\in \\im(\\phi_V)$}\\bigr\\}.\n\\]\nThen $K[x]^n = \\im(\\phi_V) \\oplus N_V$.\nWe call $N_V$ the \\emph{standard complement} of $\\im(\\phi_V)$.\nFor any $p\\in K[x]^n$, there exist $p_1\\in K[x]^n$ and~$p_2\\in N_V$ such that\n\\[\\frac{p}{a}V = (p_1V)' + \\frac{p_2}{a}V.\\]\nThis decomposition is called the \\emph{polynomial reduction} of~$p$\nwith respect to~$V$.\n\\end{defi}\n\n\\begin{prop}\\label{PROP:finite}\nLet $a\\in K[x]$ and $B\\in K[x]^{n \\times n}$ be such that $aV'=BV$, as before.\nIf $\\deg_x(B) \\leq \\deg_x(a)-1$, then $N_V$ is a finite-dimensional\n$K$-vector space.\n\\end{prop}\n\\begin{proof}\nIn addition to the proof of the assertion, we also explain how to determine\nthe dimension and a basis for $N_V$, for later use. For brevity, let\n$\\mu:=\\deg_x(a)-1$. We distinguish two cases.\n\n\\smallskip\n{\\it Case 1.}~\nAssume that $\\deg_x(B) < \\mu$. For any $p\\in K[x]^n$ of degree $s>0$, we have\n\\[\n  \\lt\\bigl(\\phi_V(p)\\bigr) = s\\lc(a)\\lc(p)x^{s+\\mu}.\n\\]\nThus all monomials $e_i x^j\\in \\cS$ with $1\\leq i\\leq n$ and $j\\geq \\mu+1$ are not in~$N_V$.\nLet $\\vec{B}_1, \\ldots, \\vec{B}_n$ be the columns of~$B$, expressed in the basis $\\cS_\\mu$.\nLet $C(B)$ be the $K$-subspace of $K[x]_\\mu^n$ generated by these column vectors.\nIf $q\\in \\im(\\phi_V)$, then $q = \\phi_V(p) = pB$ for some $p \\in K^n$, which implies that\n$\\vec{q}\\,$ is a linear combination of $\\vec{B}_i$'s. Then $K[x]_\\mu^n = C(B) \\oplus N_V$.\nSo $\\dim_K(N_V)= (\\mu+1)n - \\dim_K(C(B))$ and a basis of $N_V$ can be computed by\nlooking at the echelon form of the matrix $\\bigl(\\vec{B}_1, \\ldots, \\vec{B}_n\\bigr)$.\n\n\\smallskip\n{\\it Case 2.}~\nAssume that $\\deg_x(B) =\\mu$. For any $p\\in K[x]^n$ of degree $s$, we have\n\\[\n  \\lt\\bigl(\\phi_V(p)\\bigr) = \\lc(p)(s\\lc(a)I_n + \\lc(B))x^{s+\\mu}.\n\\]\nLet $\\ell$ be the largest nonnegative integer such that $-\\ell \\lc(a)$ is an\neigenvalue of $\\lc(B)\\in K^{n\\times n}$. Then for any $s>\\ell$,\nthe matrix $J_s = s\\lc(a)I_n + \\lc(B)$ is invertible. So any monomial $e_ix^j$ with $j> \\ell+\\mu$ is not in~$N_V$\nfor any $i=1, \\ldots, n$. Let $p = \\sum_{i=1}^n \\sum_{j=0}^{\\ell} p_{i, j} e_ix^j$.\nThen $\\phi_V(p)$ belongs to $K[x]_{\\ell+\\mu}^n$.\nIn the basis $\\cS_{\\ell+\\mu}$, we can\nexpress $\\phi_V(p)$ as a vector of length ${n(\\ell+\\mu+1)}$ with entries linear in the $p_{i, j}$'s.\nThis vector can be written in the form $M_{\\ell} \\vec{P}$,\nwhere $\\vec{P} = (p_{1, 0}, p_{2, 0}, \\ldots, p_{n, \\ell})^T$ and $M_{\\ell} \\in K^{n(\\ell+\\mu +1) \\times n(\\ell+1)}$.\nEvery $q\\in K[x]_{\\ell+\\mu}^n$ can be expressed as a vector $\\vec{q} \\in K^{n(\\ell + \\mu +1)}$.\nThen $q\\in\\im(\\phi_v)$ if and only if $\\vec{q}\\,$ is in the column space of~$M_{\\ell}$.\nTherefore,\n\\[K[x]_{\\ell+\\mu}^n = C({M_{\\ell}}) \\oplus N_V. \\]\nThis implies that $\\dim_K(N_V) = n(\\ell+\\mu+1) - \\rank({M_{\\ell}})$, and\na basis of $N_V$ can be computed by\nlooking at the echelon form of the matrix ${M_{\\ell}}$.\n\\end{proof}\n\nIn general, the condition $\\deg_x(B) \\leq \\deg_x(a)-1$ may not hold for an arbitrary basis~$V$ of~$A$.\nThe following lemma shows that we can perform a simple change of basis to make the condition hold.\n\n\\begin{lemma}\\label{LM:CB}\nLet~$W =\\{\\omega_1, \\ldots, \\omega_n\\}$ be an integral basis of~$A$ such that it is also normal at infinity. Then\nthere exist nonnegative integers~$\\tau_1, \\ldots, \\tau_n$ such that\n\\[ V := \\{\\nu_1, \\ldots, \\nu_n\\} \\quad \\text{with $\\nu_i = x^{-\\tau_i} \\omega_i$}\\]\nis a basis of~$A$ which is normal at $0$ and integral at all other places (including infinity).\n\\end{lemma}\n\\begin{proof}\nIt is clear that such a basis~$V$ will be normal at zero, because multiplying the generators by\nthe rational functions $x^{\\tau_i}$ brings it back to a global integral basis, which is in particular\na local integral basis at zero.\nIt is also clear that such a basis will be integral at every other point $a\\in\\bar K\\setminus\\{0\\}$, because the\nmultipliers $x^{-\\tau_i}$ are locally units at such~$a$.\nFinally, since the original basis is normal at infinity, there exist rational functions $u_1,\\dots,u_n$\nsuch that $\\{u_1\\omega_1,\\dots,u_n\\omega_n\\}$ is a local integral basis at infinity.\nSince $u_i$ can be written as $u_i=x^{-\\tau_i}\\tilde{u}_i$ with $\\tau_i\\in\\set Z$ and $\\tilde{u}_i$ being a unit\nin $\\bar{C}(x)_\\infty$, we see that also $V$ is a local integral basis at infinity.\nThe integers~$\\tau_i$ can only be nonnegative because the $\\omega_i$'s have no finite poles and therefore each\nof them is either constant or has a pole at infinity by Chevalley's theorem.\n\\end{proof}\n\n\n\nCombining the Hermite reduction and polynomial reduction, we get the following theorem.\n\\begin{theorem}\\label{THM:polyred}\nLet $W$ be an integral basis of~$A$ that is normal at infinity.\nLet $T := \\diag(x^{-\\tau_1}, \\ldots, x^{-\\tau_n}) \\in K(x)^{n\\times n}$\nbe such that $V = TW$ is integral at infinity.\nLet $e\\in K[x]$, $\\lambda \\in \\bN$, and $B, M \\in K[x]^{n \\times n} $ be such that\n$eW' = MW$ and $x^\\lambda eV' = BV$.\nThen any element $f\\in A$ can be decomposed into\n\\begin{equation}\\label{EQ:add}\nf = g' + \\frac{1}{d} PW + \\frac{1}{x^\\lambda e} QV,\n\\end{equation}\nwhere $g\\in A$, $d\\in K[x]$ is squarefree and $\\gcd(d, e)=1$, $P, Q\\in K[x]^n$ with $\\deg_x(P) < \\deg_x(d)$ and $Q\\in N_V$, which is\na finite-dimensional $K$-vector space. Moreover, $P, Q$ are zero if and only if $f$ is integrable in~$A$.\n\\end{theorem}\n\\begin{proof}\nAfter performing the Hermite reduction on $f$, we get\n\\[f = \\tilde{g}' + \\frac{1}{d} PW + \\frac{1}{e} UW,\\]\nwhere~$P = (v_1, \\ldots, v_n)\\in K[x]^n$ and $U = (u_1, \\ldots, u_n)\\in K[x]^n$\nwith~$u_i, v_i$ introduced in~\\eqref{EQ:h}. By Lemma~\\ref{LM:CB}, there exists\n$T := \\diag(x^{-\\tau_1}, \\ldots, x^{-\\tau_n}) \\in K(x)^{n\\times n}$\nsuch that $V = TW$ is normal at $0$ and integral at any other places (including infinity). Note that we can\nchoose~$T$ as the identity matrix if~$\\deg_x(M)\\leq \\deg_x(e)-1$.\nBy taking derivatives, we get\n\\[V' = \\left(T' + T\\frac{M}{e}\\right)T^{-1}V = \\frac{B}{a}V, \\]\nwhere $a=x^\\lambda e$ for some $\\lambda\\in \\bN$ and $B\\in K[x]^{n\\times n}$. Since $V$ is locally integral\nat infinity, $\\deg_x(B) \\leq \\deg_x(a)-1$ by Lemma~\\ref{lemma:degM}.\nBy expanding in terms of the new basis~$V$, we get\n\\[\\frac{1}{e} UW = \\frac{1}{a} \\tilde{U}V, \\]\nwhere $\\tilde{U} = x^\\lambda U T^{-1} \\in K[x]^n$. Next, we decompose $\\tilde{U}$ into\n$\\tilde{U} = \\phi_{V}(\\tilde{U}_1) + \\tilde{U}_2$ with $\\tilde{U}_1, \\tilde{U}_2\\in K[x]^n$ and\n$\\tilde{U}_2\\in N_V$. Then we get\n\\[\\frac{1}{e} UW = (\\tilde U_1 V)' + \\frac{1}{a} \\tilde U_2 V. \\]\nWe then get the decomposition~\\eqref{EQ:add} by setting\n$g = \\tilde g + \\tilde U_1 V$ and~$Q = U_2$.\n\nAssume that $f$ is integrable. Then Lemma~\\ref{LEM:d} implies that $d\\in K$.\nSince $\\deg_x(P) < \\deg_x(d)$, we have $P=0$. Then\n\\[\\frac{1}{x^\\lambda e} QV = \\sum_{i=1}^n (a_i \\nu_i)'\\]\nfor some $a_i\\in K[x]$. So $Q \\in \\im(\\phi_V)$.\nSince $\\im(\\phi_V) \\cap N_V = \\{0\\}$, it follows that $Q=0$.\n\\end{proof}\nThe decomposition in~\\eqref{EQ:add} is called an \\emph{additive decomposition} of~$f$ with respect to~$x$.\n\n\\section{Telescoping via reductions: \\hskip0ptplus1fill\\break second approach} \\label{SECT:CT-2}\n\nWe now discuss how to compute telescopers for algebraic functions via Hermite reduction and\npolynomial reduction.\n\nLet $W, V, e, \\lambda, M, B$ be as in Theorem~\\ref{THM:polyred}.\nTo construct a telescoper for~$f\\in A$,\nwe first consider the additive decompositions of the successive derivatives $\\partial_t^i\\cdot f$ for $i\\in \\bN$.\nAssume that\n\\[\\partial_t\\cdot W = \\frac{1}{\\tilde{e}} \\tilde{M}W \\quad \\text{and}\n\\quad \\partial_t\\cdot V = \\frac{1}{x^{\\tilde{\\lambda}} \\tilde{e}} \\tilde{B}V,\\]\nwhere $\\tilde e \\in K[x]$, $\\tilde M, \\tilde B\\in K[x]^{n\\times n}$, and $\\tilde \\lambda \\in \\bN$.\nSince $\\partial_t$ and~$\\partial_x$ commute, Proposition 7 in~\\cite{chen14a}\nimplies that $\\tilde e \\mid e$ and $x^{\\tilde{\\lambda}} \\tilde{e} \\mid x^\\lambda e$, as polynomials in $K[x]$.\nSo we can just take $\\tilde e = e$ and $\\tilde{\\lambda}  = \\lambda$ by multiplying $\\tilde M, \\tilde B$ by some factors\nof~$x^\\lambda e$. A direct calculation yields $\\partial_t\\cdot f = (\\partial_t\\cdot g)' + h$,\nwhere\n\\[h = \\left(\\partial_t\\cdot\\frac{P}{d}+\\frac{P\\tilde M}{de}\\right)W + \\left(\\partial_t\\cdot\\frac{Q}{x^\\lambda e}+ \\frac{Q\\tilde B}{x^{2\\lambda} e^2}\\right)V.\\]\nThis implies that the squarefree part of the denominator of $h$ divides $xde$. Applying Hermite reduction and polynomial reduction\nto~$h$ yields\n\\[ h = \\tilde g_1' + \\frac{1}{d} P_1W + \\frac{1}{x^\\lambda e} Q_1V,\\]\nwhere $P_1, Q_1\\in K[x]^n$ with $\\deg_x(P_1) < \\deg_x(d)$ and $Q_1\\in N_V$.\nRepeating this discussion, we get the following lemma.\n\\begin{lemma}\\label{LEM:idtf}\nFor any $i\\in \\bN$, the derivative $\\partial_t^i\\cdot f$ has an additive decomposition of the form\n\\[ \\partial_t^i\\cdot f = g_i' + \\frac{1}{d} P_iW + \\frac{1}{x^\\lambda e} Q_iV,\\]\nwhere $g_i\\in A$, $P_i, Q_i\\in K[x]^n$ with $\\deg_x(P_i) < \\deg_x(d)$ and $Q_i\\in N_V$.\n\\end{lemma}\nAs application of the above lemma, we can compute the minimal telescoper for $f$ by finding the first\nlinear dependence among the $(P_i, Q_i)$ over~$K$. We also obtain an upper bound for the order of telescopers.\n\\begin{corollary}\nEvery $f\\in A$ has a telescoper of order at most $n\\deg_x(d) + \\dim_K(N_V)$.\n\\end{corollary}\n\n\n\\begin{example}\nWe continue with Example~\\ref{ex:ct}, by applying the polynomial reduction\nto the Hermite remainders $h_0,h_1,h_2$. The matrix~$M$ computed before\nsatisfies the degree condition of Proposition~\\ref{PROP:finite}, so no\nchange of basis is needed. First we compute polynomials\n$u_i,v_i\\in K[x,y]$ such that for $i=0,1,2$ we have\n\\[\n  h_i = \\frac{u_i}{e} + \\frac{v_i}{d} = \\frac{u_i}{27x^2+54xt+27t^2+4} + \\frac{v_i}{x}.\n\\]\nBy noting that $\\deg_x(u_i)=1$ and $\\deg_x(e)=2$, we see that the\nmap for the polynomial reduction $\\phi(p) = ep' + pM$ can only be\napplied for $p\\in K^n$ so that it turns into $\\phi(p) = pM$.\nThis means that we reduce $xy^2$ using the third row of~$M$ and\n$xy$ using its second row. A straightforward calculation reveals\nthat $h_0$, $h_1$, and $h_2$ all reduce to~$0$. Hence we are left\nwith finding a $K$-linear combination among the $v_i$:\n\\begin{align*}\n v_0 &= \\frac{-6y^2+9yt-4}{27t^2+4},\\\\\n v_1 &= \\frac{6\\bigl(27y^2t-(27t^2-2)y+18t\\bigr)}{(27t^2+4)^2},\\\\\n v_2 &= \\frac{6\\bigl((96-1539t^2)y^2+(1215t^3-306t)y-1026t^2+64\\bigr)}{(27t^2+4)^3}.\n\\end{align*}\nAs expected, we obtain the same telescoper as in Example~\\ref{ex:ct}.\n\\end{example}\n\n\n\\section{The D-finite Case}\n\nWith algebraic functions being settled, it is natural to wonder about a possible\nreduction-based creative telescoping algorithm for D-finite functions. Recall\nthat in this setting we consider an operator $L\\in K(x)[\\partial_x]$ instead of a minimal\npolynomial $m\\in K[x,y]$ and instead of an algebraic field extension\n$K(x)[y]\/\\langle m\\rangle$ we consider the $K(x)[\\partial_x]$-left-module\n$A=K(x)[\\partial_x]\/\\langle L\\rangle$. Then the element $1\\in A$ is a solution of $L$\nbecause $L\\cdot 1=L=0$ in $A$ by construction. If $n=\\deg_{\\partial_x}L$, then\nthe general element of $A$ has the form\n$f=f_0+f_1\\partial_x+\\cdots+f_{n-1}\\partial_x^{n-1}$ for some\n$f_0,\\dots,f_{n-1}\\in K(x)$. Very much as in the algebraic case, there is a natural way\nto associate certain series objects to the elements of~$A$. Based on these\nseries objects, a notion of integrality was proposed last year~\\cite{kauers15b}, and an\nalgorithm for computing integral bases has been given for so-called Fuchsian\noperators~$L$.\n\nIt turns out that the Hermite reduction of Section~\\ref{sec:hermite} also works in this setting, if we say that\na term $(x-a)^\\alpha\\log(x)^\\beta$ in a generalized series solution is integral if and only if $\\alpha\\geq0$.\nNote that then $\\log(x)$ will then be considered integral at zero, despite the singularity of the complex\nfunction at this point. This has the somewhat counterintuitive consequence that $\\log(x)$ is integral at\nevery $a\\in\\bar K\\cup\\{\\infty\\}$ although it does not have a pole anywhere. For algebraic functions,\nthis is not possible by Chevalley's theorem, and this fact enters in an essential way in the proofs of\nSections~\\ref{SECT:CT-1} and~\\ref{sec:polynomial}. The lack of Chevalley's theorem is not an artefact of a (possibly\nwrong) treatment of logarithmic terms. Because of the Fuchs relation \\cite[p.~241]{schlesinger95} there exist operators\n$L\\in K(x)[\\partial_x]$ whose series solutions at any point $a\\in\\bar K\\cup\\{\\infty\\}$ have no logarithmic\nterms, only nonnegative exponents, and which are nevertheless not constant.\n\nFor the time being, the existence of such operators is a severe obstruction to a possible generalization of\nthe termination arguments for reduction-based creative telescoping from algebraic functions to Fuchsian D-finite\nfunctions. We hope to explore this topic further in the future.\n\n\\section*{Acknowledgements}\n\nWe would like to thank Ruyong Feng and Michael F.\\ Singer for helpful discussions.\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}