WAVELENGTH Division Multiplexing (WDM) significantly

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1 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 25, NO. 6, AUGUST 27 1 Groomin Multicast Traffic in Unidirectional SONET/WDM Rins Anuj Rawat, Richard La, Steven Marcus, and Mark Shayman Abstract In this paper we study the problem of efficient roomin of iven non-uniform multicast traffic demands on a unidirectional SONET/WDM rin. The oal is to try to minimize the network cost as iven by (i) the number of wavelenths required per fiber and (ii) the number of electronic Add-Drop Multiplexers (ADMs) required on the rin. The problem is NP hard for both the cost functions. We observe that the problem with cost function (i) can be reduced to a correspondin traffic roomin problem for unicast traffic which can then be modelled as a standard circular-arc raph colorin problem. For cost function (ii), we construct a raph based heuristic and compare it aainst the multicast extension of the best known unicast traffic roomin heuristic [1]. We observe that our heuristic requires fewer ADMs than required by the multicast extension of the unicast heuristic iven in [1]. We also develop a lower bound and compare it aainst some upper bounds to study the maximum penalty of not employin intellient wavelenth assinment and/or traffic roomin under the unidirectional SONET/WDM rin scenario. Index Terms Graph theory, SONET rin, traffic roomin, wavelenth division multiplexin (WDM). I. INTRODUCTION WAVELENGTH Division Multiplexin (WDM) sinificantly increases the available network bandwidth capacity by deliverin data over multiple wavelenths (channels) simultaneously. With each channel operatin at a hih rate (currently 1 Gb/s) and multiple channels deployed per fiber (currently 32 wavelenths per fiber), very hih transmission capacities (currently 3.2 Tb/s) can be achieved. An important issue with such a hih capacity network is that it places enormous burden on the electronic switches. Hence, it is hardly surprisin that the dominant cost in WDM based networks is the cost of the electronic switchin equipment required. Fortunately it is not necessary to electronically process all the incomin traffic at each node since most of the incomin traffic is neither sourced at that node nor destined to it. So to reduce the cost of electronic components at each node, we can selectively drop the wavelenths carryin traffic that requires electronic processin at that node and allow the remainin wavelenths to optically bypass the node. Typically in WDM based optical networks, the bandwidth available per wavelenth is much larer than the bandwidth required per session, and with the advancement of optical technoloy, it seems likely that this mismatch will continue to row in the near future. Hence for efficient bandwidth usae, Manuscript received March 7, 26. The authors are with the Department of Electrical and Computer Enineerin, University of Maryland, Collee Park, MD ( {anuj, hyonla, marcus, shayman}@lue.umd.edu). Diital Object Identifier 1.119/JSAC-OCN it is prudent to combine several low rate traffic sessions onto a sinle wavelenth. The problem of effectively packin lower rate traffic streams onto the available wavelenths in order to achieve some desired oal is called traffic roomin. If the traffic demands are known in advance, then the problem is called static, otherwise the problem is called dynamic. In static traffic roomin, usually the aim is to minimize the overall network cost. Here the network cost includes the cost of electronics (this is the dominant cost) as well as the cost of optics (wavelenths per fiber). In dynamic traffic roomin, the aim is to room the incomin traffic demands such that the blockin probability is minimum. In this work we are interested in the static traffic roomin problem. The inherent reliability and hih bandwidth capacity of a WDM based Synchronous Optical Network (SONET/WDM) rin has made it the architecture of choice in the current network infrastructure. Typically, in a SONET/WDM rin each wavelenth operates at a line rate of OC-N 1 and can carry several low rate OC-M (M N) traffic channels usin Time Division Multiplexin (TDM). The timeslots on a wavelenth are referred to as the subwavelenth channels. Electronic Add-Drop Multiplexers (ADMs) are required to add (drop) the subwavelenth traffic at the source (destination) node. On receivin a wavelenth channel, the ADM, correspondin to that particular wavelenth, can add/drop timeslots on the wavelenth channel without disruptin the onward transmission of other timeslots on the wavelenth. So if a node (say n) does not act as a source or a destination for any traffic on some wavelenth (say λ), i.e., if no add/drop of any timeslot on λ is required at n, then there is no need for an ADM correspondin to wavelenth λ at node n. Since the cost of the ADMs (electronics) form the bulk of the network cost [2], we can see that intellient roomin of low-rate traffic onto wavelenths can result in ADM savins, which results in a lower network cost. Groomin static unicast subwavelenth traffic to minimize either the number of ADMs or the number of wavelenths required per fiber in WDM rin networks is a well studied problem [1][2][3]. Different traffic scenarios such as uniform all-to-all traffic [3][4], distance dependent traffic [2] and nonuniform traffic [1][5] have been studied. Work has also been done on other cost functions such as the overall network cost [6], which includes the cost of transceivers, wavelenths and the number of required hops. Recently there has been a lot of work on roomin both static [7] as well as dynamic [8][9][1] /7$2. c 27 IEEE 1 OC-N (Optical Carrier-N) is a SONET standard desinatin a fiber optic circuit operatin at N times meabits per second, i.e., N times the operatin rate of STS-1 (Synchronous Transport Sinal-1).

2 2 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 25, NO. 6, AUGUST 27 traffic in mesh networks. References [11] and [12] provide an excellent review on the problem of roomin unicast traffic in WDM networks. While most of the earlier studies of the traffic roomin problem have dealt exclusively with the unicast traffic, it is expected that in the future a sizable portion of the traffic will be multicast in nature. This is mainly because of the increasin demand of multicast services such as multimedia conferencin, video distribution, collaborative processin, etc. Groomin multicast traffic is still an area of active research and althouh a lot of literature is available, not many results are known. Most of the work in the multicast case has focused on heuristics for roomin multicast traffic in WDM mesh networks under non-uniform static [13] as well as dynamic traffic [14][15][16][17][18][19] scenarios. Althouh multicast traffic roomin in mesh WDM networks is a eneral case of the same problem in WDM rins, the ideas that are applied for mesh networks in [13][14][15][16][17][18][19] are not very attractive for unidirectional rins. The difference between the mesh and unidirectional rin cases is that, in mesh networks there are many possible routins for each traffic demand whereas in unidirectional rins the routin is fixed and we have control over wavelenth assinment only. All of the heuristics for roomin multicast traffic in mesh networks take advantae of the multiple routins possible and the wavelenth assinment is usually trivial (first fit). This is clearly not desired for roomin in unidirectional rins, since for unidirectional rins the routin is already fixed and the only way to effectively room traffic is by usin intellient wavelenth assinment. Althouh most of the work on multicast traffic roomin looks at mesh WDM networks, there has been some work in the case of WDM rins also. More specifically, in [2] the authors look at the problem of roomin iven multicast traffic demands in a bidirectional WDM rin. They present a heuristic alorithm inspired by the alorithm to room unicast traffic demands on WDM rins iven in [1]. We shall compare this to our work in more detail once we state our exact problem. In this paper we look at the problem of static roomin of non-uniform multicast traffic on a unidirectional SONET/WDM rin. In eneral, the SONET rin nodes may or may not have SONET diital-cross connects (DXCs) and wavelenth converters. SONET DXCs and wavelenth converters are expensive components so in this work we assume that the network nodes do not have wavelenth converters and SONET DXCs. Since the ADMs do not have wavelenth conversion or timeslot chanin functionality, the absence of wavelenth converters and SONET DXCs implies wavelenthcontinuity and timeslot-continuity constraints in the network. This sort of network setup for roomin unicast traffic has been cateorized as a sinlehop rin [21]. In another type of network setup some nodes of the network use SONET DXC to consolidate or sereate subchannels (timeslots on a wavelenth). This setup is referred to as a multihop rin [21]. Hence in this work we are concerned with the sinlehop rin case for roomin multicast traffic. We consider two different cost functions (i) the number of wavelenths required per fiber and (ii) the total number of ADMs. We observe that for cost function (i), the problem can be modeled as a circular-arc raph colorin problem. Thus, the standard colorin techniques apply. We then suest a raph based heuristic for cost function (ii). We extend the traffic roomin heuristic for non-uniform unicast traffic iven in [1] to the multicast case and compare this multicast extension to our heuristic. We also develop a lower bound on the number of ADMs and compare it aainst some of the upper bounds to et interestin insihts into the problem. The problem that we study here is quite different from the problem studied in [2]. The main difference, other than the fact that we study unidirectional rins while [2] looks at bidirectional rins, is that the cost function used is different. We consider the number of ADMs and the number of wavelenths required per fiber as our cost, whereas in [2], the total number of ports of e-dac nodes in the network is considered as the cost. In [2], the authors define two different types of nodes, o-dac and e-dac nodes. When all the traffic on all the incomin wavelenths needs to be forwarded, o-dac nodes are used since the splittin can be done in the optical domain. If this is not the case then e-dac nodes are used. Note that the cost functions are not the same since we require ADMs at all the nodes where some traffic needs to be dropped whereas in [2], even the nodes where there is some drop traffic can be treated as o-dac nodes. Another important difference is that in [2], the authors consider a multihop rin whereas we look at a sinlehop rin. The rest of the paper is oranized as follows. In Section II, we state our assumptions on the network, the traffic and the node architecture. Here we also state the precise problem. In Section III we model the problem (for both the cost functions) usin raphs. We present our heuristics in Section IV. In Section V, we develop and study some lower and upper bounds. In Section VI, we present the complexity analysis for the roomin schemes introduced in this work. Section VII presents the simulation results. Finally Section VIII concludes the paper. For quick reference, Table I lists some of the improtant symbols and notations used in the paper alon with brief explainations. A. Physical Network II. PROBLEM STATEMENT The physical network is assumed to be a clockwise unidirectional WDM rin with N nodes numbered, 1,..., N 1 distributed on the rin in the clockwise direction as shown in Fiure 1(a). We assume that there is a sinle fiber between adjacent nodes, which can support W wavelenths iven by λ, λ 1,..., λ W 1 and the capacity of each wavelenth is assumed to be C units. Also, as noted in Section I, we assume that the network nodes do not have wavelenth converters and SONET DXCs. This implies timeslot-continuity and wavelenth-continuity constraints in the network. B. Assumptions On Traffic We assume that there are M iven multicast traffic requests denoted by R, R 1,..., R M 1. Every multicast request specifies a source node and a set of destination nodes. We assume that each multicast request is for r units of traffic. Also, the

3 SUPPLEMENT ON OPTICAL COMMUNICATIONS AND NETWORKING 3 Symbol N M N W min G = (V, E) χ G[C i ] TABLE I LIST OF IMPORTANT SYMBOLS Stands for number of nodes in the SONET rin number of multicast sessions roomin ratio number of nodes which act as a source or a destination for at least one multicast request minimum number of wavelenths required per fiber contention raph with vertex set V representin the multicast requests chromatic number of raph G subraph induced by vertex set C i V (representin the multicast requests roomed on wavelenth λ i ) on the contention raph G χ(g[c i ]) chromatic number of raph G[C i ] S v set consistin of source and destinations for the multicast traffic request correspondin to vertex v V σ = {C σ, Cσ 1,... } partition of vertex set V into nonoverlappin clusters such that i Cσ i = V, Ci σ C σ j = and χ(g[ci σ]) Σ set of all valid partitionin subraph induced by vertex set V i V (representin the multicast requests which G i G[V i ] contain network node i as source or some destination) on the contention raph G χ i k i α i β i γ i Ĝ i G[ ˆV i ] w i z i z min z max z av F chromatic number of raph G i number of multicast sessions which contain network node i as source or some destination number of multicast sessions which contain network node i as an intermediate destination number of multicast sessions which contain network node i as the source number of multicast sessions which contain network node i as the final destination subraph induced by vertex set ˆVi V (representin the multicast requests which contain network node i as source or final destination) on the contention raph G maximum width of interval raph Ĝi size of a i th multicast session R i minimum possible size of multicast sessions maximum possible size of multicast sessions averae size of multicast sessions c.d.f. accordin to which sizes of multicast sessions are distributed µ F mean of c.d.f. F U wc worst case upper bound on the number of ADMs U Alo A upper bound on the number of ADMs required by Alorithm A U Alo B upper bound on the number of ADMs required by Alorithm B L lower bound on the number of ADMs L 2, L 3 other lower bounds on the number of ADMs wavelenth capacity C is assumed to be an interal multiple of the required traffic rate r, i.e., C = r. We refer to, the number of subwavelenth multicast demands that can be roomed on a sinle wavelenth channel, as the roomin ratio. Assumin an interal roomin ratio is justified in case of SONET rins, since in SONET networks the capacity of a sinle wavelenth (OC-192, OC-48, etc.) is usually an exact multiple of the bandwidth requirement for an individual traffic request (OC-3, OC-12, etc.). Note that for SONET rins, we can assume without loss of enerality that r is equal to the capacity of individual subwavelenth channels (timeslots). Hence, the roomin ratio is equal to the number of subwavelenth channels available on each wavelenth. With the above traffic model we can consider multicast requests of different bandwidth requirements also. The important requirement is that each request should be splittable into individual multicast requests of ranularity r. C. Node Architecture Most of the current work on multicast traffic in optical networks uses multicast capable nodes called Splitter-and- Delivery nodes and multicast incapable nodes with drop-andcontinue capability. In this work, since we are lookin at rin networks, we do not require the nodes to split the incomin liht over multiple outoin links to form liht-trees. This is because here the liht-trees are just arcs on the rin as shown in Fiure 1(a), and the nodes with drop-and-continue capability suffice. The architecture for the drop-and-continue capable nodes used in the rin network is similar to the Tapand-Continue node architecture first iven in [22]. In fact the Tap-and-Continue nodes used in this paper are much simpler than the nodes in [22], since here we only have a sinle incomin and outoin fiber per node and therefore no optical switchin is required. As shown in Fiure 1(a), if a lihtpath is set-up between nodes i and j on wavelenth λ l, and traffic from i to an

4 4 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 25, NO. 6, AUGUST 27 N tap 2 1 switch tap bank switch bank λ fiber λ W 1 lihtpath terminated j λ l k i lihtpath tapped lihtpath initated demultiplexer multiplexer ADM bank low rate dropped traffic low rate traffic to be added (a) Unidirectional SONET rin with tap-and-continue nodes. (b) Tap-and-continue node architecture. Fi. 1. Network and node model. intermediate node k is also roomed on λ l or i has to send the same traffic to k (this is the case when i is the source and j and k are the destinations of a multicast traffic request), then instead of terminatin the lihtpath at k, we can drop a small amount of liht of wavelenth λ l at k to extract the required data packets and let the rest of the liht flow throuh, i.e., we can tap the lihtpath at any intermediate node. It should be clear that if we want to add (room) some subwavelenth traffic on wavelenth λ l at node k, then we have to tear down the lihtpath at k, carry out the roomin and then set-up a new lihtpath on wavelenth λ l at node k. Note that in case we are sendin different traffic to nodes j and k from node i on a sinle lihtpath (by tappin the lihtpath at intermediate node k), then usin the above scheme, after passin node k there would be some unnecessary traffic on the lihtpath (traffic sent from i to k). Clearly there is no such bandwidth wastae when we are sendin the same traffic to both the nodes j and k. The Tap-and-Continue node architecture that we consider in this paper is shown in Fiure 1(b). First the incomin liht is split into individual wavelenth channels usin a demultiplexer. Then each wavelenth channel passes throuh a tap bank. Here we have an option of tappin a small amount of power from the wavelenth and sendin it to the ADM bank to separate it into its constituent lower rate components. The switch bank consists of 2 1 switches for each wavelenth. If no new traffic is added on a wavelenth, it is allowed to pass throuh the switch but if a new lihtpath is bein initiated at some wavelenth then the sinal comin from the ADM banks consistin of the roomed traffic is switched forward. Finally the wavelenths are combined usin a multiplexer and sent over the outoin fiber. Note that at any node, in the ADM bank, we require the ADMs only for wavelenths which are bein processed at that particular node, i.e., we require ADMs for all the wavelenths which correspond to the lihtpaths that are bein dropped or terminated or initiated at that node. D. Objective The objective is to minimize the network cost. As noted in Section I, the cost of the network is equivalent to the cost of the network components and the dominatin cost amon all the components is the cost of ADMs. We also noted that another cost function that is usually considered is the number of wavelenths required per fiber. In this work we study the problem of traffic roomin under the followin two objectives. (i) Minimize the number of wavelenths required per fiber. (ii) Minimize the total number of ADMs required in the network. If we count the number of loical hops required by (possibly multihop) paths between all the source-destination pairs, then this ives us an estimate of the size of the loical network. Since O-E-O conversions introduce delay 2, the size of the network provides us with a measure of the overall delay in the system. Usually we also want to reduce the delay and therefore the size of the network. An important advantae of usin Tap-and-Continue nodes instead of reular nodes is that we can achieve reduction in the network size.to show this we consider the followin example. Let us assume that we have a multicast request R havin network node as its source and network nodes 1 and 2 as its destinations. If we use reular nodes in the rin then we can confiure the lihtpaths in either of the two ways listed below. (i) Set up a liht path between nodes and 1 and another lihtpath between nodes 1 and 2. To save ADMs and conserve the wavelenths used we can use the same wavelenth λ for both the lihtpaths. Fiure 2(a) depicts this confiuration. Note that this requires a total of 3 ADMs and 1 wavelenth. The network size achieved by this confiuration is equal to 3 loical hops (1 for s-d pair 1 and 2 for s-d pair 2). (ii) Set up a liht path between nodes and 2 on wavelenth λ and set up another lihtpath on a different 2 O-E-O conversions introduce the dominant delay in the network.

5 SUPPLEMENT ON OPTICAL COMMUNICATIONS AND NETWORKING 5 lihtpaths λ λ nodes 1 2 lihtpaths λ λ 1 nodes 1 2 lihtpaths λ nodes 1 2 ADMs required for λ λ λ ADMs required for λ λ 1 λ 1 λ ADMs required for λ λ λ (a) Reular nodes (confiuration I): requires 3 ADMs, 1 wavelenth, 3 loical hops. (b) Reular nodes (confiuration II): requires 4 ADMs, 2 wavelenths, 2 loical hops. (c) TaC nodes: requires 3 ADMs, 1 wavelenth, 2 loical hops. Fi. 2. Advantae Of usin TaC nodes. wavelenth λ 1 between nodes and 1. This confiuration is shown in Fiure 2(b). It is clear that since both the lihtpaths are carryin the same traffic, we are wastin bandwidth in this scenario. This confiuration requires a total of 4 ADMs and 2 wavelenths. But now the network size is equal to 2 loical hops (1 hop each for both the s-d pairs). In case we use Tap-and-Continue nodes in the rin, we can simply setup a sinle lihtpath on wavelenth λ between nodes and 2, and tap this lihtpath at node 1. Fiure 2(c) depicts this confiuration. Here we need 3 ADMs, 1 wavelenth and the network size is 2 loical hops (1 hop each for both the s-d pairs 3 ). Thus we can simultaneously achieve reduction in the number of ADMs, number of wavelenths and the network size. Therefore it makes sense to employ Tap-and- Continue nodes rather than reular nodes on the rin. A. Minimizin Wavelenths III. MODELING First we look at the case where the objective is to minimize the number of wavelenths required per fiber irrespective of the number of ADMs used in the network. Since we assume the network to be a clockwise unidirectional rin, each traffic request can be treated as an arc on the rin startin from the source and oin clockwise throuh the intermediate destinations (drop points) up to the final destination (termination point). Now the wavelenth and timeslot continuity constraint implies that each arc (traffic request) should be assined one subwavelenth channel. So if any two multicast requests share some fiber, i.e., the correspondin arcs overlap, then they cannot be roomed on the same subwavelenth channel (althouh they can still share the same wavelenth channel). We use this observation to model the problem of minimizin the number of wavelenths per fiber as a raph colorin problem. Consider a raph G = (V, E) where V = {v, v 1,..., v M 1 } is the set of vertices 4 with 3 We only require 1 loical hop for s-d pair 2 because the traffic reachin from node to node 2 remains in optical domain at the intermediate node 1, i.e., it does not undero O-E-O conversion anywhere on the path from the source to the destination. 4 Note that in this work, we refer to the nodes on the SONET rin as nodes and the nodes of any raph used in the problem modellin (such as G, where each node represents a traffic request) as vertices. each vertex v i representin a multicast request R i and there is an ede v i v j E if and only if the multicast requests R i and R j share some fiber, i.e., the arcs correspondin to requests R i and R j overlap. The raph G is refered to as the contention raph because the adjacent vertices in G represent the traffic requests which cannot be roomed on the same subwavelenth channel. Now the problem of assinin subwavelenth channels to the multicast requests such that we need the minimum number of wavelenths per fiber is equivalent to colorin the vertices of the contention raph G with the minimum number of colors such that no two adjacent vertices share the same color. This is the standard minimum raph colorin problem. Note that here each color sinifies a subwavelenth channel and not a complete wavelenth. We denote the minimum number of colors required for colorin contention raph G, also known as the chromatic number of the raph, by χ. Since the minimum number of subwavelenth channels required to room the iven traffic requests is equal to χ and since each wavelenth contains subwavelenth channels, the minimum number of wavelenths required per fiber is iven by χ W min = (1) An interestin observation is that minimizin the number of wavelenths required per fiber is independent of the fact that we are lookin at multicast traffic. This is because we are modellin the traffic requests as arcs on a circle and this model only preserves the information about the source and the final sink of the traffic requests. So if we consider the iven traffic requests to be unicast with source and sink nodes the same as the source and final sink nodes of the multicast requests R, R 1,..., R M 1, then we obtain the same contention raph G and therefore the same solution for minimizin the number of wavelenths required per fiber. Another observation is that the raph G belons to the family of circular arc raphs [23]. Since minimum colorin of circular arc raphs is NP complete [24] and any instance of minimum arc raph colorin can be reduced to the traffic roomin problem under study, roomin multicast (or unicast) traffic on a unidirectional rin to minimize the number of wavelenths required per fiber is NP complete.

6 6 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 25, NO. 6, AUGUST 27 B. Minimizin ADMs Now we consider the case when the objective is to minimize the ADMs required in the network under the timeslot and wavelenth continuity constraint. For modelin this problem we aain represent multicast requests as arcs on the rin and construct the contention raph G = (V, E) as described in Section III-A. Also to each vertex v i V (correspondin to multicast request R i ), we assin a set S vi consistin of the source and the destinations for request R i. Consider the vertex set C i V representin all the multicast traffic requests roomed on wavelenth λ i. Note that the contention raph correspondin to the traffic requests represented by C i is exactly equal to G[C i ], the subraph induced by vertex set C i on the contention raph G. Now as described in Section III-A, the minimum number of subwavelenth channels required to room the traffic requests represented by C i is iven by χ(g[c i ]), the chromatic number of the contention raph correspondin to the particular set of traffic requests. So the traffic requests represented by C i can be roomed on a sinle wavelenth only if χ(g[c i ]), i.e., the subraph G[C i ] induced on the contention raph G by the vertex set C i is -colorable. Also in this case the number of ADMs correspondin to wavelenth λ i required in the network is equal to v C i S v. So iven a set of multicast traffic requests modeled by the contention raph G = (V, E), any valid traffic roomin can be modeled as a partitionin σ = {C σ, C1 σ,... } of the vertex set V into non-overlappin clusters C σ, C1 σ,... such that i Cσ i = V, Ci σ C σ j = for all i j and G[Ci σ] (subraph induced by vertex set Ci σ on contention raph G) is -colorable for all i. Also, the cost (number of ADMs required in the network) correspondin to partitionin σ is iven by S v (2) i v C σ i Now let Σ denote the set of all such partitionins. Then our problem reduces to findin the partitionin σ Σ which minimizes the required number of ADMs as iven in (2). Note that since the problem of roomin unicast traffic on unidirectional rins in order to minimize the number of ADMs required is NP complete [2], and the multicast traffic roomin is at least as hard as the unicast case, roomin multicast traffic on unidirectional rins to minimize the ADMs is NP hard. IV. HEURISTICS A. Minimizin Wavelenths Per Fiber As described in Section III-A, minimizin the number of wavelenths required per fiber can be modeled as a circular arc raph colorin problem. The contention raph to be colored is obtained as described in Section III-A and the minimum number of wavelenths required is iven by (1). Althouh circular arc colorin is NP complete [24], there are several approximation alorithms [25][26] available in the literature. Kumar et. al. [25] ive a randomized alorithm with approximation ratio (1 + 1/e + o(1)) for instances of the problem needin at least ω(ln(n)) colors, where n is the number of arcs to be colored. In [26], Karapetian et. al. present a 3/2 approximation alorithm for circular arc colorin. Either of these two alorithms can be used to color the contention raph in our problem. Since these alorithms suffice for minimizin the number of wavelenths required per fiber and they have already been well studied in the literature, we will not discuss this cost function any further in this paper. Now we o on to the more interestin problem of minimizin the total number of ADMs required in the network. B. Minimizin ADMs We consider a raph based heuristic approach to minimize the number of ADMs required in the SONET rin. The basic idea of the heuristic is to start off by assinin different wavelenths to each of the multicast sessions. Now at every step of the alorithm we update the wavelenth assinment by selectin two wavelenths and assinin a sinle wavelenth to all the multicast requests previously assined to these two wavelenths. Obviously we can combine a wavelenth pair in this manner only if all the correspondin multicast sessions can indeed be roomed on a sinle wavelenth. The order in which the wavelenth pairs are considered for combination is based on the fact that if the multicast sessions assined to two wavelenths share several source/destination nodes, then we can save a lot of ADMs by usin a sinle wavelenth for all these sessions. In more detail, we first construct the raph G = (V, E) and determine the set S v correspondin to each node v V, as discussed in Section III-B. Now let H(n) = (Λ(n), L(n)) be the weihted raph representin the wavelenth assinment after n steps of the heuristic. Here the vertex set Λ(n) represents the wavelenths and correspondin to each wavelenth λ i Λ(n) we have a set of multicast requests C i V to which this wavelenth is assined. With sliht abuse of notation let S λi = v C i S v denote the set of nodes which act as source or destination for any multicast session bein roomed on wavelenth λ i. Now there is ede λ i λ j L(n) if the multicast requests correspondin to the two wavelenths have some common source/destination nodes and the weiht of the ede is iven by c(λ i λ j ) = S λi Sλj. Note that we can combine any two wavelenths λ i and λ j if the subraph induced by the node set C i Cj on the contention raph G is -colorable. If this is so then the two wavelenths are said to be reducible. Also note that if we combine wavelenths λ i and λ j into a new wavelenth λ k, then the number of ADMs required by wavelenth λ k is S λk = S λi Sλj = S λi + S λj c(λ i λ j ) (3) But now after combinin wavelenths λ i and λ j, we no loner need any ADMs on these two wavelenths, therefore we end up savin S λi + S λj S λk = c(λ i λ j ) ADMs. So at the (n + 1)th step we determine the reducible wavelenth pair λ α, λ β Λ(n) such that for any reducible wavelenth pair λ i, λ j Λ(n), c(λ α λ β ) c(λ i λ j ). If there is more than one such wavelenth pair, let the set of all such wavelenth pairs be ˆΛ. Now we pick the wavelenth pair λ α, λ β ˆΛ such that for any wavelenth pair λ i, λ j ˆΛ, S λα Sλβ S λi Sλj. This is motivated by the fact that if S λi is lare for wavelenth λ i then there is a hih chance of havin a

7 SUPPLEMENT ON OPTICAL COMMUNICATIONS AND NETWORKING 7 larer cost ede incident on this vertex at some later iteration in the alorithm, so we may not want to use wavelenth λ i in this step for a smaller ADMs savin. If there are still some ties left, then we select any wavelenth pair from the possible choices. Now we update the raph H(n) to raph H(n + 1) by replacin vertices λ α, λ β with a sinle vertex λ αβ and recomputin all the edes and weihts. By this we mean that we contract the ede between vertices λ α, λ β into the new vertex λ αβ with S λαβ = S λα Sλβ. Clearly the only edes affected are the edes that were incident on either λ α or λ β. We continue until there is no reducible wavelenth pair. It is clear that the maximum number of iterations is bounded by the number of multicast requests, since initially the number of wavelenths is equal to the number of multicast requests, and each iteration reduces the number of wavelenths by one. Alorithm 1 Minimizin ADMs Require: Graph G = (V, E) and for every v V, set S v as described in Section III-B. S v where σ = {C σ, C1 σ,... } is a valid partition of vertex set V as described in Section III-B and Σ is the set of all such valid partitionins. 1: Construct raph H() = (Λ(), L()) with Λ() = V and evaluate the ede weihts c(λ i λ j ) for every ede λ i λ j L(). 2: condition T RUE 3: while condition is T RUE do 4: Determine the reducible wavelenth pair π = λ α λ β L(n) havin the larest ede weiht amon all the reducible wavelenth pairs. If there are several such pairs λ i, λ j, then select one with minimum value of S λi Sλj. If there are still ties then pick any of the possible choices randomly. 5: if such π then 6: Construct raph H(n + 1) from raph H(n) as described in Section IV-B and update the ede weihts. 7: else 8: condition F ALSE 9: end if 1: end while Ensure: min σ Σ i v C σ i Note that the circular arc raphs form an intersection class of raphs [27] and are therefore closed under induced subraph [27]. Also, as described in Section III-A, raph G belons to the family of circular arc raphs, so any induced subraph of raph G is also a circular arc raph. And since colorin circular arc raphs is NP hard, to check whether an induced raph of raph G is -colorable or not (this is what we need to check in order to determine whether a iven wavelenth pair is reducible or not) is also NP hard. So instead of doin this we color the induced subraph usin Tucker s alorithm for colorin circular arcs [23] and see if we need more than colors. Clearly this is sub-optimal but we use this since in eneral Tucker s alorithm ives a ood bound on the chromatic number. The complete heuristic is iven as Alorithm 1. # arcs = α i # arcs = δ i # arcs = βi Fi. 3. Bounds: observe each network node separately. A. Lower Bound i V. BOUNDS # arcs = γ i To et a lower bound on the total number of ADMs required, we consider each node of the network separately. We look at all the multicast requests havin network node i {,..., N 1} as either the source or one of the destinations and try to use as few ADMs as possible on the network node i to support these requests. To do this we construct raph G i = (V i, E i ) where V i = {v V : i S v } is the set of vertices correspondin to multicast requests havin network node i as source or one of the destinations and there is an ede v j v k E i if and only if the correspondin multicast requests R j and R k share some fiber. Now G i is exactly G[V i ], the subraph induced by vertex set V i V on contention raph G described in Section III-A. Let χ i be the chromatic number of raph G i. Since all the requests represented by the vertex set V i have network node i as either source or some destination, node i must have ADMs correspondin to all the wavelenths on which any of these requests are roomed. So in order to use the minimum number of ADMs at node i (irrespective of the number of ADMs required at other nodes), we need to room the traffic requests represented by vertex set V i on as few wavelenths as possible. Usin the arument from Section III-A, the set of multicast requests that have network node i as either the source or a destination must be spread over at least χi wavelenths, and hence there must be at least this many ADMs at node i. Now applyin a similar lower bound on the number of ADMs required at each node of the network, we see that the total number of ADMs required in the network is at least as lare as L iven by L = χi where N is the number of nodes in the SONET rin and is the roomin ratio. Thus (4) ives a lower bound on the total number of ADMs required in the network. As noted in Section IV-B, the class of circular arc raphs is closed under induced subraph. Also, as described in Section III-A, raph G belons to the class of circular arc raphs. So the subraph G i, induced on the raph G by the vertex set V i is also a circular arc raph. And, althouh it is NP complete to determine the chromatic number of a eneral circular arc raph, as described next, the chromatic number χ i of raph G i can be easily calculated. Fiure 3 shows all the multicast traffic requests that pass throuh network node i. The solid arcs correspond to the (4)

8 8 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 25, NO. 6, AUGUST 27 requests that contain node i as source or one of the destinations and the dotted arcs represent the traffic requests that pass throuh node i but do not contain it as source or a destination. Of these, let α i be the number of requests that contain i as an intermediate destination, β i be the number of requests that contain i as the final destination and γ i be the number of requests that contain i as the source. Clearly in the raph G i, the traffic requests represented by the vertex set V i correspond to the solid arcs and therefore k i = α i + β i + γ i. Let us look at raph Ĝi = ( ˆV i, Êi) instead of raph G i, where Ĝ i is the subraph induced on raph G i (or equivalently on contention raph G) by the vertex set ˆVi representin the requests that either have network node i as source or the final destination, i.e., we are removin the α i nodes correspondin to the requests that contain network node i as an intermediate destination. Now raph Ĝi is an interval raph [28] and we know that for interval raphs the chromatic number is easily computable and is equal to the maximum width of the raph [29]. Here the width of an interval raph at some point is defined as the number of arcs overlappin at that point. Also it is clear that if vertex u V i and vertex v V i \ ˆV i, then u and v can not have the same color (since requests correspondin to vertices u and v share some fiber in the network). Therefore the chromatic number χ i of raph G i is iven by χ i = w i + α i (5) where w i is the maximum width of the interval raph Ĝi and α i = V i \ ˆV i is the number of traffic requests which contain i as an intermediate destination. Durin our simulations presented in Section VII, we observe an interestin property of the lower bound presented above. It seems that the lower bound iven in (4) does not depend on the number of nodes in the rin. To explain this, we try to calculate the expected value of the lower bound for roomin M multicast traffic requests on a SONET rin havin N nodes and roomin ratio. Let z i represent the size of the i- th multicast session R i. Here by the size of a session, we mean the total number of source and destination nodes in that session. Therefore, the size of session R i (represented by vertex v i V ) is iven by z i = S vi. For the purpose of our simulations (and hence for this analysis), we assume that the multicast session sizes z, z 1,..., z M 1 are independent and identically distributed accordin to some cumulative distribution function F havin mean µ F. We also assume that the nodes (actin as the source or any destination) in any multicast session, are selected randomly and uniformly from all the nodes of the rin, i.e., for every multicast request R i (represented by vertex v i V ) havin size z i, the probability that node j S vi is equal to zi N for every j {,..., N 1}. Moreover, the selection of source and destination nodes of different multicast sessions is assumed to be independent of each other. We first note that even thouh it is hard to estimate the expected value of the lower bound, we can estimate the expected value of the followin function which approximates the lower bound. ˆL = ki (6) Here k i is the number of multicast sessions that have node i as either source or some destination. When raphs G i are dense (which is the case in Section VII as well as the case in most of the interestin examples), χ i k i. Now it is easy to observe that the expected value of the approximate lower bound ˆL is iven by E(ˆL) = E ki = E ki = N E k (7) Here for the third equality we are usin the fact that in any multicast session, nodes are selected with equal probability, and hence k i s are identically distributed. So we can drop the subscript i and assume that the number of multicast requests that have i as either source or some destination is distributed accordin to random variable k. To et an estimate of E(ˆL), we first observe that E k k E E k + 1 (8) Also, the number of multicast sessions selectin a particular network node as source or one of the destinations can be written as k = x + x x M 1 (9) where random variable x i takes value 1 if the i-th multicast session R i selects the node under consideration as source or one of the destinations and otherwise. Now we can evaluate E(k) as E(k) = = M 1 M 1 E(x i ) = E M 1 z i = 1 N N E(E(x i z i )) M 1 E(z i ) = M N µ F (1) Here the third equality follows from the fact that iven z i (the size of session R i ), the random variable x i is distributed accordin to a Bernoulli trial with probability of success p = z i N and. Now (1) ives us E k = Mµ F N (11) E k + 1 Mµ F = N + 1 (12) Usin equations (7), (8), (11) and (12), we can easily bound the required expectation as Mµ F E(ˆL) Mµ F + N (13) Now if Mµ F / N (which is the case in Section VII and is typically the case), then from (13), we note that the expected value of our lower bound can be approximated by Mµ F /, which is independent of the number of nodes in the rin. It should be clear that this is mainly because our lower bound looks at each node of the rin in isolation. If we start considerin pairs (or triplets, etc.) of nodes at a time then our bound will depend on the number of nodes in the rin. But it is not trivial to extend the iven bound and for the purpose of our discussion the iven bound suffices.

9 SUPPLEMENT ON OPTICAL COMMUNICATIONS AND NETWORKING 9 B. Upper Bounds Now we investiate some upper bounds on the number of ADMs required in the network. We study the upper bounds for the worst case and two very simple alorithms. 1) Worst Case: The maximum number of ADMs is required when we use a different wavelenth for each multicast request, i.e., we do no traffic roomin and wavelenth reuse. In this case, the number of ADMs required U wc, is iven by U wc = k i (14) where N is the number of nodes in the SONET rin and k i is the number of traffic requests havin network node i as source or one of the destinations. Thus (14) ives an upper bound on the number of ADMs required. Trivially in (5), the value of maximum width w i is lower bounded by w i max{β i, γ i } (15) Now usin (5) and (15) we et We also know that χ i α i + max{β i, γ i } (16) k i = α i + β i + γ i (17) Usin (16) and (17), we can easily show that k i = α i + β i + γ i α i + 2 max{β i, γ i } Now (18) and (14) ive U wc = 2(α i + max{β i, γ i }) 2χ i (18) k i 2χ i 2 χi = 2L (19) This means that any sort of wavelenth assinment (and traffic roomin) solution is an approximation alorithm with approximation ratio 2. An interestin observation is that in case of no roomin ( = 1), any wavelenth assinment will be within twice the optimal as far as the number of ADMs required in the network is concerned. 2) Alorithm A: Another interestin bound that we consider is for the simple heuristic in which we randomly roup the traffic requests into clusters of requests each. We assume that requests in a particular cluster are routed on the same wavelenth. This is clearly possible since we are providin a separate subwavelenth channel for each traffic request. Then we assume that each network node that acts as a source or destination for some multicast request is provided with an ADM for all these wavelenths. Note that if network node i does not act as source or destination for any multicast request, i.e., if i / S v for every v V, then since no traffic is bein added or dropped at i, there is no need to equip i with ADM on any wavelenth. Let N denote the number of nodes that act as source or destination for at least one multicast request. Clearly N N. Now the number of ADMs in the network is iven by M U Alo A = N (2) Let z av be the averae size of multicast sessions, i.e., let Now (18), (2) and (21) ive us U Alo A = N k i N k i = z av M (21) z av χi N 2 χ i z av = N L NL (22) The second inequality holds because of the fact that z av 2. This is true since every multicast session has at least one source and one destination. Note that if we further assume a lare enouh averae multicast session size, then we can show a better upper bound. More specifically, we can show the followin U Alo A N 2 χ i 2N z av z av ( 2N + 1 z av ( 2N + 1 z av N z av 2 χ i + z av χ i + N 2N χi + N z av ) ( ) χi 2N = + 1 L z av ) L (23) Here the last inequality is due to the fact that if node i acts as a source or a destination for at least one multicast request, then the raph G i has at least one vertex and therefore χ i 1. Now observin that there are N such nodes, we et χi N. So the simple heuristic of routin any traffic requests on the same wavelenth is an approximation alorithm with approximation ratio N. And if the averae session size z av 2N, then, for the same heuristic, we can et a better approximation ratio equal to 1 + 2N z av. 3) Alorithm B: Another simple heuristic is that we try to use the minimum number of subwavelenth channels for all the traffic requests and then we randomly combine subwavelenth channels into one wavelenth. So now we may have more than requests in one wavelenth. This is equivalent to colorin the raph G described in Section III-B usin the minimum number of colors and then roupin colors (subwavelenth channels) toether to form one wavelenth. Let the chromatic number of raph G be χ. As mentioned in Section III-B, G is a circular-arc raph and therefore it can be colored by Karapetian s alorithm [26] usin at most 3 2 χ colors. Aain, as for Alorithm A, the maximum number of ADMs required by this technique is when each node that acts as a source or a destination for at least one multicast request, is provided with an ADM for all the wavelenths. The number of ADMs is iven by U Alo B = N 3 2 χ (24)

10 1 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 25, NO. 6, AUGUST 27 Now if all the sessions are multicast with size at least z min, then the minimum number of ADMs required for each wavelenth is z min. Hence a lower bound (other than our primary lower bound L iven in (4)) on the number of ADMs required in the network is χ L 2 = z min (25) Usin equations (24) and (25), we see that 3 U Alo B = N 2 χ 2χ χ N 2N 2N z min L 2 2N z min L 2 (26) 2N z min. So the approximation ratio of this simple alorithm is We can arrive at a different (better in some cases) approximation ratio by followin a separate line of analysis. Let χ = 2n + δ + ɛ (27) where n is a non-neative inteer, δ {, 1} and ɛ < 1. From (25) and (27), we et L 2 = z min 2n + δ + ɛ = z min (2n + δ + ɛ ) (28) Aain from (24) and (27), we et 3 U Alo B = N 2 χ 3χ N 2 3 = N 2 (2n + δ + ɛ) ) δ + ɛ N (3n Now only the followin two cases are possible. δ + ɛ = δ = ɛ = In this case, from (28), the lower bound becomes And from (29) and (3), we et (29) L 2 = 2nz min (3) U Alo B 3nN = δ + ɛ > In this case, from (28) we et 3N 2z min L 2 (31) L 2 = z min (2n + δ + ɛ ) z min (2n + 1) (32) And from (29), we et ) δ + ɛ U Alo B N (3n + 3 = N (3n + 3) (33) 2 where the second equality is based on the fact that since δ {1, }, ɛ [, 1) and δ + ɛ >, < δ + ɛ < 2. Now usin equations (32) and (33), we et U Alo B 3N 3N (2n + 1) N L 2 + 3N 2z min 2 3 ( ) N + 1 L 3 2 z min 3 ( ) N + 1 L 3 (34) 2 z min where L 3 is another lower bound on the number of required ADMs iven by L 3 = max{l 2, N } (35) It should be clear that L 3 is a valid lower bound because L 2 is a lower bound and since we need at least one ADM on all N nodes that act as a source or a destination for at least one multicast request, N is also a valid lower bound on the number of ADMs required in the network. From equations (31) and (34), we observe that the alorithm has an approximation ratio 3(N+zmin) 2z min. Also note that this approximation ratio is better than the previously computed 2N approximation ratio of z min whenever z min < N 3. VI. COMPLEXITY ANALYSIS In this section we present the complexity analysis for our raph based traffic roomin heuristic presented in Section IV-B and the two simple schemes presented as upper bounds for the roomin problem in Section V-B. A. Alorithm A Alorithm A described in Section V-B.2 starts by randomly roupin the iven traffic requests into clusters of size each. M This clusterin requires O(M) steps and we et clusters. Let the clusters be C,..., C M. All the traffic requests 1 in cluster C i are routed on wavelenth λ i, therefore the set of network nodes which should be equipped with ADM on wavelenth λ i is iven by S λi = S v (36) v C i M There are clusters containin traffic requests and M M clusters containin M mod traffic requests. Since S v {,..., N 1} for every v V, the number of steps required for evaluatin S λi accordin to (36) is N if cluster C i contains requests and N(M mod ) if it contains M mod requests. So the total number of steps required for determinin the placement of ADMs at each network node on all the wavelenths is ( ) M M M N + N (M mod ) M = N + N (M mod ) = NM (37) Therefore the overall complexity of Alorithm A is O(NM). B. Alorithm B Alorithm B described in Section V-B.2 first colors the contention raph G defined in Section IV-B usin Karapetian s alorithm [26]. This requires O(M 2 ) time. The total number of colors used is upper-bounded by max{χ, M} where χ is the chromatic number of raph G. The the colors are then randomly split into roups of size. Based on these roupin the iven traffic requests are partitioned into clusters such that the cluster correspondin to a particular roup of colors contains all the traffic requests that were assined colors from

11 SUPPLEMENT ON OPTICAL COMMUNICATIONS AND NETWORKING 11 N= , M=8, =4, z min =8 N=8, M=8, = , z min = LowerBound Alorithm 2 Alorithm 1 Alorithm A Alorithm B Number of ADMs 15 1 Number of ADMs 2 15 LowerBound Alorithm 2 5 Alorithm 1 Alorithm A Alorithm B Number of Network Nodes Groomin Ratio (a) Varyin network size. (b) Varyin roomin ratio. 22 N=8, M=8, =4, z min = N=8, M= , =4, z min = Number of ADMs Number of ADMs LowerBound Alorithm 2 4 Alorithm 1 2 Alorithm A Alorithm B Maximum Session Size LowerBound 5 Alorithm 2 Alorithm 1 Alorithm A Alorithm B Number of Sessions (c) Varyin session size. (d) Varyin number of sessions. Fi. 4. ADMs required by Alorithms 1, 2, A, B and lower bound L. that roup. This clusterin can be done in O(M) steps. Let there be K such clusters C,..., C K 1. Let the number of traffic requests in cluster C i be i. All the traffic requests in cluster C i are routed on wavelenth λ i and the set of network nodes which should be equipped with ADM on wavelenth λ i is iven by (36). As arued above in the complexity analysis of Alorithm A, the number of steps required for evaluatin S λi is N i. So the total number of steps required for determinin the placement of ADMs at each network node on all the wavelenths is K 1 N i = N K 1 i = NM (38) Therefore the overall complexity of Alorithm B is O(NM + M 2 ). C. Heuristic Consider the raph based traffic roomin heuristic presented as Alorithm 1. As described in Section IV-B, we start off by assinin a different wavelenth to every traffic request. In each iteration on the heuristic, we update the wavelenth assinment by first determinin the best (as described in Section IV-B) reducible wavelenth pair and then assinin all the traffic requests on the two wavelenths to a sinle wavelenth. We continue to update the wavelenth assinment iteratively till there are no more reducible wavelenth pairs left. The wavelenth assinment after completin n steps of the heuristic is maintained as raph H(n) = (Λ(n), L(n)) where the vertex set represents the set of wavelenths. For each wavelenth λ i Λ(n) we maintain S λi, the set of network nodes which act as source or destination nodes for any multicast session bein roomed on wavelenth λ i. Also for every wavelenth pair λ i, λ j Λ(n), we maintain S λi Sλj, S λi Sλj and whether the wavelenth pair is reducible or not. First we study the complexity of the n + 1-st iteration in Alorithm 1. Since in each iteration we reduce the number of wavelenths by 1, Λ(n) = Λ() n = M n. So the

12 12 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 25, NO. 6, AUGUST N= , M=8, =4, z min = N=8, M=8, = , z min =8 Alorithm 2 Alorithm Number of Wavelenths 15 1 Number of Wavelenths Alorithm 2 Alorithm Number of Network Nodes Groomin Ratio (a) Varyin network size. (b) Varyin roomin ratio. 25 N=8, M=8, =4, z min = N=8, M= , =4, z min =8 2 2 Number of Wavelenths 15 1 Number of Wavelenths Alorithm 2 Alorithm Maximum Session Size Alorithm 2 Alorithm Number of Sessions (c) Varyin session size. (d) Varyin number of sessions. Fi. 5. Wavelenths per fiber required by Alorithms 1, 2. number of wavelenth pairs to consider is (M n)(m n 1) 2. The number of steps required to determine the best reducible wavelenth pair is linear in the number of wavelenth pairs. After determinin the best reducible wavelenth pair λ α, λ β Λ(n) we update raph H(n) to H(n + 1) with vertex set Λ(n + 1) = (λ(n) {λ αβ }) \ {λ α, λ β } where all the traffic requests previously roomed on wavelenths λ α and λ β are now assined on the new wavelenth λ αβ. As explained in Section IV-B, we need to compute S λαβ = S λα Sλβ and for every other wavelenth λ i Λ(n+1) we need to evaluate S λαβ Sλi, S λαβ Sλi and whether the wavelenth pair λ αβ, λ i is reducible or not. Since S λ {,..., N 1} for any wavelenth λ, evaluatin S λαβ, S λαβ Sλi and S λαβ Sλi require O(N) steps. For any wavelenth λ k, let C k denote the set of traffic requests that are assined wavelenth λ k. To determine whether wavelenth pair λ αβ, λ i is reducible or not, we check if we can color raph G[C i Cαβ ], the contention raph of all the traffic requests that are assined wavelenths λ i or λ αβ, usin at most colors or not. We employ Tucker s alorithm [23] for colorin the circular arc raph which requires O( C i Cαβ 2 ) time. Since C i Cαβ M, checkin if wavelenth pair λ i, λ αβ is reducible or not takes O(M 2 ) time. Therefore the number of steps required in n+1- st iteration of the heuristic is (M n 2)O(N + M 2 ). As already explained after each iteration, the size of the vertex set of the raph decreases by 1, so there can be a maximum of M 1 iterations. Hence the iterations in the heuristic require O(M 2 (N + M 2 )) steps. Now we count the number of steps required to initialize the raph H() in the first step of the heuristic. Note that checkin whether a wavelenth pair λ i, λ j Λ() is reducible or not requires O(1) steps. This is because every wavelenth corresponds to just a sinle traffic request. Aain, determinin S λi Sλj and S λi Sλj require O(N) steps. Since there are M(M 1) 2 wavelenths pairs, the construction of raph H() requires O(NM 2 ) steps. Therefore the overall complexity of Alorithm 1 is O(M 2 (N + M 2 )).

13 SUPPLEMENT ON OPTICAL COMMUNICATIONS AND NETWORKING 13 VII. NUMERICAL RESULTS Since presently there is no other heuristic for roomin multicast traffic in unidirectional rins with which we can compare our heuristic, we extend the unicast traffic roomin alorithm presented in [1] to the multicast case. We do this by simply startin with multicast sessions in place of unicast sessions in the circle construction phase. More specifically, we try to put as many multicast sessions on circles without introducin aps. In [1], the authors do this for unicast sessions by assumin each unicast session to be a connection and then combinin two connections with common end points to form complete circles. After constructin the maximum possible circles in this way, they then apply Alorithm IV:Construct Circles - Non-Uniform Traffic to construct the rest of the circles. Each circle here corresponds to a subwavelenth channel. After all the connections have been assined to some circle, the circles are roomed into wavelenths. In our extension of this alorithm, we consider multicast sessions to be the startin connections and construct the circles in exactly the same way. After we have the circles, the circle roomin heuristic is exactly the same as in [1]. We refer to our heuristic as Alorithm 1 and this extended heuristic as Alorithm 2. We evaluate the performance of both Alorithms 1 and 2 in terms of the number of ADMs required. For a more complete picture, we also compare the performance of both the heuristics to our lower bound as iven in equation (4). Since the number of wavelenths required also contributes to the network cost (albeit, not as much as the ADMs), we also compare the wavelenths required by the two heuristics. For the sake of completeness we also compare the two simple multicast traffic roomin schemes presented as Alorithm A and Alorithm B in Section V-B. We identify the problem of roomin multicast traffic on unidirectional rins by the five parameters: N, M,, z min and z max. Here the parameters N, M and denote the number of nodes in the rin, the number of multicast sessions and the roomin ratio respectively. Parameters z min and z max denote the minimum and the maximum possible size of the multicast sessions. For the purpose of simulation, while eneratin a multicast session, each node is iven equal probability of bein selected as the source. The size of each multicast session is selected uniformly from z min to z max. After the source node and the size z of the multicast session are fixed, destination nodes are selected such that every subset of size z 1 of the remainin N 1 nodes (since one node has already been selected as the source) has equal probability of bein the destination set. For simulation, we consider a nominal rin network havin 1 nodes, 8 multicast sessions, with each session size selected uniformly between 2 to 8 and havin roomin ratio 4. We study the performance of both the heuristics by varyin one parameter of the problem at a time in this nominal network. More specifically, we vary the roomin ratio from 2 to 6, the network size (number of nodes in the network) from 8 to 16, the number of multicast sessions from 6 to 1 and the maximum size of multicast sessions from 2 to 1. Fiure 4 presents the simulation results comparin the number of ADMs required by the various roomin schemes as well as the number of ADMs specified by our lower bound L. The simulation results comparin the wavelenths per fiber required by Alorithm 1 and 2 are presented in Fiure 5. Each point in the plots is enerated by takin an averae of 2 randomly selected roomin problem instances with the required parameters. We can see from the plots that, as measured by the number of ADMs required, our Alorithm 1 always outperforms Alorithm 2. This is true even for unicast traffic (the case for which Alorithm 2 was desined in [1]). We also note that our Alorithm 1 usually requires more wavelenths than Alorithm 2. But the increase in the number of wavelenths is never more than 2, and is overshadowed by the savins in the number of (more expensive) ADMs. Form the plots we also observe that of the three roomin schemes presented in this work, our raph based roomin heuristic (Alorithm 1) always outperforms the simple roomin from Section V-B (Alorithms A and B). And amon the two simple schemes, Alorithm B always outperforms Alorithm A. We can justify this trend in the liht of the complexity analysis of the three schemes presented in Section VI. Assumin that the number of traffic requests to be roomed is much larer than the number of network nodes (which is usually the case and is true for our simulations as well), we observe that based on their time complexities, Alorithm A is the simplest, Alorithm 1 is the hardest and Alorithm B lies somewhere in-between the two. Since we et what we pay for, the relative performances of the three schemes is as expected. Althouh not presented in the plots, the number of wavelenths required by Alorithms A and B are also very similar to that required by Alorithms 1 and 2. Also from the plots, we can see that the lower bound L iven in (4) tracks the performance curves of the heuristics as we vary the roomin ratio, the number of sessions or the size of sessions. This suests that the bound tracks the chanes in these parameters quite well. But we observe that this is not so in the case of the size of network. This is consistent with our discussion in Section V, and it is easy to verify that the value of the lower bound (averaed over 2 runs) closely matches our estimate iven in (8). VIII. CONCLUSION In this paper we have studied the problem of roomin nonuniform multicast traffic on a unidirectional SONET/WDM rin. We consider two different costs, (i) number of wavelenths and (ii) number of ADMs. We observe that minimizin the number of wavelenths can be modeled as a standard arcraph colorin problem. We then ive a raph based heuristic for minimizin the number of ADMs. Based on extensive simulations we observe that our heuristic performs better than the multicast extension of the best known unicast traffic roomin heuristic for rins iven in [1]. We also develop a lower bound for the problem and look at some interestin relations between the lower bound and a couple of upper bounds. REFERENCES [1] X. Zhan and C. 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McKee and F. R. McMorris, Topics in Intersection Graph Theory. Philadelphia, PA: SIAM, [28] M. C. Golumbic, Alorithmic Graph Theory and Perfect Graphs. New York: Academic Press, 198. [29] S. Olariu, An optimal reedy heuristic to color interval raphs, Information Processin Lett., Anuj Rawat received the B.Tech. in Electrical Enineerin from the Indian Institute of Technoloy, Kanpur, India, in 22. From 22 to 23, he was a member of technical staff at Atrenta, in India. Since fall 23 he has been workin towards his Ph.D. deree in Electrical and Computer Enineerin at the University of Maryland at Collee Park, MD, USA. His research interests include wireless sensor networks, desin of alorithms for optical networks, traffic enineerin and routin. Richard J. La (S 98 - M 1) received the B.S.E.E. from the University of Maryland at Collee Park in 1994, and the M.S. and PhD. derees in Electrical Enineerin from the University of California at Berkeley in 1997 and 2, respectively. From 2 to 21 he was a senior enineer in the Mathematics of Communication Networks roup at Motorola. Since Auust 21 he has been on the faculty of the ECE department at the University of Maryland at Collee Park. Steven I. Marcus (Fellow, IEEE) received the B.A. deree in electrical enineerin and mathematics from Rice University in 1971 and the S.M. and Ph.D. derees in electrical enineerin from the Massachusetts Institute of Technoloy in 1972 and 1975, respectively. From 1975 to 1991, he was with the Department of Electrical and Computer Enineerin at the University of Texas at Austin, where he was the L.B. (Preach) Meaders Professor in Enineerin. He was Associate Chairman of the Department durin the period In 1991, he joined the University of Maryland, Collee Park, where he was Director of the Institute for Systems Research until He was Chair of the Electrical and Computer Enineerin Department at the University of Maryland durin the period He is currently Professor in the Electrical and Computer Enineerin Department and the Institute for Systems Research. His research is focused on stochastic control and estimation, with applications in manufacturin and communication networks. His address is: marcus@umd.edu. Mark A. Shayman received the Ph.D. deree in Applied Mathematics from Harvard University, Cambride, MA, in He served on the faculty of Washinton University, St. Louis, MO, and the University of Maryland, Collee Park, where he is currently Professor of Electrical and Computer Enineerin. His research interests are in communication networks, includin optical, wireless and sensor networks. He received the Donald P. Eckman award from the American Automatic Control Council and the Presidential Youn Investiator Award from the National Science Foundation. He has served as Associate Editor of the IEEE Transactions on Automatic Control.