ILP Formuation and K-Shortest Path Heuristic for the RWA Probem with Aocation of Waveength Converters Karcius D.R. Assis 1, A. C. B. Soares, Wiiam F. Giozza and Heio Wadman 1 UEFS- State University of Feira de Santana- DTEC-Av. Universitária, s/n-km da BR 116, Camus Universitário, CEP:441-46 UNIFACS- Savador University - Rua Ponciano de Oiveira, 16, Rio Vermeho, CEP:4195-75 Savador-BA, Brazi.E-mai: {karcius, giozza}@unifacs.br UNICAMP- State University of CaminasAv. Abert Einstein, 4, CP. 611, Caminas-SP, Brazi.E-mai: wadman@decom.fee.unicam.br Abstract Different integer inear rogramming (ILP) formuations have been roosed for the routing and waveength assignment (RWA) robem in WDM networks. An imortant goa in the design of WDM networks is to use the minimum number of converters to serve communication needs, because this is cost effective. In this aer, we study the robem of aocating the minimum number of converters in a network for soving the RWA robem. Furthermore, our formuation aows any kind (artia or fu, sarse or ubiquitous) of waveength conversion in the network. We aso roose an exerimenta comarison of the heuristic from [Swaminathan and Sivarajan ] with a K-shortest ath heuristic with imited number of converters, for a arge network. 1. Introduction The design of an otica network invoves three cosey reated tasks: dimensioning, routing, and waveength assignment. Given is a hysica tooogy of the network as we as, for every air of nodes, a demand for a number of ightaths to be estabished between these nodes [Koster and Zymoka 5], [Subramaniam, S. and Azizogu, M., 1996] Otica networks design draws an increasing amount of attention nowadays. The Waveength Division Mutiexing (WDM) technique sits the arge bandwidth avaiabe in otica fibers into mutie channes, each one oerating at different waveengths and at secific data rates (u to 4Gbs). The ightaths are ogica channes which rovide an end-to-end connectivity in the a otica network [Ramaswami and Sivarajan 1998]. The erformance of the WDM networks can be imroved by aowing waveength transation (conversion) at the routing nodes [Swaminathan and Sivarajan ], [Ramaswami and Sasaki 1998], [Ramamurthy and Mukherjee 1998]. The key issue in the a-otica network design rocess once the hysica reaization has been achieved, is to roery dimension the network with resect to waveength numbers and to determine a waveength aocation an [Swaminathan and Sivarajan ], [Murty and Gurusamy ].
For circuit switched ightath service on an otica network, each ightath wi carry traffic reated to ony one source-destination air. A session is a set of connections between a source-destination air. For exame (A,B:1), (B,C:) are sessions wherein the number of connections required from node A to node B is one and from node B to node C is two. A waveength-convertibe network, which suorts comete conversion at a nodes, is functionay equivaent to a circuit-switched network, i.e., ightath requests are bocked ony when there is no avaiabe caacity on the ath. Moreover, in most cases, it may be uneconomic to deoy waveength conversion caabiity at a nodes, but having a few nodes with waveength conversion caabiities may be desirabe [Murthy and Gurusamy ]. Then the questions are (1) How many nodes in a network shoud have conversion caabiity? () How many converters shoud a node have? In this aer we extend a heuristic from [Assis and Wadman 4] and focus on question (). In order to rovide an answer, the key oint is the formuation of the weknown Routing and Waveength Assignment-RWA robem in networks with sma number of converters er node, as a matter of fact: whie revious iterature considers conversion with imitess/arge number of converters in nodes [Swaminathan and Sivarajan ], [Ramaswami and Sivarajan 1995]; our strategy considers modifications in RWA robem to aow aocation of a imited number of converters in a node of the network. It is imortant because this number, if sma, is cost effective. Furthermore, our inear formuation and K-shortest ath heuristic aow any kind (artia or fu, sarse or ubiquitous) of waveength conversion in network. The aer is organized as foows. Section defines the traditiona RWA robem, extending an auxiiary grah aroach for a imited number of converters, and we discuss our resuts for a ring tooogy. Section, we discuss our resuts for a mesh tooogy and exain over the comutationa comexity. In section 4, we roose a heuristic with imited number of converters for arge networks. The heuristic is executed on a random mesh Nationa Science Foundation Network (NSFNET) and the erformance is comared with K-shortest ath heuristic without imited number of converters from [Swaminathan and Sivarajan ]. Finay, section 5 resents our concusions.. Routing and Waveength Assignment Probem The hysica tooogy network is reresented as G(N,E,W), in which N reresents the sets of nodes, E reresents the set of directiona fibers, and W reresents the set of waveengths on each ink. The hysica tooogy and the traffic matrix are given as inut for the robem. Our objective is maximize the number of ightaths to be estabished from the traffic matrix. Since we try to maximize connections in a given session or traffic matrix, for a fixed set of waveengths, it is caed Max-RWA robem [Swaminathan and Sivarajan ], [Ramaswami and Sivarajan 1995]. The agorithm that soves the above robem in genera shoud: a) Maximize the number of ightaths estabished using the minimum number of waveengths. In addition, our formuation tries to maximize the number of ightaths estabished using the minimum number of converters (If conversion is avaiabe). The foowing assumtions are made in our RWA robem: a) the number of waveengths in each ink of the fiber is assumed to be
same b) each ca requires a fu waveength on each ink of its ath c) simex connections are considered..1 Linear Program We formuate the Max-RWA robem as a Mixed Integer Linear Program (MILP). In the formuation the aths for a connection are not secified before hand, the inear rogram sover is aowed to choose any ossibe ath and any ossibe waveength for a source destination air. Thus the ogica tooogy design and waveength assignment are inbuit in the formuation itsef, which is not in the formuations resented in [Ramaswami and Sivarajan 1996], but is simiar to the resented in [Swaminathan and Sivarajan ]. The constraints invove the edges or arcs of the network. By this aroach the soution for the RWA robem tends to otimaity.. Definitions We use the foowing notation: i and j denote originating and terminating node of a ightath, resectivey; m and n denote endoints of a hysica ink. The arameters of the formuation are: N = Number of nodes in the network. K= The traffic matrix, i.e., K is the number of connections that are to be estabished between node i and j. P mn denotes the existence of a ink in the hysica tooogy. Here, P mn = 1, then there is ony a fiber ink between nodes m and n, otherwise P mn =. W denotes the number of waveengths the fiber can suort. C ( ): set of the waveengths into which can be converted in node. D ( ): set of the waveengths that can be converted to in node. The variabes of the formuation are: b denotes the number of ightaths estabished between node i and node j, taking ositive integra vaues. The mn variabe denotes the number of ightaths between nodes i and j being routed through fiber ink m-n. L is the maximum number of waveength channes suorted er fiber; c is the number of ightaths between node i and node j that start in the waveength ; for =1,,,...,W; d is the number of ightaths between node i and node j that finish in the waveength ; for =1,,,...,W. Furthermore, mn = 1, if the ightath between node i and j uses waveength through hysica ink m-n.. Max-RWA: Origina Mathematica Formuation Objective: Maximize: i j b (1) Remark: The objective here is to maximize the number of connections to be estabished from the given traffic matrix Routing on hysica tooogy mn : mk = kn, if k i, j () m n in = b i, j () n
mj = b i, j (4) m mn L. Pmn m, n (5) b K i, j (6) Remark: These constraints ensure that demand at origin and destination node for a source-destination air (i,j) is satisfied. On cooring ightaths with conversion resources: in = c i, j, (7) n mj = d i, j, (8) m c = d = b i, j (9) Remark: These constraints aow a ightath to be started in waveength, and finished in another waveength. m n m n nt n t C ( ) mt m t D ( ) if i,j (1) if i,j (11) Remark: conversion in intermediate nodes, exained ater. = mn i, j, m n (1) mn, mn Pmn m, n, (1) Remark: Constraints (1) guarantees that the number of waveengths resent in each hysica ink is equa to the number of ightaths traversing it. In (1) we assure that there is no waveength cash at hysica ink. Int b, mn, c, d Bin mn..1 Exanantion of (1) and (11) In a node with resources of conversion, equations (1) and (11) guarantee that a waveength that arrives in node in coor can be converted to another waveength in accordance with definition of the C ( ) and D ( ) sets. Notice that if conversion is not aowed (fig. 1b) in node =4, then: Therefore, for = 1,, : m C ( ) = D ( ) = {}, nt C ( ) = and m n
n = mt D ( ) n n m = m n. m 1 5 1 i 4 6 j b a 7 c 1 Figure. 1. (a) Otiona hysica inks that arrive and eave node 4 for a ightath from node i to node j, (b) no conversion, (c) artia conversion. This ast equaity exresses a fow conservation of -coored aths through node that is vaid when conversion is not aowed. Now, with artia conversion at node =4, as iustrated for the fig 1c. Then, C 4 ( 1 )={ 1, }, D 4 ( 1 )={ 1, } and C 4 ( )={, }, D 4 ( )={ 1, } and C 4 ( )={ 1, }, D 4 ( )={, }. Aying 1 and 11, we get: 14 1 + 4 1 + 4 1 = 1 45 + 46 1 + 47 1 + 45 + 46 + 47 14 + 4 + 4 = 45 + 46 + 47 + 45 + 46 + 47 14 + 4 + 4 = 45 1 + 46 1 + 47 1 + 45 + 46 + 47 Through the exame above we evidence that the definition of the sets C ( ) and D ( ) aows any kind of conversion in network: artia, fu, sarse, ubiquitous etc..4 Number of converters In fig., in order to aow the secification of a imited number of converters, a node with resources of conversion is sit into two auxiiary nodes a and b. After, it is created one unidirectiona arc a-b. If we woud ike to have one converter, it is created one more auxiiary node c 1 and two more auxiiary arcs a-c 1 and c 1 -b. If we woud ike to have two converters, it is created one more auxiiary node c and two more auxiiary arcs a-c
and c b and so on. Note that in auxiiary grah constraints arcs from or to c i have oad L=1 and arcs from a to b don t have traditiona cash constraints. Therefore (5) and (1) must be reaced by two other equations to guarantee the success of the strategy. These two new equations wi be shown in the next subsection. L ac 1 =1, L ac =1...L aci =1 L C1 b =1, L C b =1...L C i b =1 C i C E 1 C 1 E 1 E 1 E E a b E E a mn P mn b E E (a) (b) (c) Figure.. (a) The origina grah with back nodes (with resources of conversion) and bi-directiona arcs. (b) Back node is sitting in two nodes (a and b), (c) A auxiiary grah with auxiiary nodes a, b and c: a is incoming of arcs, b is out of arcs and c is an auxiiary node that reresents a converter..4.1 Auxiiar Grah (New formuation) In fig., a, b and c i denote nodes from the auxiiary grah (for estabished number of converters N c with i=1,...n c ). Therefore, if the auxiiary grah is made, i.e., there is the secification of a imited number of converters. Thus, we reace constraints (5) and (1) by the foowing more restrictive equations and in addition, we define the Nc. L. P for (ink m-n) (ink a-c i ) or (ink c i -b) mn (14) mn 1 for (ink m-n) = (ink a-c i ) or= (ink c i -b) mn Pmn for (ink m-n) (ink a-b) (15).4. Comexity In the formuation with auxiiary grah, based in the strategy of fig., with sarse conversion in one node, the number of nodes grows from N to N+, for 1 converter. From N to N+ for converters and so on. Therefore, the number of nodes grows from N to N+N c +1 nodes, when N c converters are ut in one node of network. Simiary, the number of inks E grows from E to E+, for 1 converter, from E to E+5 for converters and so on. Therefore the number of inks grows from E to E+1+.N c when N c converters are ut in one node of the network. If the same number of converters are ut in a N nodes of network, then the number of nodes in formuation grows from N to
N+(N.N c ) and the number of inks grow from E to E+(1+.N c ).N. Issues, ike the number of routing variabes O(N.E.W), where there are W waveengths, wi become critica in anayzing scaabiity. However, in the network ractica design the cost is minimized when the minimum number of converters is used [Yiming and Oiver ], [Assis and Wadman 4]..4. The sime exame (Nc is the number of converters in one node with conversion resources) 6 5 1 4 W =4 W =4 (a) (b) (c) Figure.. (a) Nr=6 and (b) Nr=1: two unidirectiona rings (cockwise direction). Back nodes-conversion. Cear nodes-no conversion, (c) Traffic matrix (1 connections). Notice that with the aication in formuation/strategy with N c = in one node, fig. b, we wi obtain 1 connections, what roves the efficiency and cost saving of the formuation. Notice in tabe 1, that in auxiiary grah, the comutationa cost increases, see revious subsection, and is given by: Node cost = N+Nr.(1+Nc), Link cost: = E+Nr.(1+.Nc), where Nr is the number of nodes with conversion resources.. Simuations Tabe 1: Comutationa cost for auxiiary grah Ring Nc Node cost Link cost comutationa (b) Nr=1 (a) Nr=6 1 8 9 9 11 1 1 4 11 15 1 18 4 4 6 48 4 6 6 An exhaustive aroach that enumerates a the ossibe ways of converter number acement and choosing the best one is not efficient for moderate arge networks. In this section, we roose simuations in the foowing way:
1. To aow fu conversion in a nodes of the network in the origina grah, to ay the traditiona formuation and to get the maximum number of estabished connections. This way, the RWA suose that a node has imitess or great number of converters in each node of the network, in agreement with the hysica-out-degree of this node. For exame, in a node with hysica-out-degree δ and W waveengths, the number of converters Nc wi aow Nc = W.δ (as it is aied for a nodes, we have ubiquitous conversion). This ste serves ony for comarison.. Using the auxiiary grah (roosed strategy), to ace Nc=1 in a nodes of the network. To ay the modified formuation and to verify the estabished number of connections.. If the number of estabished connections in revious ste is not same the number obtained in ste one, return to the auxiiary grah (ste ) with Nc=, and so on. Notice that when we use the auxiiary grah, the connections aways have their origin in auxiiary nodes b and aways finish in auxiiary nodes a. For exame (fig. 4b), connections that are initiated in node 1 in the origina grah, are initiated in the node b in auxiiary grah and connections which are finished in node 1 in origina grah, are finished in the node a in auxiiary grah. c i 1 a 1 b 6 5 4 6 5 4 (a) (b) Figures. 4.: (a) 6 node-mesh tooogy (b) auxiiary grah for 6 node-mesh tooogy. are finished in the node a in auxiiary grah. Tabe. Traffic matrix for 6 node mesh network 7 7
.1 Numerica Resuts for 6 node-mesh tooogy Simuations have been carried out to investigate the erformance of the roosed strategy with auxiiary grah over an 6-node mesh tooogy (Fig.4). The traffic matrix used by this network is shown in tabe. The tota number of connections to be set u is 9. The resuts with Linear Program are tabuated in tabe for the cases of No conversion/origina grah. Fu conversion with Nc=1/auxiiary grah and Sarse conversion/auxiiary grah with one or two nodes with Nc=1. In tabe, the number of connections obtained with Nc = 1 in a nodes is same to the number obtained with ubiquitous conversion with Nc imitess, for a waveength ans; therefore, the atter is not shown and the ste is run ony one time for each an. However, a more effective strategy is to ut converters ony in one/some (sarse conversion) nodes of the network; in this case the acement order of converters is the foowing: first, one converter is ut on node 1; if the number of estabished connections is ess than in the case of ubiquitous conversion with Nc=1, then another converter is ut on node, and so on. In tabe with sarse conversion, for W=, W= and W=6 with ony one converter in node 1 we wi estabish the same number of connections from Nc=1 in a nodes, and for W=4 and W=5 with converters: one converter in node 1 and one converter in node we wi aso estabish the number of the connections from Nc =1 in a nodes. However, for W=4 and W=5, if we just one converter in node 1, we wi estabish 1 and 6 connections, resectivey; so the tota number of connections can not be estabished. Therefore the strategy roosed can be used to guide the acement of converters at the design of a network with RWA using the minimum number of converters, as seen in the figure 5. W Tabe. Lightaths estabished x Waveength Number of ightaths estabished No Conversion Conversion Sarse Conversion Nc=, for a Nc=1,for a Node 1 4 5 6 1 11 15 16 4 5 5 7 6 8 9 Nc 1 11 Nc 1 16 Nc 1 1 Nc 1 1 7 Nc 1 9
5 Sarse conversion, Nc=1 for one/some nodes. Nc=1, for a nodes. ubiquitous conversion. Tota Number of Converters 4 1 4 5 6 Number of the Waveengths Figure. 5. Resuts for 6-node mesh Network. Tota number of converters (Nt). Nt = Nc for estabished connections from tabe III in function of waveength an. In ubiquitous conversion, W.δ, for each node..1. Some comments Integer Linear Programming modes are ouar in the iterature as they rovide forma descritions of the robem. In ractice, however, scaabiity to networks with at east 1 s of nodes, with 1 s of demands is required. In many cases a but trivia instances of theses ILP s are comutationay difficut with current state-of-the-art software. The comexity of our formuation grows as demonstrated in sub-section (.1.4.). In our 6-node mesh network, on average our strategy run be the otimization software CPLEX [ILOG CPLEX ] took around five seconds on an Inte Pentium IV/1.6Ghz. However, in a arge network as the Nationa Science Foundation Network - NSFNET, with matrix of traffic given for [Swaminathan and Sivarajan 1998], and Nc=1 for a nodes, our strategy exceeded the CPU memory restriction. Thus, in next subsection a heuristic wi be deveoed to find soutions to robems with size tyicay found in ractice. 4. K shortest ath Heuristic with imited number of converters (KSPNc) Here we resent a comutationay ess intensive heuristic agorithm based on K shortest aths for soving the Max-RWA robem with imited number of converters. Assuming that in the ightath request matrix, the argest ightath request between any source-destination air is m, we find the K shortest aths, wherein K is greater than m. The agorithm roceeds in foowing stes: Ste 1: Fiding K shortest aths in terms of ho-ength between a source-destination airs in traffic matrix:
The K shortest aths are store in ightaths1, ightaths,, ightathsk arrays. Consider the first shortest ath array i.e., ightaths1 for rocessing and go to Ste. Ste : Waveength assignment to the ightaths: For the ightath which is not waveength assigned in the traffic matrix, choose the ath for that from the chosen K shortest ath array. A tyica ightath between nodes (1) and (N) is reresented as node[1], node[],,node[q],,node[n]; where nodes node[],, node[q] are nodes aong the ightath. The hysica fiber inks of the ightath (node[1], node[]) abeed as ink 1, (node[], node[ ]) as ink and the ast ink (node[n-1], node [N]) as ink N-1. The first hysica ink of the ightath (here node[1], node[]) is taken and scanned for a free waveength. If a waveength j is free, then we try to find in a inks of that ightath for the avaiabiity of the waveength j. Then the agorithm roceeds further, differenty for the foowing cases: i. For the case of no conversion of waveength aong the ightath, if the waveength j is avaiabe in a hysica inks aong the ightath, then we aocate that waveength for the ighath. If the continuity of the waveength j aong the inks in the igthath is not ossibe then, we scan for the next free waveength j on the ink (node[1],node[]). As before the avaiabiity of the waveength j on a the hysica inks on the igthat is checked, if the waveength is avaiabe then it is assigned, ese we scan for the next free waveength on the ink (node[1], node[]) and the above rocedure for waveength assignment is reeated ti the ightath is waveength assigned ot the waveength in the ink (node[1], node[]) is exhausted. ii. For the case of imited waveength conversion, if waveength j is bocked on any ink n, then go back one hysica ink towards the source node of the ighath n- 1 and try to obtain a free waveength by waveength conversion. If waveength conversion is not avaiabe go back further one ink n- and try to obtain a free waveength by waveength conversion. Reeat the above rocedure ti a free waveength is obtained or ink is reched on the ightath. If a free waveength is not avaiabe at ink then go to ink 1 and choose a new free waveength and traverse the hysica inks of the ightath towards the destination node assigning waveengths with or without conversion. Whie back tracking for a free waveength if at any intermediate ink if we get a free waveength after conversion, then traverse from that ink towards the destination node assigning waveength with or without conversion. The waveength conversion aowed at any node deends on the degree of conversion aowed and number of converters Nc aowed. Ste : Reeat the ste ti a the ightaths in the chosen array are exhausted. Ste 4: If any of the ightaths in given traffic matrix is not waveength assigned, then choose the next of the K shortest ath arrays and go to ste. If a the ightaths are waveength assigned or a the K shortest ath arrays are exausted, then sto the agorithm.
4.1 Numerica Resuts The NSFNET shown in figure 6 is a 14 node network with 1 edges. In this network each edge reresents a air of fibers, one in each direction. The traffic matrix that has to be reaized over the NSFNET is shown in the tabe V. We assumed that at most mutie connections were ermitted for a source-destination air. The number of connections are chosen from,1,, with equa robabiity for a source destination air. The tota number of connections to be set u is 68. For comarison with [Swaminathan and Sivarajan ], we assumed: i. A nodes in the network are equied with waveength converters with imited conversion caabiity. Therefore, ony the case ii in ste from heuristic is considered. ii. The degree of conversion is. For he the reader, see fig.1b, it the degree of conversion is. First, we reax the integer constraints of mathematica formuation, as [Swaminathan and Sivarajan ]. For a given waveength, we find LP uer bound. After, we executed the KPSH and KPSNc agorithm on the NSFNET network with the vaue of K=5. The greater the vaue of K, more wi be number of connections reaized, because there are more aternative aths avaiabe for waveength assignment. The KPSNc resuts based on the number of converters are catured in tabe aong with resuts of LP (uer bound) and KPSH Heuristic with arge Nc from [Swaminathan and Sivarajan ]. Comaring in tabe 5, we find that in terms of erformance (Estabished Connections); LP (uer bound), KPSH, KPSNc with Nc=7 and KPSNc with Nc=5 gives better erformance in that order. In LP, this haens because of the reaxed integer constraints. In KSPH, this haens because of arge number of converters. The KSPNc had imited number of converters, therefore the erformance is ess or the same from W=1 to W=. However, in the network ractica design the cost is minimized when the minimum number of converters is used. Then, KSPNc is a heuristic costeffective. 5. Concusions In this aer, we resented ILP formuation and a K-shortest ath heuristic aroach for the imited number of converters and kinds of conversion in otica networks. The formuation/strategy roosed in this aer have a significant imact to the we known RWA robem. First, it can he understand the reationshi between the number of waveengths required and the number of converters. Second, it can be used to guide the acement of converters at the design of a network. Third, a new feature of the roosed formuation is that any kind of conversion can be made in each node of the network. This is obtained by the more genera constraints (1) and (11). By recenty deveoed heuristic technique advancements [Koster and Zymoka 5], the best known soution coud be imroved for more instances with minimum converter waveength assignment, again roving otimatiy. Besides further founding
the benefit of our aroach, this observation aso indicates that the heuristic agorithm are sti imrovabe. References Koster, A.M.C.A and Zymoka. (5) Provaby good soutions for waveength assignment in Otica networks Otica Network Design and Modeing- ONDM 5, Mian, Itay. Assis, K.D.R and Wadman, H. (4) An Efficient Strategy for the RWA Probem with Aocation of Waveength Converters Otica Network Design and Modeing-ONDM 4, Gent, Begium., v. 1,. 551-564. Yiming, Z. and Oiver W.W. Y. () An Effective aroach to the connection routing robem of a-otica waveength routing DWDM networks with waveength conversion caabiity IEEE-Internationa Conference on Communications. ILOG CPLEX version 9., (). htt://www.iog.com/roducts/cex Swaminathan, M.D. and Sivarajan, K.N. () Practica Routing and Waveength Assignment agorithms for A Otica networks with Limited WaveengthConversion IEEE Internationa Conference on Communications - ICC. Murthy, C.S.R and Gurusamy, M. () WDM Otica Networks: Concets, Design, and Agorithms. Prentice Ha, ISBN -1-667-5. Ramaswami, R. and Sivarajan, K.N. (1998) Otica Networks: a Practica Persective, Morgan Kaumann Pubishers, ISBN 1-5586-445-6, San Francisco, USA. Ramaswamy, R. and Sasaki, G. (1998) Mutiwaveength Otica Networks with Limited Waveength Conversion, IEEE/ACM Transactions on Networking Vo. 6. No. 6. December. Ramamurthy, B. and Mukherjee, B. (1998) Waveength Conversion in WDM Networking, IEEE Journa on Seected Areas in Communication vo 16, No 7, 161-17. Banerjee, D. and Mukherjee, B. (1996) Pratica aroaches for routing and waveength Waveength assignment in arge a-otica waveength-routed networks. IEEE Journa on Seected Areas in Communications, 14 (5): 9-98, June. Ramaswami, R. and Sivarajan, K.N. (1996) Design of ogica tooogies for waveength-routed A Otica Networks, IEEE/JSAV, vo. 14,. 84-851, june. Subramaniam, S., Azizogu, M. and and Somani, A.K. (1996) A-Otica Networks with Sarse Waveength Conversion, IEEE/ACM Transactions on Networking, 544-557. Ramaswami, R and Sivarajan, K.N. (1995) Routing and waveength assignment in a otica networks, IEEE/ACM Trans. Networking, vo., n.5,. 489-5, Oct.
Tabe 4. Session Matrix for NSFNET (68 connections) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 Figure 6. NSFNET
Tabe 5. Resut obtained for NSFNET by LP formuation and heuristics Estabished Connections Uer Bound Heuristic W (LP) KSPH KSPNc Nc=W.δ Nc=7 Nc=5 1 198 187 187 18 11 8 196 196 191 1 18 9 7 1 8 18 14 14 8 9 7 4 15 48 8 6 16 58 46 4 9 17 6 5 47 47 18 67 55 5 51 19 68 58 56 58 68 6 59 59 1 68 64 61 6 68 66 65 64 68 67 67 67 4 68 68 68 68 5 68 68 68 68 6 68 68 68 68