Reducing Network Cost of Many-to-Many Communication in Unidirectional WDM Rings with Network Coding

Transcription

1 1 Reducin Network Cost of Many-to-Many Communication in Unidirectional WDM Rins with Network Codin Lon Lon and Ahmed E. Kamal, Senior Member, IEEE Abstract In this paper we address the problem of traffic roomin in WDM rins with all-to-all and its eneralization to many-to-many service by usin network codin. We consider minimizin the number of Line Terminatin Equipment (LTE on two types of unidirectional rins, namely, sinle-hub and unhubbed rins, as our objective. In sinle-hub rins, we investiate the minimum cost provisionin of uniform all-to-all traffic in two cases: where network codin is used to linearly combine data, and where it is not used and data is transmitted without codin. We eneralize the service mode to many-to-many and evaluate the cost of provisionin. In un-hubbed rin, we propose a multi-hub approach to obtain the minimum cost provisionin in the case of all-to-all and many-to-many traffic. In each type of rin topoloy, two network scenarios are considered: first, the distinct communication roups in the rin are node-disjoint and second, the different roups may have common member nodes. From our numerical results, we find that under many-to-many traffic pattern for both scenarios, network codin can reduce the network cost by 10-20% in sinle-hub rins and 1-5% in un-hubbed rins in both network scenarios. Index Terms WDM rins, Traffic roomin, Line Terminatin Equipment (LTE, Network codin I. INTRODUCTION WAVELENGTH-division multiplexin (WDM technoloy allows an areate traffic on the order of Tbps to be carried on a sinle fiber, with each wavelenth carryin traffic in the tens of Gbps order. However, the traffic demands of network applications are at much smaller ranularity than wavelenth bit rates. In order to utilize wavelenth capacity more efficiently, a number of flows from multiple network connections with sub-wavelenth ranularity may be packed onto the same wavelenth. This process of allocatin low bit rate tributary streams to wavelenths with hih bandwidth is referred to as traffic roomin. There are two types of traffic roomin problems, static and dynamic. The objective of the static problem is usually to minimize the overall network cost, iven the traffic demands, whereas in the dynamic problem, maximizin the throuhput or minimizin the blockin probability of connections. The static traffic roomin problem of unicast traffic has been widely studied in the literature [3], [4], [18]. But recently, This research was supported in part by rant CNS from the National Science Foundation. Lon Lon and Ahmed E. Kamal are with the Department of Electrical and Computer Enineerin, Iowa State University, Ames, IA, USA, ( lonlon@iastate.edu; kamal@iastate.edu Copyriht (c 2009 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sendin a request to pubs-permissions@ieee.or. multipoint traffic has become more important in a number of application environments, and this is why a number of studies addressin multipoint traffic roomin have recently appeared in the literature [15], [16], [17]. Amon a number of network architectures, rin topoloies drew sinificant attention in the research community due to the availability of leacy SONET equipment [12], [19], [20]. In rin networks, both all-to-all [1], [2], [11], [13], multicast [9], [14] and arbitrary[21] traffic scenarios have been studied. Most of the literature addressin this problem focuses on evaluatin and reducin the dominant cost in the optical network, namely, Electronic Add-Drop Multiplexers (ADM, which is required at a node if it either has data to transmit to or receive from another counterpart. The number of ADMs required at a node is only a function of the number of lihtpaths established and terminated at the node. Another cost function that is similar to the number of ADM is the number of e-dac roomin ports presented in [9], which refers to Electronic Drop-and-Continue roomin ports, in which part of the traffic is dropped off at a node, and the rest of the traffic continues. In this paper, we address the static traffic roomin problem of a class of multipoint traffic in unidirectional rin networks with the number of electronic Line Terminatin Equipments (LTE ports, be it ADM ports, as the network cost. The number of wavelenths and the cost of other optical equipment, such as the optical splitter (which is neliible compared to the electronic LTEs are not factors to be considered here. We consider uniform all-to-all traffic roomin, in which all users in the network exchane data. We also consider a eneralized case where there are multiple communication roups and each node in a roup has to receive the data from all the other users in the same roup. A user may belon to multiple roups. We consider two types of unidirectional rins, namely, sinle-hub and un-hubbed rins. In a sinle-hub rin [10], all the traffic has to be sent to the hub and then forwarded to the destinations by the hub. In an un-hubbed rin, there is no such hub. All-to-all traffic can be implemented usin two approaches, unicast and multicast. In unicast mode, traffic duplication can only be implemented in the electronic domain, whereas in multicast mode, traffic duplication can be done in the optical domain by usin optical splitters. If a node needs to send a traffic stream on two outoin links, the node requires two LTE ports in unicast mode but only one in multicast mode. All-to-all communication will benefit sinificantly in terms of the network cost by dividin it into multiple multicast sessions. However, it requires multicast capable nodes to be deployed in

2 2 the network. The correspondin node architecture is referred to as the Tap-and-Continue node, and is introduced in [14], in which a node is capable of replicatin optical sinals in the optical domain usin optical splitters, and one copy is dropped locally, while the remainin sinal continues on the fiber without the additional cost of LTEs. Since each node on the unidirectional rin has only one incomin link and one outoin link, and routin of all lihtpaths is fixed alon the direction of the rin, there is only one possible way to multicast - drop and forward. We use such node architecture to implement any multicast needed in the network. Network codin [5] is a promisin new technique that enables network nodes to perform alebraic operations on the multiple received packets besides simply forwardin them. It has been applied to a variety of network applications in order to improve the performances, such as in multi-hop wireless networks [28], network tomoraphy [24], network protection [25], [26] and content distribution in peer-to-peer networks [27]. However, it is rarely applied to optical networks for the cost savin objective. We will study two unidirectional rin networks, sinle-hub and un-hubbed, and investiate whether applyin network codin will reduce the network cost to provision iven traffic requests. The paper is structured as follow. In Section II, we introduce network codin and its benefit in savin LTE ports in optical networks. We will explore the network costs with or without applyin network codin in sinle-hub and un-hubbed rins in Section III and IV, respectively. Numerical results of multiple many-to-many communication will be shown in Section V. Finally in Section VI, we conclude the paper. II. NETWORK CODING IN OPTICAL NETWORK Network codin is a novel technique which was oriinally proposed for improvin network capacity, particularly in multicast scenarios[5], [6]. Besides the traditional routin functions, network nodes are desined to linearly combine packets arrivin at input edes and transmit those combinations on output ports. By carefully choosin codin coefficients for each codin node in a network, where a codin node refers to a network node that has the capability of formin linear combinations of packets, a network can achieve the maximum multicast throuhput for a iven multicast request, which is equal to the min-cut max-flow of the network. In other words, iven a traffic demand, employin a network codin scheme may provision the traffic by utilizin fewer network resources than the traditional routin scheme. Work has been done in recent years to explore the efficient codin schemes. For example, a classic polynomial time alorithm of codes construction for multicast traffic is proposed in [7] and a random codin scheme is proposed in [8]. In an optical network, traffic flow is carried and conveyed in a lihtpath as an optical sinal. To establish a lihtpath, one LTE port is needed at the source to oriinate the sinal and one is required at the destination to terminate it, and then the total number of LTE ports 1 required at each node is the total number of lihtpaths terminated and oriinated at this node. 1 In rest of this paper, we use LTE to represent LTE port for short. Fi. 1. S λ 2 λ 2 A B C λ 2 x ( 1 λ 2 λ 2 T1 T2 T3 (a Traditional lihtpaths S λ 2 λ ( 3 x 1 + x2 A B C λ ( x + 2 λ x x 3( 1 x2 λ 2 x ( 2 λ 2 T1 T2 T3 (b Network Codin An example of LTE cost reduction throuh network codin Given a multicast service request, if network codin scheme can be employed to reduce the total bandwidth used, the total number of lihtpaths required may be reduced, which will consequently result in a savin in the total network cost. An example is shown in Fiure 1 to illustrate the benefits of usin network codin to save network cost in terms of LTE in a multicast traffic scenario. In the iven network, one multicast session needs to be implemented, in which one source S enerates two traffic flows and λ 2 and both of them should be received by all three destination nodes T 1, T 2 and T 3. We also assume that each traffic flow takes an entire wavelenth to accommodate. Thus, Fiure 1.(a shows the provisionin of the multicast session by usin the traditional routin scheme. In order to achieve the multicast throuhput, which is 2 in this case, four lihtpaths have to be set up at source S and one of its outoin links need to carry two traffic flows. In this case, links (S, A and (S, C carry one optical sinal each and link (S, B must carry two optical sinals. We assume that each intermediate node is deployed with the capability of both multicastin and forwardin functionality in optical domain and thus no LTE is required at node A, B and C. Therefore, source node S requires four LTEs to establish the lihtpaths and each destination node needs two LTEs to receive the traffic. Network codin is introduced in Fiure 1.(b to provision the same multicast service. Instead of eneratin four optical sinals at source S in (a, only three lihtpaths need to be set up as shown, in which two of them carry the oriinal traffic flow and λ 2 and the third one carries the encoded data + λ 2 enerated by performin bitwise operation Exclusive OR on the two oriinal sinals. In this case, node T 1 will receive the traffic and + λ 2 while node T 3 receives λ 2 and + λ 2. Clearly, both nodes perform XOR operation on the received flow in order to recover the oriinal traffic λ 2 and, respectively, and hence multicast service is fulfilled. Therefore, employin network codin in the example in Fiure 1.(b reduces the cost by one LTE at the source node S when compared to the cost required in 1.(a. Althouh a recent study has introduces ways to conduct the network codin operation in optical domain[22], [23], our scheme is based on the current network infrastructure where network codin can only be implemented in an electronic domain, which requires the traffic that participates in the network codin operation to o throuh O-E-O conversion at the codin nodes. Such conversion may not be necessary if

3 3 traditional routin is used. Thus, employin network codin may require more LTEs to terminate oriinal traffic and reenerate encoded sinals at codin nodes, hence makin the problem a trade-off between reduction in the number of codin nodes and the reduction in LTEs in order to use fewer lihtpaths in total. Moreover, applyin network codin to optical networks also introduces new issues such as where and how should network codin be performed? The answer to this question is straihtforward in a sinle-hub rin network. Since all traffic is collected by the hub, the hub is the perfect node to combine packets. However, it is not as clear in an unhubbed rin. In addition, achievin network codin requires determinin the codin scheme and the finite field size GF (q from which we choose codin coefficients. Of course, network codin does not come for free, and all the codin nodes should be equipped with the capability of codin, which introduces extra computation cost in the application layer. Compared to the cost of LTEs in physical layer, however, such resource consumption is almost neliible. Therefore, we only consider the number of LTEs as the network cost in this paper. III. COST ANALYSIS IN SINGLE-HUB UNIDIRECTIONAL RING In this section, we address the problem of roomin all-toall traffic in a sinle-hub unidirectional rin and then extend the problem to address many-to-many traffic service. In manyto-many communication, multiple roups are required to fulfill the all-to-all service where each roup consists of two or more nodes. We consider two different network scenarios where roups are node-disjoint and when a node may belon to different roups. In each scenario, the cost of provisionin traffic is derived usin two approaches: traditional routin and by applyin network codin. For many-to-many communication, we prove that the traffic roomin problem is NP-complete and also propose an Inteer Linear Prorammin (ILP formulation to solve the problem optimally in the case where the roups are non-disjoint. A. Uniform all-to-all traffic Under all-to-all service, each node should receive data from all the other nodes on the rin. The problem can be stated as follows: Given a roup of n nodes and roomin factor 2, each node i, enerates and transmits traffic at constant rate, r, and must receive the traffic sent by other nodes such that the network resources, the LTEs in particular, are minimized. We assume each data unit has to be transmitted to the hub before bein relayed to the destination(s. As mentioned earlier, each node is equipped with optical splitters such that traffic can be duplicated in the optical domain. Moreover, we allow traffic bifurcation, which refers to splittin the traffic from a session over multiple lihtpaths, since this may result in the minimum number of lihtpaths and achieve the minimum overall cost in the case when is not a multiple of r. The all-to-all communication process involves two steps. The first step is to deliver traffic upstream from nodes to the 2 Groomin factor refers to the maximum number of low-rate traffic demands that can be multiplexed into one wavelenth channel. hub. In the second step, the hub rooms the traffic into the minimum number of wavelenths and multicasts the roomed traffic downstream to every node on the rin. In the upstream direction, each node enerates r units of low-rate traffic stream, which requires r wavelenths to accommodate the data as well as r LTEs to send it, and the hub needs r LTEs to receive the traffic from one node. Either roomin the traffic from different nodes before sendin it to the hub, or sendin data to the hub directly by each node, will not chane the total number of LTEs durin the upstream process. This only effects the number of wavelenths used. However, the number of wavelenths is not a factor of the network cost accordin to our assumptions. Thus, there are n r unidirectional lihtpaths established from each node to the hub in the upstream process and the total cost includes the LTEs used by the nodes to transmit traffic and the hub to receive traffic, which is n r + n r = 2n r. Let us consider downstream now. The total amount of traffic units collected at the hub is nr. Since traffic bifurcation is allowed, the minimum number of wavelenths can be used to pack all the traffic, denoted by nr, which is also equal to the number of LTE ports required by the hub to transmit and by each node to receive. Each node employs a tap-andcontinue function which splits the optical sinal and receives a small portion of power that is just enouh to be detected and leave the rest of power to continue propaatin on the rin. Such a power splittin function, performed by optical splitters, enables a broadcast service to be fulfilled by usin only n + 1 LTEs, one required at the hub and one required at each receivin node. Thus, broadcastin all of the traffic, which requires nr wavelenths, will use a minimum cost of (n + 1 nr LTEs for the downstream traffic delivery. To sum up, the resources consumed in both the upstream and downstream direction result in an overall minimum cost of 2n r nr +(n+1 LTEs to achieve all-to-all communication. B. Application of Network Codin When usin network codin, it is obvious that the hub is a perfect place to perform network codin, since all data has to be delivered to the hub first and then converted into electronic sinals for roomin, and hence no additional LTEs will be needed for O-E-O conversion to perform network codin. Therefore, the encodin operation is performed at the hub and the decodin is done at each node. We can also consider this problem in the upstream and downstream contexts. Since upstream is unicast and no network codin is needed, the number of LTEs required in the upstream process remains the same. In order to save sub-wavelenth channels in the downstream data delivery, we use the followin codin scheme. Since each node needs to receive data from different n 1 nodes, then without implementin network codin but usin splitters, each node has to receive all the data units from the hub, which is denoted by nr, in order to achieve minimum network cost. However, if a node receives linear combinations of the traffic instead of the oriinal data, only n 1 linearly independent combinations are needed. By countin its own data in, each node has n linearly independent combinations,

4 4 from which oriinal data of all other nodes can be decoded, iven the codin coefficients are known. This is the basic idea for usin network codin to save lihtpaths and hence LTEs. Fi. 2. a + b b + c a A (a+b, b+c Hub a + b b + c b B C a + b b + c c OXC Application of network codin in downstream in a sinle-hub rin The example shown in Fiure 2 illustrates how to use network codin in the downstream process. We suppose that the hub has received the traffic from nodes A, B and C whose data units are denoted by a, b, c, respectively. The roomin factor is 2 and transmission rate at each node is 1. Hence, each wavelenth is able to accommodate the traffic transmitted by two nodes. Instead of sendin all the traffic a, b and c to each node on the rin, the hub encodes the data and enerates code words a + b and b + c usin modulo 2 addition and broadcast them. Hence, node A will have combinations a, a+b and b+c, where a a + b b + c = a b c Therefore, the coefficient matrix shown above has a full rank such that a, b and c can be decoded from the combinations. Followin the same method, node B and C are also able to obtain all the oriinal data a, b, c. Apparently, this codin scheme can be applied to the n-node case where the hub needs to enerate n 1 linearly independent combinations which are also independent from all raw data units and broadcast them to each node. Hence, in the case where the number of nodes is n and the traffic rate associated with each node is r, the cost of upstream transmission does not chane from the case without network codin, which is denoted by 2n r. In the downstream direction, however, the total traffic that the hub has to deliver is reduced to (n 1r which requires (n 1r wavelenths. The total cost of LTE ports in the downstream is (n + 1 (n 1r. Therefore, the overall cost savins by the application of network codin in a sinle-hub rin with all-to-all traffic demand is (n + 1( nr (n 1r 0. The savins can be either in LTEs or network bandwidth, dependin on the specific network scenarios as indicated above. C. Multiple Many-to-Many Groups In practice, the most common applications which use the all-to-all service mode are multimedia conferences and cooperative processin. Usually, there is more than one multimedia LTE conference roup simultaneously in the network. Thus, it is essential to consider multiple roups. In this scenario, the network nodes are divided into multiple roups, and within each roup, nodes enae in all-to-all communication, while the traffic rates can be different in different roups. 1 Disjoint Groups: The definition of disjoint roups is straihtforward - it refers to the case in which different roups do not share any common node. The problem of optimizin the network cost in a disjoint roup is stated as follows: minimize the number of LTEs used in a sinle-hub unidirectional rin, iven m disjoint roups where each roup i (1 i m has n i (n i 2 nodes and each node has to transmit r i 1 units of traffic to all the other nodes within the same roup. Let us consider the case without network codin first. The process is similar with the all-to-all case, which includes upstream and downstream process. The analysis still beins with the upstream process. Every roup i is independent from each other such that the total number of LTEs used is the sum of LTEs consumed by each roup individually. Followin the analysis in Section III.A, for each roup i, the upstream consumes 2n i r i LTEs, resultin in m 2n i r i in total. In the downstream direction, the hub first rooms the traffic from the same roup toether. For roup i, the number of wavelenths used to carry the traffic is rini rini, and of them are filled up and sent back to the nodes belonin to the same roup directly. Every lihtpath used here is fully loaded and carries data for only one roup. Thus, we only need to pay attention to the remainin portion of areate traffic for each roup that cannot fill up a sinle wavelenth if it exists, since it may need to be roomed with the traffic from other roup(s before delivery in order to save the lihtpaths and hence LTEs. Each roup has at most one piece of such traffic. This piece of traffic of roup i is equal to n i r i n ir i. We denote this portion of the traffic by p i, where 0 p i <. Thus, the problem can be stated formally as follow: Problem GMP: Given m pieces of traffic, each of them with p i units and has to be received by the correspondin n i nodes, room and multicast the traffic at the hub such that the total LTE ports used is minimized. 2 NP-completeness: In the optimal solution, any p i must not be bifurcated and packed into more than one wavelenth. We can prove it by contradiction. Assume that one piece p i is split and assembled into two different wavelenths, which costs each node of roup i two LTE ports to receive. This results in 2n i LTE ports in total. If the hub uses a separate wavelenth to send p i, it only takes n i LTEs at the nodes and 1 more LTE at the hub. Since 2n i (n i + 1 = n i 1 0, it means that any optimal solution can be transformed to be the solution without traffic bifurcation for the problem. Therefore, this problem turns into a special case of eneral traffic roomin problem in a rin network, which has been proven to be NP-complete by reduction from Bin Packin problem in polynomial time in [1]. 3 Solutions without Network Codin: We now analyze the minimum network cost of many-to-many roups communication without employin network codin. The network cost of upstream transmission for n roups has been derived in 2, which is m 2n i r i. The cost of downstream transmission

5 5 consists of two parts. The first part is the number of LTE ports used for broadcastin the lihtpaths that are fully loaded for each individual roup; the second part is the number of LTEs used for transmittin all the remainin traffic p i. Since problem GMP is equivalent to the Bin Packin problem, which has been solved by many approaches in the literature, we consider two methods to obtain the minimum wavelenths in GMP. The first method is a heuristic alorithm based on First Fit Decreasin (FFD [29]; In the second one, it is formulated by usin Inteer Linear Prorammin [30]. In the downstream transmission, for each roup i, n ir i wavelenths are fully utilized to pack the data, which takes (n i +1 niri LTEs. To sum up, the total number of LTEs used to transmit this portion of the traffic is m (n i + 1 niri. We use W to denote the number of wavelenths used at the hub to accommodate the p i units of traffic of all the roups, which can be solved by either the heuristic or ILP described above. Hence, we need W LTEs at the hub to transmit and 1 LTE for each node of roup i to receive p i if it exists, which is determined by a binary number, niri niri. Therefore, the total cost of transmittin all the p i data units is equal to W + m ( n ir i n ir i n i. Thus, the total network cost in both upstream and downstream for m roups without network codin is: m (2n i r i + (n i + 1 n ir i + ( n ir i n ir i n i + W. 4 Solutions with Network Codin: To apply network codin to sinle-hub rin networks, the hub acts as an encoder and enerates n i 1 code words for each roup followin the same codin scheme described in Fi.2 with coefficients from GF (2. The combinations of oriinal data has the same rate as the oriinal traffic, which is r i. Thus, network codin can reduce the total traffic broadcast rate for each roup i by r i. The upstream transmission consumes the same amount of resources as in the case without network codin, but the total traffic rate turns out to be (n i 1r i units for each roup i in the downstream process. Followin the same computation rules used above, the network cost in this case also includes two parts. The first part is denoted by m (n i+1 (ni 1ri. In the second part, let W denote the minimum number of wavelenths used at the hub to pack all the remainin traffic p i from each roup after network codin. Combinin the cost spent in both upstream and downstream process ives us the total number of LTE ports with the application of network codin in a many-to-many traffic scenario, which is expressed as: m (2n i r i + (n i + 1 (n i 1r i + ( (n i 1r i D. Non-disjoint Groups (n i 1r i n i + W. Intuitively, non-disjoint roups denote that in a network, communication roups are not node disjoint and hence some nodes may belon to multiple roups. It is clear that in sinle-hub rin network, non-disjoint roup scenario is the eneralization of disjoint roup case. Therefore, it follows that the traffic roomin problem with non-disjoint roups is also NP-complete by reduction to the special disjoint roup case. Since the network cost is exactly the same for both cases with and without network codin in the upstream direction, we will just focus on the cost analysis of the downstream process. Assume after the hub rooms the data sent from the same roup, there may be a remainin traffic p i for each roup i that can not be able to fill up a wavelenth at the hub and the hub will room them toether. The difference between this case and the node-disjoint roup case is that by roomin the p i of the roups that share some common nodes toether, it saves some LTEs at the common receivin nodes of these roups. Therefore, the hub may have preference to room the p i of the roups that have more common nodes toether. In order to achieve the optimal solution of the total number of LTE required in downstream direction, we formulate the problem by usin inteer linear prorammin. Given the network topoloy, traffic rate and number of nodes of each roup, as well as the number of nodes shared by any pair of roups, we could achieve the optimal solution for both cases with and without usin network codin. The only difference between the two cases is the remainin traffic p i of each roup i. Without employin network codin in the network, for each roup i, the total number of sub-wavelenth circuits needed to broadcast at the hub is denoted by: tr i = N k=1 mi k r i, i M, where N is the total number of nodes, M denotes the total number of roups and binary m i k is equal to 1 if node k is a member of roup i. Whereas in the case where network codin is applied, the total number of circuits for each roup i collected at the hub is equal to tr i = ( N k=1 mi k 1r i, i M. Accordinly, for both cases, the remainin portion of traffic flow for roup i is denoted by: p i = tr i tr i, (0 p i <. Therefore, the total number of LTEs required at the hub and nodes for remainin traffic communication can be obtained by solvin the followin ILP: Symbol W p w i h w s w i,j ns i,j Meanin the maximum number of the wavelenths used in the formulation the remainin circuit p i from roup i that is allocated into wavelenth w at the hub a binary variable that equals 1 if a LTE is used at the hub to transmit the remainin traffic carried on wavelenth w a binary variable that equals 1 if wavelenth w carries circuits for both roups i and j the number of nodes shared between roup i and j, a constant iven in the problem TABLE I VARIABLES AND PARAMETERS IN THE ILP FORMULATION FOR DOWNSTREAM PROCESS IN SINGLE-HUB RING

6 6 Objective: W M W W M M Minimize h w + p w i n i Subject to: w=1 w=1 h w 1 M w=1 j>i s w i,jns i,j M p w j, w; (1 M p w i p i, w; (2 W p w i = 1, i; (3 w=1 p w i + p w j 1 s w i,j 1 2 (pw i + p w j, i < j M, w W. (4 The objective function consists of three terms. The first term denotes the total number of LTEs required at the hub. the second term sums up the number of LTEs required for all the nodes of roup i to receive p i without considerin node sharin. It means that we may count a LTE twice if a node k belons to both roup i and j and p i and p j happen to be roomed in the same wavelenth. Therefore, in the third term, we subtract the portion of the cost counted redundantly. Constraint 1 ensures that if wavelenth w is used to accommodate traffic of any roup, the correspondin lihtpath should be set up at the hub. The wavelenth capacity constraint is satisfied by equation 2. Constraint 3 makes sure that each remainin traffic p i should be allocated into a wavelenth. Constraint 4 ensures that if the circuits of roup i and j are roomed into the same wavelenth w, then s w i,j should be 1. Therefore, addin up the LTEs cost obtained from ILP to the network cost of provisionin the rest of the traffic will yield the total cost in this downstream process, represented by M (n i + 1 tr j + LT E ILP. where LT E ILP denotes the number of LTEs obtained by solvin the ILP for the remainin traffic communication. IV. COST ANALYSIS IN UN-HUBBED UNIDIRECTIONAL RINGS Followin the same sequence of the previous section, we will first investiate the traffic roomin problem in an unhubbed rin with all-to-all traffic and then eneralize it to many-to-many roup communication. All the assumptions made in the sinle-hub rin case remain except that a hub is not used. A. Uniform all-to-all Traffic The problem can be defined as follows: iven a roomin factor and a roup of n nodes, each of which has r units of traffic, find the minimum number of LTEs required to fulfill all-to-all communication. First, the lower bound and upper bound of the special case of unitary traffic, i.e. r = 1, can be derived if optical splitters are allowed. Each node needs at least n 1 LTEs to receive and one LTE to send traffic. Thus, the lower bound of network cost is (1 + n 1 n. However, if no traffic is roomed, the maximum number of LTEs requested is equal to n 2, which can be considered as the upper bound on the cost of this special case. For the eneral case of arbitrary r, we propose a multihub approach to serve all-to-all demands while minimizin the total number of LTEs. The approach can be done in two steps: 1 Divide the nodes into a number of sub-roups such that the areated traffic of each sub-roup is just enouh to fill up a wavelenth; 2 Choose one node of each sub-roup as a hub to room the traffic from other roup members and broadcast the roomed traffic to all the nodes on the rin. Unlike the situation of a sinle-hub rin where all nodes send traffic to the same hub, in an un-hubbed rin, once enouh traffic is roomed to fill up a wavelenth at a node, then this node will set up a lihtpath and broadcast the data to the other n 1 nodes on the rin. This node is called a hub and we may have multiple hubs on the rin. The number of such hubs is equal to the number of wavelenths to accommodate all the traffic. The minimum number of wavelenths that can be achieved equals nr. In addition, we only consider the case where r <, since if r, the excess traffic (r r mod of each node can fill up separate r wavelenths without traffic roomin. In this case, we only need to consider the remainin traffic, denoted by r r, which is less than. Thus, there is no need to consider the case where r >. We analyze the network cost in two different cases: when is a multiple of r or not. In fact, the case when is a multiple of r is a special case of the other case in terms of the solution. Thus we will only discuss the case when is not a multiple of r in detail. In the case when is not a multiple of r, a wavelenth cannot be filled up without traffic bifurcation. Minimizin the total number of wavelenths used for broadcastin with traffic bifurcation will result in the minimum number of broadcast cycles. However, each traffic split introduces two additional LTEs, because when a piece of traffic is allocated into two wavelenths instead of one, this requires two lihtpaths to carry and results in two additional LTEs (one transmitter and one receiver. Alternatively, if we do not split any traffic, we cannot uarantee that the number of broadcast cycles is at a minimum. Each broadcast uses n LTEs, which means that savin one wavelenth will save n LTEs. Therefore, there is a trade-off between the number of traffic splits and the total number of broadcast cycles. In the multi-hub approach, the minimum network cost is obtained by takin the minimum value of the solutions obtained from the two scenarios where we split and we do not split the traffic. First, we consider the case without traffic bifurcation. Each wavelenth can accommodate traffic from at most r nodes,

7 7 denoted by k. A small amount of bandwidth equal to, kr, is wasted on each wavelenth. Then, the total number of broadcast cycles is n k. Amon them, each of n k broadcast cycles fully utilizes a wavelenth with a small wastae of bandwidth, and each of such cycles requires 2(k 1 + n LTEs for transmittin and receivin, which results in a total n k (2(k 1 + n LTEs for the first n k broadcast cycles. However, the number of nodes remainin in the last cycle is ( n k n k (n k n k where ( n k n k is a binary number to indicate if there are some nodes left in the last cycle whose number is less than k and their areated traffic cannot fill up one wavelenth. Hence, the number of LTEs needed in the last broadcast cycle is ( n k n k (2(n k n k 1 + n. Therefore, the number of LTEs in this scenario is iven by combinin the cost in all broadcast cycles, namely: n k (3n 2k n k 2 + n k (2k n 2n + 2k, k where k = r. This eneral solution can also be applied to the case when is a multiple of r. In the second scenario where traffic bifurcation is applied, the minimum number of wavelenths to accommodate all the traffic can be achieved, which is iven by nr, which also equals the minimum number of broadcast cycle. Let nr = w min. Hence, the number of LTEs employed for a broadcast can by obtained by nw min. However, we know that traffic splittin was used to achieve this due to the assumption that is not a multiple of r. Since each traffic split uses two additional LTEs, the problem of minimizin the total number of LTEs actually turns out to be a problem of minimizin the number of traffic splits in order to room the traffic on the minimum number of wavelenths, w min. This problem has been solved by an iterative alorithm proposed in [1]. In each iteration, three steps are processed: 1 Fill each of w min wavelenths with r r loads. Therefore, the unused capacity left on each wavelenth becomes = r r and the number of nodes whose traffic has not been assined is equal to n = n r w min; 2 We have n < w min and r > from the first step. And then allocate units of traffic of each unassined node to n wavelenths, respectively, to fill them up. 3 As a result, only w min n wavelenths still have units of capacity available. Update w min := w min n, r := r and n = n, and then repeat the three steps until all the traffic is assined. We use the alorithm here to obtain the minimum number of traffic splits in this situation, denoted by sp min, iven the minimum number of wavelenths used. The number of broadcast cycles determines the number of hubs in the rin, which is also equal to w min. If no traffic split happens, collectin traffic at those hub nodes from other nodes before broadcast requires 2(n w min LTEs. However, each traffic split increases the number of LTEs by 2. Thus, the total number of LTEs in this collection process is 2(n w min + sp min. In addition to the LTEs used for broadcastin, denoted by nw min, the total number of LTEs used in this scenario with traffic bifurcation is 2(n w min + sp min + nw min. Therefore, takin the minimum of the two solutions obtained in the two scenarios above will ive us the overall minimum network cost. Thus, the number of LTEs of all-to-all traffic without network codin is: min{ n k (3n 2k n k 2 + n k (2k n k 2n +2k, 2(n w min + sp min + nw min }, where k = r, w min = nr, and sp min is the minimum number of traffic splits obtained by the iterative alorithm proposed in [1], iven w min. B. Application of Network Codin In order to save network cost by performin network codin, a node should be chosen to collect all the oriinal data. Thus, we propose a one-hub scheme in which only one node acts as a hub. The traffic is athered and encoded at this node followin the same codin scheme proposed in Section III.B, where the network context is a sinle-hub rin. The hub can be selected from any node in the rin. Hence, in the upstream direction, every node sendin traffic to the hub consumes 2(n 1 LTEs. n 1 linearly independent code words with traffic rate r are enerated and packed into r(n 1/ wavelenths. In the downstream direction, the minimum number of broadcast cycles can be achieved, which also equals to r(n 1/. Each broadcast costs n LTEs such that the transmission in the downstream direction takes total n r(n 1/ LTEs. Therefore, the total network cost with network codin usin one-hub scheme is: 2(n 1 + n r(n 1/. Thouh one-hub scheme can save LTEs in downstream direction, it uses a few more LTEs in the upstream process compared to the multi-hub approach. Thus, the total number of LTEs consumed in both upstream and downstream directions may not always saved by employin one-hub scheme, dependin on the specific network scenario. However, iven the traffic demands, we can always use the multi-hub approach to solve the problem without applyin network codin. Therefore, by comparin the solutions yielded by the one-hub and multi-hub approaches, we choose the solution of the minimum value. Thus, assumin that LT E multi denote the solution obtained from the multi-hub approach, the total network cost in unhubbed rins with n nodes and all-to-all traffic demands r while applyin network codin is: min{2(n 1 + n r(n 1/, LT E multi }. C. Multiple Many-to-Many Groups We now extend the all-to-all communication to multiple many-to-many disjoint roups on an un-hubbed unidirectional rin. Since no node is shared by more than one roup, there is no common hub bein able to room the traffic from different roups toether. Thus, each roup can be provisioned independently, and the total network cost is the sum of the cost of all roups. Suppose there are m roups in an un-hubbed rin and each roup i has n i nodes with each node in the roup sourcin r i traffic units, for 1 i m. The minimum network cost in

8 8 terms of LTEs can be represented based on whether network codin is employed or not: In the case where no network codin is employed, the total network cost in terms of the number of LTEs is: LT E i multi = m (2k i n i k i 2n i + 2k i, min{ n i k i (3n i 2k i n i k i 2 + n i k i 2(n i w i min + sp i min + n i w i min}, where k i = r i, wmin i = n ir i, and spi min is the minimum number of traffic splits of roup i iven wmin i. In the case of usin network codin, the total network cost in terms of the number of LTEs is: m min{2(n i 1 + n i r i (n i 1/, LT Emulti}, i D. Non-disjoint Groups We also consider roup non-disjointness in an un-hubbed rin network where multiple roups may overlap. Two network scenarios, either employin network codin or not, will be discussed, respectively. We will first consider the case of applyin network codin. 1 With Network Codin: In order to perform network codin on the traffic for a roup, one member node should be chosen to play the role of the hub as in hubbed-rins. The encodin operation remains the same as in the disjoint roups case and the total number of circuit needed to broadcast for each roup i is: tr i = ( k mi k 1r i, where m i k is equal to 1 if node k is a member of roup i. The traffic requires tri wavelenths to accommodate roup i. Amon them, tr i are filled up and the correspondin number of LTEs used to broadcast this portion of traffic for roup i is n i tr i. Let p i = tr i tr i denote the remainin piece of traffic of roup i, and each hub needs to broadcast p i to the nodes within the same roup. Since some nodes are shared between distinct roups, if one node can act as the hub for mutilple roups and the p i of those roups happen to be packed into the same wavelenth, LTE cost is reduced by sharin this hub. Notice that if two roups cannot share a hub node, thouh if their p i are small enouh to be allocated into one wavelenth, we should not do it. The reason is as follow: iven two roups, i and j, a node of roup i needs one LTE to send the p i to the hub node of roup j in order to room the traffic toether, which results in another LTE at the receivin node. Thus, such roomin action results in two more LTEs for an extra delivery. In order to reduce the cost, we do not perform the traffic roomin in such case where two roups have no common node even if their remainin traffic can be packed into the same wavelenth. Since the cost of establishin the fully loaded lihtpaths is fixed, we only need to concentrate on the cost of provisionin p i of each roup i. In order to minimize the number of LTEs used to broadcast all p i, we propose a heuristic alorithm, Hub Sharin Minimization (HSM, to allocate all the p i based on the situation where the nodes are assined to different roups, from which we can calculate how many LTEs can be saved by sharin the hub nodes and receivin nodes, and consequently the total number of LTEs required. Prior to the description of HSM, we introduce some symbols used in the alorithm and the preliminary processes. 1 First, we create a bipartite raph G = (V, E, where V = X Y and X Y =. Partition X represents the roups and partition Y represents the nodes. An undirected link e E is established between i X and j Y if node j is a member of roup i in the rin; 2 Each vertex i X is assined a weiht value equal to p i. The deree of vertex i is denoted by d i and its neihbors, defined as the vertices in Y that connect to it, are denoted by N i ; 3 All the vertices in X will be radually divided into multiple sets. Each set, denoted by S k, (1 k X, is composed of a number of vertices that must have a common neihbor in Y, say j, and their correspondin traffic p i are assined in one wavelenth toether. We use SN(S k to denote the total number of times that the vertices in S k share common neihbors. Let us take an example as shown in Fi.3. If the first set of X is S 1 = {1, 2, 3}, and then SN(S 1 = = 3, since vertex 1, 2 and 3 share a common neihbor B in Y and vertex 2 and 3 also share node C in Y. We say that vertex B is shared twice while vertex C is shared once. Thus, the total number of sharin is equal to 3. We assume that the rin network has been transformed to a bipartite raph G = (V, E described in 1 and the procedure of alorithm HSM is shown in Alorithm 1: Alorithm 1: Heuristic alorithm HSM Input: G = (X Y, E Output: G = (X Y, E k = 1, C = ; while C X do select j Y with max(d j where i N j p i ; if Num(j > 1 then select the vertex j Y with max(sn(n j ; if Num(j > 1 then select the vertex j Y with max( i N j p i ; end end Let S k = N j and C = C S k ; add S k, i S k N i and e E between them into G ; remove S k, i S k N i and e E connected to them from G; k++; end Basically, heuristic HSM is a reedy based alorithm and the idea behind it is to divide all the roups into several sets, and each set represents a combination of roup(s that room their traffic p i into the same wavelenth and share the same hub node. For each roup set S i, the alorithm tries to choose the roups that share the most number of nodes such that the reatest number LTEs can be saved at both hubs and receivers. which is equal to SN(S i. The example shown in Fiure 3 illustrates how HSM works

9 9 and how LTEs can be saved. Fiure 3 shows the bipartite raph G mappin from a rin topoloy, in which we assume there are five roups represented by the vertices 1, 2, 3, 4 and 5 in partition X and each p i is indicated by the value above the vertex. The vertices in partition Y represent the network nodes. Both solid and dash edes show which node belons to which roup in the rin. In the first iteration, vertex B Y is picked with the larest deree, since this node is shared by the most number of roups. Since p 1 + p 2 + p 3 = 7 <, roups 1, 2 and 3, defined as set 1, room their traffic toether in one wavelenth at the common hub node B and the total number of LTEs saved by selectin this set is equal to SN(S 1, which is 4. Then the set 1 is separated from G by removin two dash lines, (3, G and (4, D. In the second step, vertex F is picked and the vertices 4 and 5 are rouped as set 2. Since node F is the only common node for them, only one LTE is saved at this node. Thus, the bipartite raph G is divided into two roup sets and the total number of saved LTEs is equal to four. A B C D 4 5 E F G Fi. 3. An example of the procedure of HSM in which = 8 We now derive the time complexity of this alorithm. We assume that the number of roups is denoted by X whereas the number of nodes is denoted by Y and each roup has a constant number of members and each node is shared by a constant number of roups. Hence, each N i and d i is constant number. Thus, the complexity of the alorithm is dominated by the computation in line 3, since line 4-13 take a constant computation time, denoted by c. The worst scenario is when no roup shares any common node and the computation cost of line 3 is Y and the total number of iterations in the while loop is X. Therefore, the complexity of alorithm HSM is O( X ( Y + c = O( X Y. If we denote total number of saved LTEs obtained from HSM by h s, we can then derive the total number of LTEs in both upstream and downstream processes: In the upstream process, i 2(n i 1 LTEs are required to collect the data at the hub of each roup. In the downstream process, without considerin roup sharin, each roup i uses n i tr i to broadcast the encoded data, which costs i n i tr i LTEs in total. However, sharin member nodes amon roups will save h s LTEs iven by the heuristic HSM. Therefore, the total number LTEs used in this scenario is iven by: 2(n i 1 + n i tr i h s, where tr i = ( m i k 1r i. i k 2 Without Network Codin: In the scenario where no codin is employed, we can use the same approach employed to derive the cost for node disjoint case without application of network codin, described in Section.IV.C, to obtain the X Y cost for this scenario. The only difference between two cases is that we can further reduce the cost of LTEs in this scenario by sharin the common hubs amon distinct roups if they have common nodes by usin the alorithm HSM to allocate all the remainin traffic p i for all the roups, where p i is derived as follows: For any roup i, In the case where no traffic bifurcation is applied, p i = r i (n i k i n i k i, where k i = r i. In the case where traffic bifurcation is applied, p i = tr i tri, where tr i = k mi k r i. Thus, we assume the number of LTEs saved by sharin the common hubs amon different roups is denoted by h s and h s, solved by HSM, for the cases without and with traffic bifurcation, respectively. By subtractin h s and h s from the costs in node disjoint roup case for both scenarios and takin the minimum value, we obtain the number of LTEs required for the many-to-many communication in this nondisjoint roup case: m min{ ( n i (3n i 2k i n i 2 + k i k i n i (2k i n i 2n i + 2k i h s, k i k i m (2(n i wmin i + sp i min + n i wmin i h s} where k i = r i, wmin i = n ir i, spi min is the minimum number of traffic splits of roup i iven wmin i. V. NUMERICAL RESULTS Based on the theoretical solutions obtained above, we compare the results of two different cases - with or without network codin - in various network scenarios and under different traffic conditions. Since all-to-all communication is a special case of multiple many-to-many roup cases where the number of roups is equal to 1, in this section we consider more eneral cases in which m 1. Each network scenario is enerated accordin to the followin assumptions: 1 The number of many-to-many roups in the rin network is fixed at m = 10; 2 The number of nodes per roup is lower bounded by 2 and upper bounded by a iven positive inteer value n max and is uniformly distributed in [2, n max ]; 3 The traffic rate of each node is lower bounded by 1 and upper bounded by a iven positive inteer value r max and is uniformly distributed in [1, r max ]; A. Sinle-hub Rin Since network cost in the upstream direction in sinle-hub networks is always the same whether or not network codin is used, we only compare the cost factors in the downstream direction. Table II shows the heuristic and exact solutions of the number of LTE ports in different network scenarios. Three