Optical Amplifiers Placement in WDM Mesh Networks for Optical Multicasting Service Support

Transcription

1 1 Optcal Amplfers Placement n WDM Mesh Networks for Optcal Multcastng Servce Support Ashraf M. Hamad and Ahmed E. Kamal Abstract The problem of placng the optcal amplfers (OAs) n wavelength-routng mesh networks has been studed n the lterature n two contexts: network provsonng [1] and connectons provsonng [2]. In ths paper, we ntroduce optmal and heurstc solutons for the network provsonng problem. The soluton s based on constructng a multcast forest for each multcast connecton wth the goal of mnmzng the total number of OAs needed n the network, hence reducng ts cost. The optmal soluton s formulated as a Mxed Integer Lnear Program (MILP). On the other hand, the heurstc soluton s obtaned by dvdng the problem nto subproblems and solvng them separately whle takng the nterdependency between these subproblems nto consderaton. The results obtaned from both solutons are compared and they are found to be n good match. Index Terms Optcal Amplfers Placement Problem, All- Optcal Multcastng, Power Aware Multcastng. I. INTRODUCTION ROutng and Wavelength Assgnment (RWA) s a fundamental problem n wavelength routng optcal networks whch has been nvestgated extensvely n the lterature [3]. However, most studes concentrate on the classcal vew of the problem n whch the best routng structure and wavelength(s) are computed for all or some sessons whle optmzng the network throughput, wavelength usage or blockng probablty. Wth ths classcal vew, most solutons do not consder practcal ssues, such as the power loss and nose. The mportance of these ssues stems from the fact that when a soluton may exsts for the classcal RWA problem, t may not necessarly be feasble n practce. We focus n ths paper on the effect of power loss on RWA under All Optcal Multcastng (AOM) scenaro [4][5]. In a nutshell, AOM s about supportng multcast servce n the optcal doman by elmnatng any converson of the transport sgnal between the electronc and optcal domans at the ntermedate nodes. To acheve ths goal, branchng nodes of the multcast structures (called, lght-trees or lghtforests) are equpped wth passve optcal spltters [6] whch are confgured to splt the power strength of the ncomng sgnal nto two or more outgong lnks. Ths operaton scheme has the advantage of: 1) Achevng sgnal transparency wth respect to traffc type, bt rates, and protocol, 2) Smplfyng the logcal network stack structure, and Ashraf Hamad s wth Mcrosoft Corporaton, One Mcrosoft Way, Redmond, WA , USA; Emal: ahamad@mcrosoft.com. Dr. Ahmed Kamal s wth Department of Electrcal and Computer Engneerng; Iowa State Unversty, Ames, IA , USA; Emal: kamal@astate.edu. Ths research was supported n part by the Natonal Scence Foundaton under grants ANI and CNS ) Reducng the cost of the swtchng nodes by elmnatng the extra hardware of sgnal converson. A man source of power loss s the attenuaton due to propagaton n optcal fber; hence called the Propagaton Loss. However, under AOM, the optcal sgnal faces an addtonal source of power loss due to splttng at the branchng nodes of the lght-trees/forests. Ths s called Splttng Loss. Because of ths extra power loss, the tradtonal Optcal Amplfers Placement (OAP) problem becomes more challengng. OAP problem s nvestgated n the lterature n two contexts. In the frst context, namely, Network Provsonng, the problem s formulated as a network desgn problem wth the objectve of mnmzng the network cost. In [1], we ntroduced a Mxed Integer Lnear Program (MILP) that solves the Routng (R), Wavelength Assgnment (WA) and Optcal Amplfer Placement (OAP) subproblems n an ntegrated way for AOM traffc. The network cost n [1] s represented as the total power amplfcaton needed n the network. For the uncast traffc, the authors n [7] addressed the OAP problem n the smpler broadcast-and-select archtecture where no routng s performed. The problem was solved usng Mxed Integer non-lnear Programmng (MInLP) wth the objectve of mnmzng the total number of OAs. The work n [8] generalzed the problem n [7] by ncorporatng dfferent layout topologes (stars, trees and/or rngs) and by takng nto account the fact that the cost of OAs s locaton depend. The proposed soluton s based on smulaton annealng. Whle the studes n [1], [7] and [8] consdered the case of havng unequally poweredsgnals at the entry pont of the OAs, reference [9] solved the equal powered-sgnals nstance of the problem proposed n [7] usng MILP. The second aspect of the OAP problem, namely, Connecton Provsonng, studes the mpact of power constrants on the operaton of already provsoned networks. In ths context, we proposed an MILP n [2] that provdes the optmal RWA solutons for AOM. We also desgned a heurstc algorthm that reles on a specal lnk cost functon that relates the routng decsons wth varous power constrants and produces comparable near optmal solutons. The authors n [10] followed an teratve heurstc approach n whch an ntal tree s modfed by replacng a set of adjacent splttng nodes by a sngle splttng node. Another heurstc algorthm s proposed n [11] that ensures a mnmum sgnal qualty and farness among all destnatons. Ths s acheved usng balanced lght-trees. The goal of ths paper s to nvestgate the Network Provsonng aspect of the OAP. Our study takes two drectons. In the frst drecton, we formulate the OAP problem as a MILP wth the objectve of mnmzng the total number of OAs. The MILP soluton provdes the optmal soluton for the problem

2 2 by solvng all the consttuent subproblems jontly. However, the MILP formulaton cannot provde fast soluton for large nstances of the problem. Therefore, n the second drecton of our study, we solve the problem usng a greedy heurstc that provdes faster, yet near optmal solutons for the OAP problem. The algorthm s based on formulatng lnk and path cost functons that enable us to base the routng decsons of the sessons on the requred number of OAs. The remander of the paper s organzed as follows. We frst defne the OAP problem and the system model under OAM. Ths s followed by the MILP formulaton n Secton III. Secton III also ncludes an extenson to our orgnal formulaton to handle the case of asymmetrc splttng loss at the MC nodes. However, ths extenson results n a non-lnear formulaton. The heurstc s then ntroduced n Secton IV. Some numercal results are presented n Secton V. Fnally, we conclude the paper n Secton VI. II. PROBLEM DESCRIPTION A. Power Constrants and Optcal Amplfer Model There are two man system power constrants we consder n ths paper. Frst, the ndvdual wavelength s power level must be detectable at any pont n the network by ensurng t does not fall below a certan threshold, called P Sen. Second, the total power values of all wavelengths must not ncrease beyond an upper bound, called P MAX. However, an addtonal algorthm-drven power-constrant s consdered n our model n whch all the channels over any lnk must be equally powered. Ths symmetrc constrant does not only smplfy the OAP soluton, t also ensures far utlzaton of the deployed OAs. Ideally ths can reduce the number of OAs by avodng stuatons where delta between nput powers s bg and OAs are saturated by hgh-powered nput sgnal(s) whch results n small gan and yelds to small span between OAs. Also, we use a smple model for the OA gan whch s determned by: G(P n ) = MIN{G 0, (P MAX P n )} (1) where P n represents the aggregate power of the nput sgnal, and G 0 s the small-sgnal gan n db. Both P MAX and P n are n dbm. Assumng flat gan over all the channels, ths gan apples to all nput wavelengths. B. System Model and Assumptons The network s an all-optcal wavelength-routed WDM network and s modeled as a connected undrected graph. Each vertex represents an optcal cross connect wth Dropand-Contnue (DaC) capablty. However, the nodal splttng capablty s sparse such that nodes equpped wth power spltters are called Multcast capable (MC) nodes; otherwse, they are called Multcast Incapable (MI) nodes. Moreover, the spltters have complete (.e., maxmum splttng fanout of the node equals, at least, ts out-degree) and fxed splttng rato (.e., each copy of the sgnal acqures the same porton of the sgnal power) capabltes. On other hand, each undrected edge s equvalent to two fbers carryng traffc n opposte drectons and all fbers support the same set of wavelengths. The network does not support wavelength converson; hence, wavelength contnuty constrant should be mantaned. Also, OAs can be placed ether on-ste or on-lnk. The on-ste placement s sparse and t can be at the node s nput, called Pre-Amplfcaton, or node s output, called Post-Amplfcaton. Accordngly, and based on the notaton ntroduced n [4], our system model s characterzed as S s F c R x -M s where the frst term conssts of three components that represent the sparse, complete and fxed splttng settng, respectvely, whle the second term represents the sparse on-ste amplfcaton. In addton, our solutons assume the followngs: The symmetrc power constrant over each lnk s acheved by equppng each output port of all cross connects wth an equalzer. Each node s equpped wth an array of a suffcent number of fxed-tuned transcevers (transmtters/recevers). The general delvery structure of each multcast sesson s lght forests where each lght-tree s rooted at the source node usng a separate transmtter. We employ the As Late As Possble (ALAP) OA placement polcy [9], [7]; yet, any other polcy can be used n our solutons. For sake of smplcty, our study deals wth propagaton, splttng and tappng losses only and t gnores other loss sources. Imparments due to non-lneartes and nose are outsde the scope of ths work. All power levels are n dbm, whle power gan/loss are n db. The value of P sen s assumed to be hgh enough to cope wth the varous types of noses and to guarantee an adequate Bt Error Rate (BER) [12]. C. Problem Defnton The OAP problem we are studyng here s formally defned as follows: Defnton: Gven the network topology, maxmum number of wavelengths, maxmum number of spltters, and statc traffc demand matrx, the Optcal Amplfers Placement (OAP) problem s a network provsonng problem and ts soluton s a feasble allocaton of OAs and spltters such that all traffc demands and power constrants are satsfed whle mnmzng the number of OAs. III. MILP PROBLEM FORMULATION A. Network Parameters The followng parameters are used n the formulaton.

3 3 N, E, Λ Sets of nodes, lnks, wavelengths, respectvely., j, k Node dentty, where, j, k N λ Wavelength dentty, where λ Λ e(, j) Fber lnk drected from node to node j. S Maxmum number of spltters. β Propagaton loss rato. γ Tappng power loss value at each node. L Length of e(, j) n Km. K Number of multcast sessons. a Multcast sesson dentty, 0 a K 1. src a Multcast sesson source node. D a Multcast sesson destnaton set. Φ Set of connectons n whch node s a member. Γ a Bnary-ndcator: 1 f D a ; 0 otherwse. P Sen Mnmum detecton power level per channel. P Max Maxmum aggregate power on a lnk. P 1, P 2 Negatve constants, where P 1 < P 2. δ Very small number used for SL lnearzaton. v, w Integer constants, such that v > w. Out Degree of node. B. MILP Varables The followng varables are used n the formulaton: n Number of OAs on e(, j). Bnary-ndcator: 1 f e(, j) s used by sesson a over λ; 0 otherwse. I a,λ Bnary-ndcator: 1 f λ s used by sesson a on any outgong tree lnk from node ; 0 otherwse. Υ a Bnary-ndcator: 1 f sesson a uses at least one output lnk from node. H a,λ Number of hops between src a and node over λ. SL a,λ Splttng loss on λ at node for sesson a. P Ω,a,λ Power level (n dbm) at the begnnng (Ω = beg) or end (Ω = beg) of e(, j) for λ used by a. f Number of outgong tree lnks. A f Bnary-ndcator used for power loss lnearlzaton. α Bnary-ndcator: 1 f node s MC node. M Very large number. C. MILP Formulaton The objectve functon s to mnmze the network cost n terms of the number of OAs, and t s expressed as follows: Mnmze e() E n (2) The objectve functon s subject to the followng constrants: 1. Routng and Wavelength Assgnment Constrants: I a,λ I a,λ j,j,e() E j,j,e() E M N; 0 a < K; λ Λ (3) N; 0 a < K; λ Λ (4) Υ a λ Λ Υ a λ Λ I a,λ M N; 0 a < K; λ Λ (5) I a,λ N; 0 a < K; λ Λ (6) Constrants (3) and (4) compute I a,λ as the dsjuncton between varables of all the neghbor nodes j of node. Smlar dsjuncton relatonshp s mantaned between Υ a,λ and I a,λ varables usng constrants (5) and (6) over all λ s., src a,e(,src a ) E λ Λ,src a = 0 0 a < K (7) Equaton (7) prevents any loop back to the source node from any of ts neghbor nodes n the lght-tree at any λ. I a,λ = k, {Γ a (1 Υ a )} λ Λ λ Λ k,k,e(k,) E N; src a ; 0 a < K (8) The above constrant guarantees that the number of the ncomng channels equals the number of the dstnct outgong channels of node, except for the case when node s a leaf destnaton node. k, 1 k,k,e(k,) E N; src a ; 0 a < K; λ Λ (9) Equaton (9) prevents multple traversals of nodes on each lght-forest. 1, j N; j; λ Λ (10) 0 a<k,e(k,) E The above constrant guarantees that e(, j) s used by at most one lght-tree over wavelength λ for each connecton. j, j,e() E M α + 1 N; src a ; 0 a < K; λ Λ (11) Constrant (11) prevents branchng at MI nodes. It ensures that node has at most one outgong tree lnk f t s an MI node. j,j,e(j,) E j, k,k,e(,k) E,k M N, src a ; 0 a < K (12) Ths last constrant guarantees wavelength contnuty by ensurng that there s an ncomng tree lnk ncdent on node on wavelength λ f node has at least one outgong lnk employng the same wavelength. α S N (13) Equaton (13) s needed to ensure that the number of used spltters does not exceed the number of avalable spltters. 2. Loop Avodance Constrants:

4 4 H a,λ src a = 0 0 a < K; λ Λ (14) 1 Ha,λ + 1 H a,λ j M 0 e(, j) E; 0 a < K; λ Λ (15) Γ a + Ha,λ ( N 1) 1 M e(, j) E; 0 a < K; λ Λ (16) Intally, the number of hops from any source node to tself over any λ s zero. Ths s guaranteed by constrant (14). Then, constrant (15) ensures that f e(, j) s used by sesson a, then node j s one more hop away from source node than node. Fnally, constrant (16) ensures that a tree s generated by ensurng that the destnaton nodes are reached by at most N 1 hops. 3. Power Constrants: In order to ensure that the total power constrant s met, we assume that the power value of each wavelength cannot exceed P Max Λ, where Λ s the number of wavelengths. Although ths can result n usng more OAs per lnk than needed (as more power can be used to reach longer dstance over lnks wth more free channels), ths helps n smplfyng the MILP formulaton. Moreover, ths does not contradct wth the man purpose of the MILP formulaton whch s used to bascally determne the goodness of the greedy solutons wth respect to optmal counterparts. P Ω,a,λ P Ω,a,λ P Sen + P 1 (1 ) e(, j) E; Ω {beg, end}; 0 a < K; λ Λ (17) P Max K + P 2 (1 ) e(, j) E; Ω {beg, end}; 0 a < K; λ Λ (18) Constrants (17) and (18) ensure that each power level at the begnnng and end of e(, j) s wthn the vald ranges based on whether the lght-forest uses e(, j) or not. In the former case, the power value should be between P Sen otherwse, ths value should equal to, theoretcally, dbm (.e., 0 mw). In order to represent ths case, we use two small negatve constants,.e., P 1 and P 2 wth a value of ( 5) M and ( 2) M, respectvely, such that the power s a very small negatve number. It s worth mentonng here that the value of P beg,a,λ P end,a,λ and P Max Λ ; s measured before on-ste Post-Amplfcaton, whle s measured after on-ste Pre-Amplfcaton, f any. P beg,a,λ1 P beg,b,λ2 = 3 M (1 T b,λ2 ) e(, j); 0 a, b < K; λ 1, λ 2, Λ, λ 1 λ 2 (19) Constrants (19) are used to enforce the power symmetrc constrant by ensurng that all the actve sgnals at the begnnng of each lnk have the same power strength. As the propagaton loss and power gan are both lnear, ths condton s suffce to ensure that all the power sgnals over any lnk are symmetrc. Please note that when lght-forests lnks (of the same connecton or dfferent connectons) use lnk e(, j), the rght hand sde of constrants (19) becomes zero and both power values at the begnnng of the lnk should be equal. The same occurs n the case when both lght-forests lnks do not use the same lnk. However, when only one of them exst over the lnk, the dfference n ther power values s guaranteed to be n the range between P 1 and P 2, whch s n complance wth constrants (17) and (18). (1 ) M + P end,a,λ = (1 beg,a,λ ) ( M) + P + LG β L e(, j) E; 0 a < K; λ Λ (20) Equaton (20) s used on lnk e(, j) to express the power on wavelength λ at the end pont of the lnk n terms of the power at the begnnng of the lnk and the gan and loss due to amplfcaton and attenuaton, respectvely. It should be noted that when the lnk s not part of the lght-forest,.e., the correspondng equals 0, the power value at the end of the lnk s guaranteed to be between P 1 and P 2. (1 ) v + P end,a,λ SL a,λ j γ (1 j,k ) w + P beg,a,λ j,k e(, j), e(j, k) E; 0 a < K; λ Λ (21) (1 ) w + P end,a,λ SL a,λ j γ (1 j,k ) v + P beg,a,λ j,k e(, j), e(j, k) E; 0 a < K; λ Λ (22) Constrants (21) and (22) are used to relate the values of the power levels between the end of an edge, say e(, j) and the begnnng of the followng hop, say e(j, k), f any. In order to mantan consstency wth equatons (17)-(20), and to be able to handle all the cases of the usage of the lnks e(, j) and e(j, k), the values of v and w are chosen such that v P 1 + M, and w < 0. The ratonale of choosng these values s demonstrated wth an ad of an example. Consder the case when both and j,k equal 0. Recall that equatons (17) and (18) ensure that P beg,a,λ and P beg,a,λ j,k are between P 1 and P 2. The value of v s chosen to be 6 M to guarantee that left hand sde of nequalty (21) s stll greater than rght hand sde even wth the case when both power values equal P 1. The same hold for the left hand sde of nequalty (21). Choosng the value of w to be negatve helps n ensurng ths too 1. SL a,λ M α N; 0 a < K; λ Λ (23) Equaton 23 ensures that the splttng loss value at an MI node s 0 db for any connecton carred at any channel. What remans s to determne the value of ths loss f sgnal splttng happens at an MC node, whch s determned n db by ths 1 Provng that ths crteron for choosng v and w hold for all the other cases of lnks usage s straghtforward.

5 5 formula: SL a,λ = 10 log 10 f. Snce the splttng degree, f, s a varable, ncorporatng ths loss drectly n the formulaton wll make t non-lnear. Here, we ntroduce an elegent way to fnd SL a,λ at MC nodes usng a set of lnear equatons that are equvalent to the prevous non-lnear one. j,j src a,e() E f + δ A f M N; 0 a < K; λ Λ; 2 f < Out (24) A f {10 log 10 f} SL a,λ N; 0 a < K; λ Λ; 2 f < Out (25) SL a,λ 0 N; 0 a < K; λ Λ (26) In ths context, the value of A f s 1 for all the values of f that are less than or equal the actual tree fanout at the node; otherwse t can be ether 0 or 1. However, snce the objectve functon mnmzes the amplfer gan, t attempts to mnmze the fanout, and hence the splttng loss, SL a,λ. Therefore, the value of A f n ths case wll be set to 0. LG g max n e(, j) (27) LG > g max (n 1) e(, j) (28) These constrants determne the relaton between the total gan and the needed number of OAs per lnk such that the mnmum number of OAs are used. Smlar constrants were used n [9] where g max determned the maxmum gan avalable at any amplfer whch occurs when all the nput sgnals are at P Sen. However, as the number of occuped channels per lnk s determned by the MILP soluton, we use an approxmate approach to compute g max n whch we assume that all the channels over all the network lnks are occuped. Ths assumpton enables us to precompute g max usng the OA model defne n (1) such that the total nput power (P n ) s calculated as 10 log10(λ 10 P Sen 10 ). Ths approxmaton provdes an exact value for g max when Λ s small, whch s the case wth our numercal results. D. Extenson to the Asymmetrc Splttng Case In ths subsecton we ntroduce an extenson to our optmal formulaton to allow asymmetrc splttng. That s, a node can splt the sgnal unequally, provded that the sum of splttng ratos s equal to 1. Whle ths extenson can lead to a more optmal desgn, t has two problems. Frst, the mplementaton of asymmetrc splttng may not be techncally feasble, especally f the splttng rato s arbtrary. It can, however, be approxmated usng a number of stages of spltters and combners. Second, the formulaton, as we wll see below, becomes nonlnear, and ths wll further ncrease the complexty of solvng t optmally. However, for the sake of completeness we ntroduce ths formulaton here, but we do not provde any results based on ths formulaton. We defne the followng non-negatve varables: Splttng loss (n db) of the sgnal on wavelength λ at node for sesson a, whch s then transmtted on the outgong lnk e(, j). SL a,λ SR a,λ S a,λ The splttng rato correspondng to SL a,λ. A bnary ndcator whch s 1 f the wavelength λ used by sesson a s splt at node (ths ncludes the specal case of a splttng rato of 1,.e., sgnal forwardng wth no splttng). Note that the set of SL a,λ varables, j, replaces the SLa,λ varables. Note also that the relaton between SL a,λ and SRa,λ varables s gven by the followng relaton: SL a,λ = 10 log 10 SR a,λ whch s the source of non-lnearty n the formulaton. In addton, based on the above defntons, constrants (21)- (26) wll be replaced by the followng constrants: (1 ) v + P end,a,λ SL a,λ j,k γ (1 j,k ) w + P beg,a,λ j,k e(, j), e(j, k) E; 0 a < K; λ Λ (29) (1 ) w + P end,a,λ SL a,λ j,k γ (1 j,k ) v + P beg,a,λ j,k e(, j), e(j, k) E; 0 a < K; λ Λ (30) Constrants (29) and (30) are smlar to constrants (21) and (22) and they relate the power levels between the end of a tree lnk and the begnnng of the next tree lnk usng the new varables SL a,λ. S a,λ e() E S a,λ e() E M N; 0 a < K; λ Λ (31) N; 0 a < K; λ Λ (32) These constrants compute S a,λ as the dsjuncton between all the outgong tree lnks of node. They ensure that S a,λ s 1 f at least one of ts outgong lnks from node s used by the tree of sesson a on λ. e() E SL a,λ SR a,λ = Sa,λ N; 0 a < K; λ Λ (33) = 10 log 10 SR a,λ e(, j); 0 a < K; λ Λ (34) Constrants (33) guarantee that the varous splttng rato does not exceed 1 when the rght-hand sde s 1,.e., f node s used to splt the power of sesson a on channel λ. If not, all these power ratos are guaranteed to be zero. Fnally, constrants (34) determne the relaton between SL a,λ as explaned earler. SR a,λ and

6 6 IV. GREEDY ALGORITHM The problem of Optcal Amplfers Placement (OAP) defned n subsecton II-C conssts of three man subproblems. These subproblems are: Routng (R), Wavelength Assgnment (WA), and Power Assgnment/Amplfers Placement (PAAP) subproblems. The soluton of one subproblem mpacts the solutons of the other subproblems. Hence, the MILP formulaton presented above solves the problem optmally by solvng these subproblems jontly. Despte ts optmalty, the MILP formulaton s not scalable as t cannot solve bg-szed problem nstances n a tme effcent manner due to ts hgh complexty. Such complexty s represented n terms of the numbers of constrants and varables whch equal O( N 4 C Λ ) and O( N 2 C Λ ), respectvely. Therefore, there s a need for a heurstc approach that produces fast solutons wth hgh qualty degree. Such heurstc must be able to capture the man characterstcs of the problem under nvestgaton. In ths secton, we present a heurstc soluton, referred to as Optcal-amplfers Placement (OP) algorthm. A. Greedy Algorthm Motvaton and Man Characterstcs The man goal of the OP algorthm s to acheve the balance between the produced soluton qualty and the computaton tme. In order to acheve ths goal, the operaton of the OP algorthm reles on the Dvde-and-Conquer concept by dvdng the problem nto ts natural subproblems that are solved separately. However, the mpact between these modules are taken nto account by employng a specal set of cost functons for the lnks, network and sessons. The sgnfcance of these cost functons stems from the fact that they are defned n terms of optcal amplfers numbers. As these cost functons are used for the lght-forest constructon and sesson routng, ths allows us to capture the nfluence between R and PAAP subproblems and results n effcent solutons. Moreover, the desgn of the OP algorthm realzes the nfluence of the Power Sharng concept [2] whch s a result of sharng the avalable power by wavelengths at the entry pont of the lnks and OAs. Such nfluence s translated as connecton blockng (called Power Sharng Blockng) [2] durng network operaton phase. As a desgn problem, the power sharng concept stll holds, but t has dfferent nfluence as all the connectons must be accommodated (.e., no connecton drops are allowed). In ths context, power sharng concept results n changng the Network Power Status (NPS), whch defnes the network condton n terms of ts power values at the begnnng of each lnk 2, as well as the number, locatons and gan values of the OAs. The change n the NPS s a result of any of followng behavors: 1) As optcal sgnal hops from one lnk to another, ts power strength may decay below P Sen anywhere n the lghtforest, even wth the use of the source node s maxmum avalable power. Ths results n addng more n-lne, pre-amplfcaton and/or post-amplfcaton OAs, whch 2 whch s suffcent to determne the power values everywhere n the network. ncreases the network cost. We refer to ths behavor as Power Shortage Behavor. 2) Routng a new connecton that shares lnks wth some already provsoned connectons n the network can change the NPS by ether: a) Droppng the gan of at least one OA to a level that causes a servce dsrupton for at least one sesson. Such servce dsrupton occurs f the gan drop yelds a sequence of changes n the optcal sgnals strength and other OAs gans whch results n volatng the power constrants defned n subsecton II-A. We refer to ths behavor as Gan Droppng Behavor. b) Changng the power values assgned to an already provsoned lght-forest(s) n order to mantan the power symmetrc and maxmum total power constrants defned by constrants (18) and (19), respectvely. Therefore, we call ths behavor, the Power Adjustment behavor. The NPS s hghly dynamc and senstve to any change ntroduced to the NPS from these behavors. Ths s because any adjustment made to the network condton at one pont n the network may propagate to other network locatons and can affect multple connectons. Ths can create a complcated management ssue, especally f large number of connectons are nvolved. Nevertheless, the OP algorthm s desgned to tackle ths dynamc nature by allowng changes to occur to the NPS whle ensurng that ther mpacts are handled n an effcent manner. Fnally, the OP algorthm s desgned to optmze the solutons at two levels. At the lowest level, namely, the lght-forest constructon level, t allows the destnatons to be attached to the sub-forest usng multple alternatve paths, nstead of a sngle path. Usng alternatve routng n ths manner allows the OP algorthm to explorer bgger soluton space and the lghtforest can expand to new destnatons usng the path of the mnmum (present) path. In addton, the OP algorthm allows constructng more than one lght-forest per sesson. The one wth the least cost s then chosen to be placed n the network. At the lght-forest placement level, the algorthm defnes two operaton modes, namely, Fxed and Adaptve Modes. In the Fxed mode, lght-forests are constructed once based on the ntal NPS and then they are placed n the network accordng to ther costs. Wth the Adaptve mode, however, placng each lght-forest n the network s followed by reconstructng the remanng lght-forests that are not yet provsoned based on the latest NPS. Reroutng the remanng sessons allows the OP algorthm to account for the mpact of lght-forests on each other whch mproves the soluton accuracy, yet, wth the cost of extra computaton. B. Cost Functons Defntons The followng set of cost defntons are adopted by the OP algorthm: The current cost of lnk e(, j) s defned as the current number of OAs needed over the lnk, namely c e() =

7 7 n. Ths ncludes any Post-Amplfcaton at node, and any Pre-Amplfcaton at node j. The current network cost s computed as the total number of OAs over all the lnks n the network. In other words, C = e() c e() The cost of the path that connects a destnaton to the subtree-forest s defned as the change n the network cost that results f such a path s used for expandng the subtree-forest to that destnaton. The sesson cost s also calculated as the change n the network cost f ts lght-forest s placed n the network. Usng these cost functons has the followng advantages: 1) It s possble to base the routng decsons of the lghtforests on the system power budget. Ths establshes a connecton between the R and PAAP subproblems and can result n better solutons. 2) These cost functons are dynamc as ther defnton s based on the the most recent NPS. Ths s mportant to ensure the correctness and goodness of the produced solutons. 3) Usng the defntons of the path and sesson costs s effectve n relatng the routng and placement decsons to ts future consequences. Ths post-nfluence scheme help n capturng the nfluence between the connectons. 4) Fnally, the lnk cost functon s a postve ncreasng functon. Therefore, lnk costs ncrease wth the ncrease of the lnk usage. Ths s useful n balancng the load n the entre network. C. OP Algorthm Detals We present the detals of the OP algorthm n ths subsecton. The OP algorthm s desgned as an teratve algorthm such that the fnal soluton s the result of a set of optmzed sub-solutons. Fgure 1 depcts the basc operaton of the algorthm whch conssts of three man stages. These stages are: the Lght-Forests Constructon (LFC) Stage, the Lght-Forests Placement (LFP) Stage and the Lght-Forest Reconstructon (LFR) Stage. The core operaton of the LFC and LFR stages s the Lght-Forest Constructon Module whch s depcted n Fgure 2 whle the core operaton of the LFP stage s the the Lght-Forest Placement Module whch s depcted n Fgure 3. The LFC stage starts by ntalzng all the data structures, whch nclude: the Network status (NS), the Network Power Status (NPS) and the set S. Whle NPS s defned earler n subsecton IV-A, NS defnes the channels and lnks status n the network and set S determnes the set of sessons whch are not yet provsoned (placed). Intally, the set S ncludes all the multcast sessons. Then, the Lght-Forest Constructon Module s nvoked n order to construct the lght forest for each multcast sesson n S. NS and NPS are then re-ntalzed and the set S s sorted accordng to ts sessons costs. The second (.e., LFP) stage s then nvoked for the frst sesson, a, n the sorted S and ts lght-forest obtaned from the LFC stage s placed n the network. Accordngly, the algorthm updates NS, NPS, and S and t proceeds wth the remanng multcast sessons n S n two fashons based on ts operatonal modes,.e., Fxed or Fg. 1. Intalze NS and NPS S: Set of connectons Lght-Forests Constructon Module (S) Sort (S) a Frst Connecton n S Lght-Forests Placement Module (a) No Intalze NS and NPS S S-{a} S >0? Yes Adaptve Mode? Yes No Lght-Forests Constructon Module (S) Sort (S) Basc Operaton of the OP Algorthm. Lght- Forests Constructon Stage Lght- Forests Placement Stage Lght- Forests Reconstructon Stage Adaptve. In the Fxed mode, the algorthm places all the ntal lght-forests constructed n the LFC Stage wthout changng them. On the other hand, the Adaptve mode nvolves the use of the LFR stage n whch new lght-forests are constructed for all the sessons n S based on the current NS and NPS. Therefore, the Lght-Forest Constructon Module s nvoked agan n ths stage wth the latest NS, NPS and S. After sortng S, the LFP stage s nvoked wth the frst sesson n set S. The algorthm stops when all sessons are placed. For sake of completeness, the detals of all the Lght-Forest Constructon and Placement Modules are explaned below. D. Lght-Forest Constructon Module As ndcated by ts name, the purpose of the Lght-Forest Constructon Module s to produce a lght-forest for each multcast sesson accordng to the recent NS and NPS. Each lght forest s constructed as f t s the only lght-forest n the system. The purpose of ths constructon scheme s to determne the cost of each sesson (n terms of the change n the network cost) usng the recent NS and NPS. In addton, due to the randomness nvolved n ts operaton, the Lght-Forests Constructon module generate more than one

8 8 Fg. 2. No More Connectons? More Constructon Trals? Set of connectons Yes, sesson a Yes Restore Latest NS and NPS T {src a }; EXP(T ) {}; R(T ) D a R(T ) > 0? Yes No No k, d Compute P(T,d,EXP) as { p } for every d n R(T ) exp k,d k,d Compute WA( p ) for every p P exp exp k,d k,d Compute PAAP( p ) for every p P exp exp k,d k,d Compute COST( p ) for every p P exp exp T T U MIN(P); R(T ) R(T ) {d mn } Update EXP; Update NS and NPS Tree Constructon Module. lght-forest for each sesson such that the best (least cost) lght-forest s then chosen 3. The nput to the Lght-Forest Constructon module s the set S. As shown n Fgure 2, each constructon tral for the multcast sesson starts by restorng the latest NS and NPS n order to ensure the correctness of the lght-forest nstance constructon. The lght-forest constructon s performed teratvely usng an extended verson of the Member-Only Heurstc (MOH) [13]. Intally, the lght-forest structure, called sub-forest or T, ncludes the source node only, scr a. After each teraton, T s expanded such that a new (.e., unconnected) member s attached to T va one lght-forest node. The lght-forest growth s permtted through specfc set of nodes, called the expandable nodes or EXP(T ) whch conssts of the source node, all the lght-forest nodes whch have Multcast Capablty (MC nodes) or/and leaf nodes. Instead of a sngle path, the Lght- Forest Constructon Module computes k alternatve paths to 3 The number of constructon trals per connecton s an nput parameter. each remanng destnaton. Among all these computed paths, T s then expanded usng the path of the least cost. The module stops when all remanng nodes, R(T ), are ncluded n the lght-forest. All relatve data structures are updated at each teraton durng whch the followng operatons are performed: 1- Path Computaton (PC): In ths step, the set of k-shortest paths from each unconnected destnaton, d, to each expandable node n T s computed. 2- Path Wavelength Assgnment (PWA): The PWA step s performed for each computed path from step 1. Dependng on the expandable node, two scenaros are possble n ths operaton. On one hand, we employ the Frst-Ft scheme (n whch the frst avalable common wavelength over all lnks n the path s chosen) f the path under nvestgaton connects the destnaton node to the source node tself (.e., new forest branch s created). On the other hand, f the attachng node s not the source node, the new forest segment from the expandable node to the destnaton should contnue usng the same channel (f avalable) used over the forest segment connectng the source node to the expandable node. If such a wavelength s not avalable, the path nstance s gnored. 3- Path Power Assgnment/Amplfers Placement (P- PAAP): For each path nstance passed the PWA step, the P-PAAP operaton s responsble for determnng the power values and the OAs placement over each of ts lnks. The P-PAAP module reles on usng a queue structure, called Q, whch conssts of unque enttes of the lnks denttes and ams to separate the lnks denttes from ther power values. Usng Q proves to sgnfcantly reduce the computaton and management overheads n [2], hence, we adopt the same technque here. The P-PAAP operaton starts by markng the sesson under nvestgaton as affected and addng the frst lnk of the path to Q. Then P-PAAP runs teratvely such that at each teraton, the lnk at Q s head s processed by performng the followng operatons: 1) Intal Power Determnaton Operaton. Ths operaton s responsble for determnng the power values at the begnnng of the head lnk n Q such that no power constrant s volated and the power symmetry constrant s mantaned. Two factors determne these power values, namely, whether the head lnk s connected from the source and/or t has more than one channel. For nstance, f the lnk s launched from the sesson s source node and t s the only channel on the lnk, the maxmum power value can be assgned to the channel. Otherwse, f the head lnk s not launched from the sesson s source, ths step adjusts these power values accordng to the power and symmetrc constrants. 2) OA Placement Operaton. Usng the power values at the begnnng of the head lnk, the OAs are placed over the lnk based on the ALAP

9 9 Fg. 3. B Tree Placement Module. B > 0? WA(B) F a: Lght-forest of sesson a Set of Branches n F a Yes PAAP(B) Update NS and NSP polcy. Ths step nvolves changng the prevous OAs locatons and maybe number over the lnk n order to accommodate for the change n NPS. 3) Update Q Operaton. Q s then updated by addng more lnks to t, f any. The set of potental lnks to be added to Q are chosen from the set of outgong lnks, R, launched from the current head lnk s snk node. In order to prevent Q from beng unnecessarly modfed, Q s updated wth those outgong lnks that satsfy the followng condtons: They are part of the lght-forest of the current sesson under nvestgaton. Ths should guarantee the contnuaton of power nvestgaton for the current sub-forest. They are part of the lght-forest of all the other sessons over the lnk, provded that a new power value s observed at the end of the head lnk whch s dfferent than the one from prevous teraton. Such change n power values from teraton to teraton ndcates a change n the system status that needs to be propagated. Hence, we use Q to trace such a change n the NSP. Once processed fully, the lnk at the Q s head s removed and the P-PAAP operaton contnues wth the next lnk n Q and t stops when Q becomes empty. Please note here that a lnk can be revsted more than once durng the teraton lfetme 4. Ths occurs because the same lnk can be at dfferent depth of the varous lght-forests. Durng each lnk traversal, more power values over the lnk become avalable. P-PAAP deals wth those power values that are currently avalable whch enables t to work even wth partal knowledge of the power values. However, allowng several traversals of the lnks ensures the complete avalablty of the power values at the lnk. 4 Although multple traversals of the lnk s permtted durng the P-PAAP, Q contans at most one nstance of the lnk at each algorthm step. Preventng multple copes of the same lnk n Q elmnates any unneeded calculatons snce power values are separate from the lnk denttes. No E. Lght-Forest Placement Module Ths module s responsble for placng the lght-forest constructed n the Lght-Forest Constructon module n the network and then change the NS and NPS accordngly, as shown n Fgure 3. Therefore, there s no routng effort n ths module and t focuses on solvng the WA and PAAP subproblems at the forest level. These two operatons are smlar to the correspondng ones ntroduced n the Lght- Forest Constructon Module. However, the delvery structure unt here s determned n terms of lght-forest s branch nstead of a path. For nstance, the WA s performed for each branch of the lght-forest such that t always fnds the frst avalable wavelength over all the branch s lnks (.e., no contnuaton of usage of upstream channels nvolved). Smlarly, the PAAP entty n the Lght-Forest Placement Module operates lke the PAAP entty n the Lght-Forest Constructon module (.e., P-PAAP module). However, buldng Q starts from the frst branch s lnk launched from the source node tself rather than from the frst lnk of the path. Please note that we can skp ths module n the Adaptve operaton mode as we can use the fnal NS and NPS nformaton from the LFR stage when the chosen lght-forest s placed n the network. However, t s essental to apply ths module for the fxed operatonal mode n order to determne the WA and PAAP results for each placed lght-forest n the network. V. NUMERICAL RESULTS We present some numercal results n ths secton. These results are obtaned usng CPLEX [14] for solvng the MILP formulaton and usng C++ Programmng Language for mplementng the OP algorthm. We frst examne the qualty of the solutons produced by the OP algorthm by comparng them wth the optmal solutons of the MILP formulaton usng the sample 6-nodes mesh network shown n Fgure 4. After establshng such quanttatve comparson, we present varous results that llustrate dfferent aspects of the proposed OP algorthm usng the 14-nodes NSFNET shown n Fgure 5. These results are obtaned wth the numercal values presented n Table I and under the followng assumptons: 1) The multcast groups sze follows a unform dstrbuton between 1 and N, where N s the number of nodes n the network. 2) Node membershp n each multcast sesson s determned unformly from all nodes after excludng the source node. 3) Descendng-order polcy s used n OP algorthm for sortng the set of connectons constructed n the LFC stage. 4) The OP algorthm runs for at least 10 tmes per problem nstance and the best soluton that produces the mnmum number of OAs s chosen. A. Comparatve Results Between the Optmal and Suboptmal Numercal Results Tables II and III compare the results obtaned from CPLEX to ther counterparts obtaned from the OP heurstc for the

10 TABLE II COMPARISON OF OAS NUMBERS OBTAINED BY CPLEX ( OA C ) AND OP HEURISTIC ( OA ; WHERE = 1, 2, 3, 4 REPRESENTS THE NUMBER OF ALTERNATIVE PATHS) FOR THE 6-MESH NETWORK. K AND Λ REPRESENT NUMBER OF SESSIONS AND AVAILABLE LAMBDA, RESPECTIVELY. Fg Fg. 5. Sx Nodes Mesh Network NSFNET TABLE I TYPICAL VALUES FOR THE SYSTEM PARAMETERS. Symbol β P Sen P Max P sat G 0 Value 0.2 db/km 30 db 0 db mw 20 db Symbol P 1 P 2 δ v w Value 5 M 2 M M M 6-nodes mesh network. It s mportant to note that the MILP formulaton ams to solve the OA Placement (OAP) problem wth more restrctve constrants than the OP heurstc. In ths context, the optmal solutons from CPLEX are obtaned whle the number of spltters and ther locatons are not determned and the number of avalable channels n the network s predetermned. These constrants are relaxed n the OP heurstc solutons as the number/locaton of the spltters s fxed and no upper bound s mposed on the number of avalable channels. Therefore, n order to make a meanngful comparson, the followng three actons are taken nto account: 1) The OP heurstc experments are carred out wth the same number/locaton of spltters obtaned from CPLEX for the same problem nstance. 2) The qualty of the obtaned solutons wll be determned not only by the delta of the number of OAs (.e., the objectve functon), but also by how much extra network resources (f any) needed by the OP algorthm. 3) Due to Constrants (18), we also make sure that the ndvdual sgnal strength produced by OP algorthm does not exceed P Max Λ dbm 5. Table II determnes the number of OAs obtaned from CPLEX ( OA C ) compared to those obtaned from OP heurstc ( OA ). The ndex determnes the number of alternatve paths used n constructng the lght-forest n LFC stage. Each OA soluton takes the (x/y) format to determne the 5 As ths acton s taken for comparson purposes only, we do not take ths lmtaton nto account for all the results presented n Subsecton V-B K Λ OA C OA 1 OA 2 OA 3 OA / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / /161 OA when the Fxed and Adaptve schemes are employed, respectvely. The same symbolc notaton and (x/y) format s used n Table III to determne the network resources consumed by the produced solutons. Bascally, there are two network resources that are of nterest to us and whch are computed at the network-wde scale. These resources are: 1) The maxmum number of dstnct channels consumed over any lnk; referred to as ψ. 2) The number of lnks used n constructng all the lghtforests (.e., lnks wth at least one used channel); referred to as Ł. From the results n Table II, t s clear that the qualty of the solutons produced by the OP heurstc s determned by ts computaton complexty. Two factors contrbute to ths complexty, namely, the number of alternatve paths used for constructng each lght-forest, and whether or not reroutng s performed to reconstruct the remanng unplaced lght-forests. The OP heurstc permts the use of any combnaton of these factors such that the computaton complexty ranges from mnmum computaton (namely, Fxed scheme wth one alternatve path for routng) to maxmum computaton (namely, Adaptve scheme wth maxmum number of alternatve paths for routng). The followng conclusons can be drawn from the results n

11 11 TABLE III COMPARISON OF USED NETWORK RESOURCES FOR THE 6-MESH NETWORK. ψ C (Ł C ) AND ψ (Ł ) REPRESENT THE MAXIMUM NUMBER OF WAVELENGTHS (NUMBER OF LINKS) USED BY CPLEX, AND OP HEURISTIC, RESPECTIVELY. = 1, 2, 3, 4 REPRESENTS THE NUMBER OF ALTERNATIVE PATHS, WHILE K AND Λ REPRESENT NUMBER OF SESSIONS AND AVAILABLE LAMBDA, RESPECTIVELY. K Λ ψ C ψ 1 ψ 2 ψ 3 ψ 4 Ł C Ł 1 Ł 2 Ł 3 Ł /2 1/1 1/1 1/1 4 6/5 5/5 5/5 5/ /2 1/1 1/1 2/2 6 7/7 7/7 7/7 7/ /2 2/2 2/2 2/ /12 12/12 12/11 12/ /2 2/2 2/2 2/ /12 12/12 12/12 12/ /2 2/2 2/2 2/ /12 12/12 12/12 12/ /2 2/2 2/2 2/ /12 12/12 12/12 12/ /3 3/2 3/2 3/ /12 13/12 12/12 12/ /4 4/3 4/3 4/ /13 13/13 13/13 13/ /1 1/1 1/1 1/1 5 5/5 5/5 5/5 5/ /2 2/2 2/2 2/2 8 8/8 8/8 8/8 7/ /2 2/2 2/2 2/ /12 12/12 12/12 12/ /2 2/2 2/2 2/ /12 12/12 12/12 10/ /2 2/2 2/2 2/ /12 12/12 12/12 10/ /2 2/2 2/2 2/ /12 12/12 12/12 10/ /3 3/3 3/3 3/ /11 11/10 11/10 10/ /4 4/4 4/4 4/ /11 12/11 11/11 11/ /5 5/5 5/5 5/ /13 13/13 13/13 13/ /1 1/1 1/1 1/1 5 5/5 5/5 5/5 5/ /2 2/2 2/2 1/ /11 11/11 11/11 11/ /2 2/2 2/2 2/ /11 11/11 11/11 11/ /2 2/2 2/2 2/ /11 10/10 11/11 10/ /2 2/2 2/2 2/ /10 11/10 10/10 12/ /2 2/2 2/2 2/ /10 11/10 11/10 10/ /3 3/3 3/3 3/ /10 11/10 10/10 10/10 Tables II: OP heurstc remarkably succeeds to obtan the optmal soluton n most cases, even wth mnmum computaton (e.g., check the results for all the cases when Λ = 3 and 4). For the other cases, on the other hand, the msmatch between the optmal solutons and the OP heurstcs soluton s 1 or 2 OAs only. Ths s relatvely too small dfference, especally for Wde Area Networks whch s the focus of ths study. Better solutons wth lower msmatch are produced n most cases by ncreasng the computaton complexty of the OP heurstc. For nstance, the optmal soluton for the case when Λ = 2 and K = 7 s 159 whle the OP heurstc solutons mproved from 161 to 160 when more alternatve paths and Adaptve schemes are used. However, there s a trval trade-off between the computaton tme/resources and the soluton qualty. Usng more alternatve paths alone (.e., Fxed mode wth multple paths routng) or allowng reroutng scheme alone (.e., Adaptve mode wth sngle path routng) proves to provde good soluton, especally when the system traffc s lghtly loaded. For example, for the case of Λ = 2 and K = 5, usng the Fxed mode wth multple paths routng mproves the soluton from 160 to 159 OAs (whch s the optmal soluton) startng from usng two alternatve paths and wthout the need for applyng the Adaptve mode. On the other hand, for the case of Λ = 2 and K = 2, usng the Adaptve mode wth sngle path routng produces 158 OAs whch s the optmal soluton. The results also llustrate the fact that usng alternatve routng and allowng lght-forest reconstructon do not conflct wth each other when both are employed by the OP heurstc. On the contrary, they complement each other s work whch results n savng more OAs, especally when the system traffc load s hgh. For example, when Λ = 2 and K = 8, the OA 2 equals 160 OAs when the Adaptve mode s employed, whch s an mprovement from the 161 OAs soluton acheved when = 1 (for both Fxed and Adaptve schemes) and when = 2 (for the Fxed scheme). Ths means that usng more alternatve paths alone dd not help mprovng the soluton qualty n ths case untl t was accompaned by usng the Adaptve mode. Please note that the nature of 6-nodes network as beng a small network wth lmted number of lnks and nodal degrees, makes t hard to dstngush between the ndvdual mpact of each of these mprovement schemes on the qualty of the fnal soluton. Such a dstncton wll be addressed n Subsecton V-B when the results from NSFNET are presented. Fnally, t s worth notng that because of Constrants (18), the ( OA C ) can ncrease for the same number of sessons, K, when more number of channels becomes avalable n the system as the maxmum power of each channel wll decrease. For nstance, OA C = 159, 160, and 161 for K = 7, whle Λ = 2, 3, and 4, respectvely. In order to complete our comparatve nvestgaton, we focus on Table III to determne the amount of resources used by the OP heurstc compared to those consumed by CPLEX. Wth respect to ψ, we note that OP heurstc succeeded to use the same maxmum number of wavelengths over any lnk s most cases (e.g., for all the cases when Λ = 2 and K = 3 to 6) whch gves more credblty to these solutons as they are produced wth smlar expermental condtons as CPLEX. Interestngly, the value of ψ can be even less than ψ C whch s absolutely fne as t s not the objectve of the MILP formulaton to reduce ψ. Ths can happen especally when the number of avalable channels, Λ, s hgh enough wth respect to the traffc load as t s the case when Λ = 3 and K = 2 to K = 6. On the other hand, especally when the traffc load s hgh wth respect to the number of avalable wavelengths, we note that the OP heurstc tends to use more wavelength channels per lnk than CPLEX. Ths s true for example when Λ = 2 and K = 8 and when Λ = 3 and K = 8 and 9. However, these extra resources are stll wthn acceptable ranges (.e., 1 or 2 extra wavelengths only) especally n Wde Area Network envronments where the cost of addng extra channels to the system s comparatvely much less than addng extra OAs. Wth respect to Ł, we note that there s no drect relaton between the CPLEX solutons and the OP heurstc solutons