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(CDNs), cloud computing, IP television, video streaming,. Internet of Things [1]. Most of these bandwidth-demanding services can be provisioned in the network ...
Joint Optimization of Multicast and Unicast Flows in Elastic Optical Networks Krzysztof Walkowiak, Róża Goścień, Michał Woźniak Department of Systems and Computer Networks Wroclaw University of Technology, Poland [email protected] Abstract— Nowadays, multicasting is increasingly popular, due to the ability to provision in efficient way such desirable services like streaming, IP television, software distribution, etc. At the same time, the research in the field of optical networks concentrates on the very promising elastic optical networks (EONs) approach. In this paper, we focus on the joint optimization of multicast and unicast flows in EONs. We propose to model multicast flows in EONs using pre-generated candidate trees. We present two new ILP models and a heuristic method, dedicated to solve the optimization problem with joint multicast and unicast flows. We report results of the numerical experiments carried out to compare proposed ILP models and the heuristic method as well as to evaluate potential benefits of multicasting in EONs. We show that the proposed modeling approach and heuristic method are highly elastic in terms of their applicability to different network scenarios, whilst multicasting can bring significant spectrum savings compared to unicasting. Keywords— elastic optical network, multicast, optimization

I. INTRODUCTION In recent years, the Internet has changed from a simple best effort packet forwarding network towards an advanced system offering many services like, e.g., Content Delivery Networks (CDNs), cloud computing, IP television, video streaming, Internet of Things [1]. Most of these bandwidth-demanding services can be provisioned in the network in a cost-effective way using multicasting defined as one to many transmission. In this paper, we focus on the multicasting in the context of Elastic Optical Network (EON) that is a recent approach proposed for optical networks. Elastic optical networks are considered as a potential successor of the currently deployed Wavelength Switched Optical Networks (WSONs) implementing the Wavelength Division Multiplexing (WDM) technology. On the contrary to WSONs, EONs combine advantages of flexible frequency grids [2] and advance transmission and modulation techniques, allowing for distance adaptive transmission [3], [4]. Hence, the EONs are spectrally efficient and they can support ever increasing, high-rate traffic demands (100 Gb/s and above). The paper focuses on static (offline) optimization of multicast and unicast flows in EONs. The importance of the problem addressed in this work follows directly from growing popularity of many bandwidth-demanding services in the contemporary Internet that can be efficiently provisioned by

Mirosław Klinkowski National Institute of Telecommunications Warsaw, Poland [email protected] multicast transmission. The main contribution of this work is fourfold. First, we propose to model multicast flows in EONs using pre-generated candidate trees. Second, we formulate two new ILP (Integer Linear Programming) models that enable joint optimization of multicast and unicast flows in an EON. Third, we propose an efficient heuristic greedy algorithm (called AFA) as a scalable approach able to outperform the ILP method for larger problem instances, since AFA can effectively make use of larger sets of candidate paths and trees. Finally, we present results of wide experiments run to compare both ILP models and the heuristic as well as to show potential benefits of using multicasting in EONs. To the best of our knowledge, this paper is the first one that addresses the problem of joint multicast and unicast traffic optimization in EONs. The remainder of the paper is organized as follows. In section II, we discuss the multicasting in EONs. Section III, includes formulations of ILP models. In section IV, we propose a new heuristic algorithm for joint optimization of multicast and unicast traffic in EONs. Section V presents results of experiments and in section VI we conclude the work. II. MULTICAST MODELING IN EONS In this section, we introduce a new approach to the multicasting modeling in EONs. Note that we consider static multicast demands, i.e., all requests of multicast connections to be established in EON are known a priori, and they are not subject to blocking. The optimization concerns the amount of network resources (spectrum) required to allocate the demands. A. Approaches to Multicast Modeling The most popular approaches to model multicast flows proposed in the literature are: canonical formulation and flow formulation. In the canonical formulation, a Steiner tree is used to model the multicast transmission by analyzing all cuts of the network graph [5]. The main weakness of this approach is that the number of all possible cuts increases exponentially with the number of nodes. Therefore, the canonical formulation is mostly not applied in modeling multicast flows in computer networks. The second approach – flow formulation – is founded on unicast multi-commodity node-link formulation [6]. In a nutshell, the path from the source (root) node of the multicast session to each multicast receiver (client) node is modeled as a traditional unicast path with node-link flow conservation constraints [6]. Next, all unicast paths connecting

the source node to each receiver form a multicast tree, i.e., each link included in at least one of the unicast paths, is included in the tree. Note that a type of this approach is used in [7] for modeling the multicasting in EONs, however the authors of [7] use a link-path modeling of unicast paths included in the tree instead of the node-link modelling. B. Candidate Tree Modeling in EONs The approach that we propose to multicast modeling in EONs is based on the candidate tree (CT) formulation studied in [8] in the context of p-Cycle based multicast protection. The CT model is an analogy to the link-path formulation of unicast flows [6]. More specifically, for each multicast session (source and set of receivers) there is a set of candidate trees that originate at the source node and include all receiver nodes. For a session to be established, one of the candidate trees is selected. This approach is highly elastic when compared to previous approaches, especially in the context of EONs with multiple modulation formats and flexible spectrum allocation. The reasons are the following: 1) When applying CTs, one can easily tune scalability of the optimization process by selecting the number of candidate trees for each demand according to problem characteristic and other requirements. For instance, if the considered network and number of multicast sessions are quite small, we can use a large number of candidate trees for each session and still keep the problem to be solved in a reasonable time. However, if the considered instance includes a larger network with more multicast session, we can reduce size of the optimization problem by limiting the number of candidate trees. 2) A constraint that can occur in optical multicasting is that multicasting capable optical cross connects (MC-OXCs) have a limited fanout defined as the number of outgoing signals [9]. The CT modeling can be easily adapted to this requirement, i.e., in the tree generation process only trees fulfilling a defined fanout are allowed. In contrast, previous approaches to optical multicast modeling (e.g., [7]) need adding new constraints, what can significantly increase complexity of the model. 3) In optical multicasting there is a concept of sparse splitting networks assuming that only some network nodes are multicast incapable (MI), i.e., there are not supplied with MC-OXCs. Notice that to include the requirement of using the MI nodes, the CT approach again does not need any modification in the ILP model, since the requirement of using MI nodes can be simply included in candidate trees generated as an input of the problem. In contrast, in the previous multicast formulations, additional constraints are required to model sparse splitting networks, what can significantly increase size of the models and the processing time. For further discussion, we introduce the following notions and notations. The considered EON is denoted as a directed graph G = (V,E) where V is a set of nodes and E is set of directed links. We assume that all network nodes are equipped with MC-OXCs that are able to replicate the input data stream to multiple outputs [9]. The multicast demand (session) is described by a source node (root) and set of client nodes (receivers) that participate in the multicast transmission. For each demand d we are given the demand volume (bitrate) hd

expressed in b/s. To model routing of multicast demand, we apply the candidate tree approach. For each demand d we are given a set P(d) of candidate trees. Set P(d) contains t candidate trees. Each of the trees starts in the root (source) of the session and all receivers of the session are included in the candidate tree, i.e., there is a path in the tree from the root to each of the receivers. Constant edp denotes if link e belongs to candidate tree p of demand d. The candidate trees can be found by any dedicated algorithm. C. Modulation Level and Spectrum Allocation The EONs can support multiple modulation formats. In this paper, in the analysis we assume that the following formats are available: BPSK, QPSK, and x-QAM, where x belongs to {8, 16, 32, 64}. These modulation formats provide some trade-off between spectrum efficiency and transmission range, i.e., more spectrally effective modulation formats provide shorter transmission range [4]. For each demand, we can select one of the available modulation formats according to some strategy providing the best performance according to selected metrics. In this work, we assume that the optimization objective is to minimize the spectrum usage defined as the number of frequency slices required in the network to allocate all demands. However, the minimization of the spectrum usage often leads to the selection of a more spectrally efficient modulation format that needs less spectrum, but at the same time provides shorter transmission range. As a consequence, the number of required signal regenerators increases and, accordingly, the cost and power consumption grows [4]. Therefore, to select the modulation format we use – as in [7] – a distance-adaptive transmission (DAT) rule. The main idea behind the DAT approach is to preselect a modulation format for a particular path based only on the transmission distance. In more detail, all available modulation formats are analyzed and the most spectrally efficient modulation format is selected, for which the considered transmission distance does not exceed the modulation transmission range. Accordingly, additional regenerators are not required, while the spectrum usage is kept on a reasonable level. Note that in the case of a unicast demand, the transmission distance is simply the path length. In the case of a multicast demand, the length of the tree is defined as the maximum distance from the root node to the receiver calculated for all receiving nodes in the session. The regenerator placement problem in multicasting is not addressed in this paper and is left for future work. Let ndp denote the number of required slices for demand d if tree p is used. In this paper we assume that the maximum transmission range of a modulation format depends on the demand bitrate as in [10]. Here, we can easily spot the next advantage of the CT modeling. Since all candidate trees are pre-calculated in advance, for each tree we can find the tree length and accordingly apply the DAT rule to find the modulation format and next compute values of ndp. In contrast, in the modeling proposed in [7], the selection of the modulation format and number of required slices must be included in the ILP model. III. ILP MODELS In this section, we present and discuss two ILP formulations of Routing and Spectrum Allocation for Joint

Multicast and Unicast flows (RSA/JMU) problem, namely slice-based (SB) model and channel-based (CB) model. Both models differ in the way how the optical spectrum usage is modeled. The SB model uses the spectrum modeling from [7], the CB model applies the concept of spectrum channels [11]. Real optical networks usually carry various types of traffic. Therefore, to make the considered optimization problem more realistic, we assume that there are two types of demands to be routed in the network: multicast and unicast. Set D contains both types of demands. Note that in [7], only optimization of the multicast traffic is taken into consideration. To model unicast routing, we apply the link-path approach, i.e., for each demand d we are given set P(d) of k candidate paths. RSA/JMU/Slice-based Model sets D demands (multicast and unicast) P(d) candidate structures for demand d. If d is a unicast demand, structure p is a path connecting end nodes of the demand. If d is a multicast demand, structure p is a tree with the root in the source node and including all client nodes E directed network links S slices constants edp = 1, if link e belongs to structure p realizing demand d; 0, otherwise ndp requested number of slices for demand d on structure p variables xdp = 1, if structure p is used to realize demand d; 0, otherwise (binary) yde = 1, if demand d uses link e; 0, otherwise (binary) ud number of slices required for demand d (integer) odi = 1, if the starting slot of demand d is smaller than that of demand i; 0, otherwise (binary) cdi = 1, if demands d and i use common link(s); 0, otherwise (binary) wd starting slice used for demand d (integer) zd ending slice used for demand d (integer) z maximum slice index used in the network (integer) objective minimize  = z (1) subject to pP(d) xdp = 1, dD (2) pP(d) edpxdp  yde, dD, eE (3) pP(d) ndpxdp  ud dD (4) cdi ≥ yde + yie – 1, d, iD, d  i , eE (5) odi + oie = 1, d, iD, d  i (6) zi – wd + 1  |S| (1 + odi – cdi), d, iD, d  i (7) zd – wi + 1  |S| (2 – odi – cdi), d, iD, d  i (8) zd – wd + 1 ≥ ud, dD (9) z ≥ zd, dD (10) wd, zd, z (0, |S|], dD (11) The objective function (1) has the goal to minimize the maximum slice index used in the network to realize all multicast and unicast demands. Equation (2) assures that

exactly one candidate structure (path or tree) is selected to realize each demand d. Constraint (3) is used to check if demand d uses link e (variable yde). Equation (4) defines the number of sliced required for demand d. Constraints (5)-(11) are used to control the spectrum usage and are based on the formulation proposed in [7]. In more detail, (5) checks if two different demands d and i use the same link e. Constraints (6)(8) are in the model to avoid spectrum overlapping of two different demands d and i. Equation (9) assures that the requested number of slices for each demand is satisfied. Constraint (10) defines the maximum slice index used in the network. Finally, constraint (11) limits the variable ranges. The second ILP model uses the concept of channels [11]. Channel c is defined as a contiguous subset of slices in ordered set S, i.e., cS. Let C(d,p) denotes a set of channels accessible for demand d on candidate structure p. Channel cC(d,p) includes enough adjacent slices to support traffic volume of demand d using a modulation selected with the DAT rule. RSA/JMU/Channel-based Model sets (additional) C(d,p) candidate channels for demand d allocated on structure p constants (additional) dpcs = 1, if channel c associated with demand d on candidate structure p uses slice s; 0, otherwise variables (additional) xdpc = 1, if channel c on candidate structure p is used to realize demand d; 0, otherwise (binary) zes = 1, if slice s is occupied on link e; 0, otherwise (binary) zs = 1, if slice s is occupied on any network link; 0, otherwise (binary) objective minimize  = z (12) subject to pP(d)cC(d,p) xdpc = 1, dD (13) dDpP(d)cC(d,p) dpcsedpxdpc  zes eE sS (14) eE zes  |E|zs, sS (15) sS zs  z, sS (16) The objective (12) is – as in (1) – to minimize the width of spectrum required in the network. Equation (13) assures that for each demand d exactly one candidate structure and one candidate channel are selected. To avoid slice overlapping and to assure that a slice on a particular link can be allocated to at most one lightpath, we use constraint (14). The next constraint (15) defines that slice s is used in the network (ys = 1) only if there is at least one link on which the slice s is allocated. Finally, the last equation (16) defines the value of maximum slice index used in the network. IV. HEURISTIC ALGORITHM The heuristic method we propose is called Adaptive Frequency Assignment – Joint Unicast and Multicast (in this paper called briefly “AFA”). The algorithm is based on the standard AFA approach, proposed initially for the unicast

traffic in [12]. Due to space limit, we will briefly describe the algorithm without providing its pseudo-code. The key idea behind the AFA method is adaptive ordering of demands which are subject to allocation. Therefore, for each traffic demand d, a basic metric nd is calculated. For a unicast demand, metric nd is a minimum number of slices required to realize this demand on any of its candidate paths [12]. For a multicast demand we introduce the following metrics: (1) bitrate, (2) bitrate multiplied by the number of receivers, (3) bitrate multiplied by the number of slices required to realize transmission with first candidate tree, (4) bitrate multiplied by the number of slices required to realize transmission with first candidate tree multiplied by the number of receivers. Moreover, a special metric is calculated for each network link ce = dDpP(d) edpnd and each candidate structure lp = ep ce. Finally, a metric for each demand is calculated as ld = |Pd|1 pP(d) lp. Unicast and multicast demands are allocated in separate loops of the algorithm, wherein the multicast demands are served first. The allocation process is as follows. Initially, demands with the same value of the basic metric equal to m are grouped together in to a subset Bm={d: dB, nd = m}. Next, sets Bm are allocated one by one in decreasing value of m. For a particular set Bm its demands are allocated iteratively. In each iteration, we use a function MinD(d:nd=m) to find a demand, that currently provides allocation with the smallest value of the objective. If more than one demand satisfies this condition, the demand with lower value of metric ld is selected. Further, for this demand we use function MinS(d) to find a candidate structure that currently provides allocation with the smallest value of the objective. If more than one structure satisfies this condition, the structure with the smallest value of lp metric is used. The allocation process for multicast demands is repeated for each of the considered metrics, while in the case of unicast demands only the bitrate metric is applied for ordering. The final solution of AFA is selected as the best one obtained for all examined multicast ordering metrics. V. RESULTS In this section, we present results of the numerical experiments. The first goal of experiments is to compare both ILP models presented in this paper. Next, we evaluate performance of the heuristic algorithms in comparison to ILP modeling. Finally, we report some interesting results obtained by the AFA algorithm for larger networks in terms of various parameters related to multicasting in EONs. A. Main Assumptions Due to low scalability of ILP models, for the ILP experiments we use three smaller networks: German backbone network DT14 (14 nodes, 46 directed links), US backbone network NSF15 (15 nodes, 46 directed links), and panEuropean backbone network Euro16 (16 nodes, 48 directed links). For further experiments with heuristics, we apply a US backbone network US26 (26 nodes, 84 directed links) and a pan-European network Euro28 (28 nodes, 82 directed links).

The traffic matrices include both unicast and multicast demands and are generated at random. The bitrate of an unicast demand is selected from the range 10-400 Gb/s, while the bitrate of a multicast session is selected from the range 10-200 Gb/s. If not otherwise stated, we assume that each traffic matrix includes 50% of unicast and 50% of multicast traffic. However, it must be underlined, that volume of multicast traffic is calculated as the traffic received by all participants of the session (receivers). More specifically, if the multicast session bitrate is 200 Gb/s and the session includes 5 receivers, it makes 1 Tb/s of received traffic. For smaller networks, each multicast session includes 5 receivers, while for larger networks each session has from 5 to 15 receivers. In the case of smaller networks, for each of three values of overall traffic (4 Tb/s, 5 Tb/s and 6 Tb/s), we generate 25 sets of traffic patterns. To generate candidate paths for unicast demands we use the k-shortest path algorithm. In the case of candidate trees we apply a more complicated approach. In a nutshell, for a particular multicast session defined by a root node and set of receivers, we take into consideration k-shortest paths connecting the root node and each receiver. Using these paths, we generate various structures by taking one of k paths for each receiver. Each obtained structure provides connectivity required for multicasting, i.e., it connects the root to receiver. Having the structures, we eliminate graphs that are not trees (i.e., graphs that include loops). Finally, we sort all structures to obtain the best candidates. As the main sorting criterion, we apply the tree length (length of the longest path from the root to receivers), what assures that according to the DAT rule, the spectrum usage will be as low as possible. We assume that the EON uses bandwidth-variable transponders (BV-Ts) implementing the PDM-OFDM technology with multiple modulation formats selected adaptively between BPSK, QPSK, and x-QAM, where x belongs to {8, 16, 32, 64}. The spectral efficiency is equal to 1,2,...,6 [b/s/Hz], respectively, for these modulation formats. The network operates within a flexible ITU-T grid of 6.25 GHz granularity [2]. Three types of BV-Ts are applied, each with a different capacity limit, i.e., 40 Gb/s, 100 Gb/s, and 400 Gb/s. The BV-Ts allow for bitrate adaptability with 10 Gb/s granularity. We make use of the transmission model from [10], which estimates the transmission distance in a function of the selected modulation level and transported bitrate. We introduce a 12.5 GHz guard band between neighboring connections. Moreover, we assume that the model does not contain additional physical impairments following from using optical splitting to provide multicasting, since according to our best knowledge there is still no reliable research on this issue in the context of EONs. However, it should be underlined that the presented ILP models and heuristic are generic and can be easily adapted to new physical multicasting models that will be developed in the future. B. Comparison of ILP Models Both ILP models are implemented in CPLEX 11.1 solver [13]. We limit the execution time of CPLEX to 1 hour, so not all presented results have the guarantee of optimality. Due to low scalability of the ILPs (CPLEX) in the considered problem, we decided to run the experiments with the following

settings: number of candidate paths for unicast demands is k=2 and number of candidate trees for multicast sessions is t=10. As the main performance metric in the comparison between SB and CB models we use the execution time of the CPLEX solver. During the preliminary experiments, we have noticed that performance of the CB model strongly depends on the amount of slices available in the network denoted by parameter |S|. If we set a large value of |S|, then the CB model in most cases was not able to find an optimal solution within 1 hour limit, while the SB model was generally not vulnerable to value of |S|. Therefore, we decided to first run AFA to obtain a reasonable upper bound on the objective function and next set |S| to this value for further experiments. In Table I, we show the comparison of both ILP models. For each model, we report the average value of the objective function as well as we compare execution times in three categories: SB