Dynamc Lghtpath Protecton n WDM Mesh etworks under Wavelength Contnuty Constrant Shengl Yuan* and Jason P. Jue *Department of Computer and Mathematcal Scences, Unversty of Houston Downtown One Man Street, Houston, TX 7700, yuans@uhd.edu Department of Computer Scence, Unversty of Texas at Dallas, Rchardson, TX 75083, ue@utdallas.edu Abstract-- Path protecton requres fndng a workng path and a protecton path that are lnk dsont. In ths paper, we consder the dynamc lghtpath protecton problem n WDM mesh networks under the wavelength contnuty constrant. Exstng polynomal tme algorthms can be appled to fnd a par of lnk-dsont lghtpaths on a sngle wavelength; however, such algorthms fal f the workng and protecton lghtpaths are on two dfferent wavelengths. We prove the problem s P-complete for both dedcated protecton and shared protecton. We develop an ILP formulaton and heurstc solutons for the problem. Computer smulatons are conducted to evaluate the performance of the heurstc algorthms. Index terms-- Lghtpath protecton, wavelength contnuty constrant, optcal network, nteger lnear program (ILP) I. ITRODUCTIO In wavelength dvson multplexng (WDM) networks, end users can communcate wth one another va all-optcal WDM channels called lghtpaths [][]. Because of the hgh data rate on lghtpaths, t s mperatve to develop approprate protecton and restoraton schemes [3] [4] to prevent or reduce data loss. In protecton schemes, backup resources are pre-computed and reserved for each connecton before a falure occurs [5][6]. In restoraton schemes, an alternate route s dscovered dynamcally for each nterrupted connecton after a falure occurs [7][8]. Compared to restoraton schemes, protecton schemes have faster recovery tme and provde guaranteed recovery ablty but requre more network resources. Protecton schemes can be dvded nto path protecton and lnk protecton based on the level of network resources nvolved n the protecton. In path protecton, a workng path and a dsont protecton path are establshed for each connecton. In lnk protecton, separate backup resources are reserved for each ndvdual lnk on the workng path. Path protecton usually has lower resource requrements and lower end-to-end propagaton delay for the recovered route [5][7]. Protecton schemes can be further dvded nto dedcated protecton and shared protecton based on whether backup resources are shared by more than one connecton. In dedcated protecton, each lnk or node can be reserved as a backup resource for at most one connecton. In shared protecton, a lnk or node can be reserved as a backup resource for multple connectons, as long as those connectons do not fal smultaneously. Dedcated protecton requres more network resources but s smpler to mplement, whle shared protecton s more resource effcent but requres more complex sgnalng and network management []. The path protecton problem can be consdered under ether statc or dynamc traffc. Under statc traffc, the entre set of connecton requests s known. The routes and the wavelengths for the workng and protecton lghtpaths of all connecton requests must be determned[5][9][0][][]. Under dynamc traffc, connecton requests arrve one at a tme and each connecton exsts for only a fnte duraton. Routes and wavelengths must be determned ndvdually for the workng and protecton lghtpaths of each connecton request. In an optcal network wthout wavelength converson capablty [3][4], the establshment of a lghtpath s subect to the wavelength contnuty constrant,.e., a lghtpath s requred to be on the same wavelength channel throughout ts entre path n the network. Under ths constrant, the workng lghtpath and ts protecton lghtpath may both be on the same wavelength, or each may be on a dfferent wavelength. A network falure may be caused by ether a lnk falure or a node falure. Most modern node devces have bult-n redundancy whch greatly mproves ther relablty. Therefore lnk falure s more of a concern than node falure, and we only consder lnk falure n ths paper. In order to fnd the two lnk-dsont lghtpaths, we may ntutvely use a smple two-step soluton,.e., we fnd one shortest path on one wavelength frst, then remove all lnks on that path and fnd the second shortest path on the same wavelength or on a dfferent wavelength. These two paths are guaranteed to be lnk dsont. However, ths soluton fals n so-called trap topologes (Fg. ). s e a b Fgure. A sngle-wavelength WDM network. The numbers ndcate lnk costs. For sngle-wavelength networks, a feasble soluton can be found usng Suurballe's algorthm and ts varatons [5][6]. The total cost of the resultng two lnk-dsont lghtpaths s mnmal among all such path pars. The algorthm runs n O(n log n) tme, where n s the number of nodes. For networks wth multple wavelengths, we can apply ths algorthm on every wavelength n order to fnd the two lghtpaths on the same wavelength. However, f such paths do not exst, the problem s to fnd two lnk-dsont lghtpaths on two dfferent wavelengths. In [7], t s proven that, for the f d IEEE Communcatons Socety Globecom 004 09 0-7803-8794-5/04/$0.00 004 IEEE
specal case n whch the total cost of the two lghtpaths s to be mnmal, the problem s P-complete. In ths paper, we prove that n a more general case, the problem s stll Pcomplete for both dedcated protecton and shared protecton, regardless of the total path costs. To solve the Routng and Wavelength Assgnment (RWA) problem of lghtpath protecton, fxed alternate paths heurstcs have been proposed n [8] and [9]. Wth these heurstcs, alternate routes are predefned for each sourcedestnaton par. When a connecton request arrves, the predefned routes are searched to fnd a workng path and a protecton path wth free bandwdth on the entre route. In ths paper, we develop heurstc algorthms that select routes and wavelengths based on the real-tme network status. The rest of the paper s organzed as follows. Secton II proves that, under the wavelength contnuty constrant, the problem of fndng a workng lghtpath and ts protecton lghtpath, each on a dfferent wavelength, s P-complete, regardless of the lghtpaths total cost. Secton III formulates the problem as an ILP and gves heurstc solutons. Secton IV presents computer smulatons results for the heurstc solutons and compares the performance of the heurstcs. Secton V concludes the paper. II. P-COMPLETEESS OF THE DYAMIC LIGHTPATH PROTECTIO PROBLEM UDER THE WAVELEGTH COTIUITY COSTRAIT For dedcated protecton, the problem s formally defned as follows. Gven optcal network G = (, L), where s the set of optcal swtchng nodes and L s the set of fber lnks, and gven the number of wavelengths on each fber lnk, fnd two lnk dsont lghtpaths from source node s to destnaton node d such that each lghtpath s on a dfferent wavelength. A. Proof of P-Completeness for Dedcated Protecton We reduce the 3SAT problem, whch s known to be Pcomplete [0], to the target problem. The 3SAT problem s stated as follows. Gven a collecton C = C, C,, C M of clauses on a fnte set V = v, v,, v of varables such that C = 3 for M, where clause C s the boolean or of three lterals (a lteral s ether a varable or the boolean not of a varable) and s satsfed by a truth assgnment f and only f at least one of the three lterals s true, s there a truth assgnment for V that satsfes all the clauses n C? We construct a graph G for an arbtrary nstance of 3SAT C, such that the graph contans two lnk-dsont lghtpaths, P on wavelength λ and P on wavelength λ, from node s to node d f and only f there s a truth assgnment satsfyng all clauses n C. In ths proof, the graph contans only two wavelengths, λ and λ, but t can easly be expanded to the case of more wavelengths. Followng are the steps for the graph constructon:. Create source node s and destnaton node d.. Correspondng to the varables n V, create + nodes z, 0. There s a lnk from s to z 0 and from z to d. Between z - and z, there are nodes x, y, x, y,, x M, y M, and x, M M, whch correspond to the M clauses n C. There are lnks z - x, x y, y x, x y,, x M y M, y M z and lnks z - x y x, M M M x y z. Lnks x y and x y each contan two wavelengths, λ and λ. All other lnks created n ths step contan only wavelength λ. 3. Correspondng to each clause C, create nodes u and w, M. There s a lnk from s to u and from w M to d. There s also a lnk from w to u +. Other lnks are formed accordng to the followng rules: a. A lnk from u to x exsts, and a lnk from y to w exsts, f and only f varable v s n clause C. b. A lnk from u to exsts, f and only f varable x exsts, and a lnk from y to w v s n clause C. All lnks constructed n ths step only contan wavelength λ. An example s gven n Fg. In ths example, we construct graph G for a 3SAT nstance C = C, C, V = v, v, v 3, C = v v v v v 3, C = v v v v v 3. The dotted lnks contan wavelength λ and the dashed lnks contan wavelength λ. The sold lnks contan both wavelengths λ and λ. For a truth assgnment v =, v =, v 3 =, the correspondng dsont paths are: p (s-z 0 - z - z - x - 3 3 - z 3 3 -d) on wavelength λ and p (s -u - x - y - w - u - x -y -w -d) on wavelength λ. s x y x y x y x y y 3 y 3 z 0 z z z 3 x y x y x y x y x3 y 3 x 3 y 3 u w u w Fgure. A graph constructed from a 3SAT nstance. Lemma : If C s satsfable, then there exst two lnk-dsont lghtpaths of dfferent wavelengths from node s to node d n graph G. Proof: Let boolean values be assgned to v, v,, v that satsfy C. The two paths should be routed as follows: P s on wavelength λ. It traverses all z nodes for 0. Between node z - and z, the path s routed va nodes x and y ( M) f and only f v = 0. Otherwse t s routed va nodes x and y. P s on wavelength λ. It traverses all u, w nodes for M. Between node u and w, the path s routed as follows. By constructon, lnk u w corresponds to clause C whch has three lterals. Each of the lterals corresponds to a path from u to w that goes ether through nodes x and y f the lteral s n the form of v, or through nodes x and y f the lteral s n the negaton form, v. Because C s satsfed, at least one of the three lterals n C must be. Let the varable n that true lteral be v. Then d IEEE Communcatons Socety Globecom 004 00 0-7803-8794-5/04/$0.00 004 IEEE
f the lteral s n the form of v, then v =, and route P passes through nodes x, y ; f the lteral s n the form of v, then v = 0, and route P passes through nodes y. If more than one lteral s true, then randomly pck one of the true lterals and route P accordngly. Thus, P doesn t traverse any of the nodes u, w for M, and P doesn t traverse any of the nodes z for 0. Furthermore, f P traverses node x, y, then P traverses and vce versa. Therefore P and P are lnk dsont, and each s on a dfferent wavelength. Lemma :If there exst two lnk-dsont lghtpaths of dfferent wavelengths from s to d n the constructed graph G, then C can be satsfed. Proof:. Snce there are only two lnks orgnatng from the source node s, the two lnks must each belong to a separate path. Let sz 0 be part of P and su be part of P.. Snce P s already on wavelength λ, P must not traverse any of the nodes u, w for M, otherwse t would also be on wavelength λ and volate the wavelength contnuty constrant. Therefore, f P traverses x for, then t must also traverse y, x, y,, x M, y M, z. Smlarly f P traverses x for, then t must M M also traverse, z. 3. Snce P s already on wavelength λ, P must not traverse any of the nodes z for 0, otherwse t would also be on wavelength λ and volate the wavelength contnuty constrant. Furthermore, f P traverses node u ( M) and x ( ), t must also traverse y and then back to w. Smlarly, f P traverses node u and t must also traverse y and then back to w. 4. Loops are not allowed. Therefore once P reaches w ( M), t must go to u + f < M, or to d f = M. 5. If P traverses nodes x, y, M,, t must not also traverse nodes k x and k k, and vce versa; otherwse P s blocked and cannot reach the destnaton node d wthout volatng the lnk dsont constrant. 6. If P traverses nodes x, y, M,, then P must traverses nodes traverses x for, then t must also traverse y M,. Smlarly f P traverses nodes, M then P must traverses nodes x, y, x, y,, x, y,, x M, y M. 7. Assgn values to v, v,, v as follows: If P traverses nodes x, y, M,, then assgn v =, makng clause C to be true. If P traverses nodes M,, then assgn v = 0, makng clause C to be true. Varables that are not assgned a value n the frst two steps are randomly assgned ether or 0. Ths assgnment satsfes C. Combnng Lemma and Lemma, we see that the 3SAT problem s reducble to the problem of fndng dsont lghtpaths on dfferent wavelengths. Therefore ths problem s P-complete, regardless of the paths costs. B. Proof of P-Completeness for Shared Protecton Wth shared protecton, one or more protecton lghtpaths may traverse a common wavelength on a fber lnk. If the problem wth shared protecton s solvable, then the problem wth dedcated protecton can also be solved snce t s a specal case of that wth shared protecton. III. ILP FORMULATIO AD HEURISTIC ALGORITHMS We now develop ILP formulaton and heurstc solutons for the P-complete problem of fndng lnk-dsont lghtpaths on dfferent wavelengths. A. ILP Formulaton The ILP formulaton should be solved for each ncomng connecton request. The obectve s to fnd any two lnkdsont lghtpaths. An alternatve obectve s to mnmze the total hop count of the two lnk-dsont lghtpaths. The followng are gven as nputs to the problem. : number of nodes n the network L: collecton of all fber lnks n the network. Λ : collecton of all free wavelengths on fber lnk L. s, d: source node and destnaton node. The ILP solves for the followng varables: α : f wavelength w on lnk s taken by the workng lghtpath from source s to destnaton d; 0 otherwse. β : f wavelength w on lnk s taken by the protecton lghtpath from source s to destnaton d; 0 otherwse. Obectve: Fnd a workng lghtpath and a protecton lghtpath that satsfy the wavelength contnuty constrant. L α + L Constrants: Flow-conservaton constrant: = l α - = = Lnk dsont constrant: βl - = α = l l β = 0, β > 0 (), f l = d -, f l = s otherwse l, w W, (), f l = d -, f l = s 0, otherwse l, w W, (3) IEEE Communcatons Socety Globecom 004 0 0-7803-8794-5/04/$0.00 004 IEEE
α + B. Heurstc Algorthms β, L (4) In ths secton we ntroduce heurstc algorthms for fndng lnk-dsont lghtpaths n WDM networks. The frst algorthm s named the Route-Frst Algorthm. In ths algorthm, we use a standard routng and wavelength assgnment (RWA) approach. We frst try to fnd two dsont routes, and then assgn free wavelengths to them. The second algorthm s named the Wavelength-Scan Algorthm. In ths algorthm, we frst scan through each wavelength for a par of lnk-dsont lghtpaths usng Suurballe s algorthm. If Suurballe s algorthm fals, we search through each par of wavelengths for a par of lnk-dsont lghtpaths on dfferent wavelength usng a two-step approach. At the begnnng of both algorthms, we scan all fber lnks and ncrease the cost of a lnk lnearly to the number of wavelengths already n use on the lnk. Ths step ncreases the lkelhood of fndng free wavelengths on selected routes. It also balances traffc load among the network lnks, thus mproves the blockng probablty. For the Route-Frst Algorthm, the runnng tme s O(n log n + Wn), where n s the number of nodes and W s the number of wavelengths n the network. The pseudo code s gven n Fg 3. Intalze c l on every lnk to the cost of the lnk; for ( all network lnks) ncrease c l on lnk l accordng to the number of wavelengths n use on l; f ( Suurballe s algorthm(s, d) succeeds and fnds two routes r and r ) for ( all wavelengths λ and λ n the network and λ λ ) f ( a wavelength λ s avalable on route r and a wavelength λ s avalable on route r ) assgn λ to r and make t the workng lghtpath p ; assgn λ to r and make t the protecton lghtpath p ; return( p and p ); return(failure); Fgure 3. Route-Frst Algorthm For the Wavelength-Scan Algorthm, the runnng tme s O(W n log n), where n s the number of nodes and W s the number of wavelengths n the network.. The pseudo code s gven n Fg 4. Comparng the two algorthms, the Route-Frst Algorthm obtans the routes frst before t assgns free wavelengths to the routes. If the algorthm returns successfully, the total cost of the two lghtpaths s mnmal among all lnk-dsont paths from s to d. On the other hand, the Wavelength-Scan Algorthm scans through all avalable wavelengths, searchng for the two lnk-dsont paths, frst on a sngle wavelength then on dfferent wavelengths. Thus the runnng tme of ths algorthm s hgher. When the traffc load s low, the Route- Frst Algorthm should have lower blockng probabltes because free wavelengths are readly avalable and the routes are optmal n total cost. When the traffc load s hgh, the Wavelength-Scan Algorthm should have lower blockng probablty because t searches through all avalable wavelengths. Intalze c l on every lnk to the cost of the lnk; for ( all network lnks) ncrease c l on lnk l accordng to the number of wavelengths n use on l; for ( all the wavelengths ) run Suurballe s algorthm and return the two dsont paths wth the mnmum total cost f they are found; //If the prevous step fals for ( all wavelength λ n the network ) f ( Dkstra s algorthm(s, d) on λ succeeds and fnds the frst shortest path p from s to d) for ( all λ n the network and λ λ ) remove lnks on the frst shortest path p ; f ( Dkstra s algorthm(s, d) on λ succeeds and fnds the second shortest path p from s to d) return( p and p ); //Return the lghtpaths return(failure); Fgure 4. Wavelength-Scan Algorthm To ncrease resource utlzaton and thus reduce blockng probablty, we can modfy these two algorthms to support shared protecton []. The tradeoffs are complex network management and addtonal sgnalng protocol. IV. SIMULATIOS We have dscussed two heurstc algorthms for the dynamc path protecton problems under the wavelength contnuty constrant,.e., the Route-Frst Algorthm and the Wavelength-Scan Algorthm. We can also develop shared protecton support for each of the algorthms. Computer smulatons were developed to evaluate the performance of these four algorthms. In these smulatons, the prmary performance metrc s the blockng probablty. We use the 6-node, 5-lnk SFET backbone topology (Fg. 5) for the smulatons. Other network topologes are also used and yeld smlar results. The cost of every lnk s assumed to be, and the capacty on each lnk s 8 unts. Workng paths and protecton paths each takes one unt of capacty. Connecton requests arrve accordng to a Posson process, and holdng tmes are exponentally dstrbuted. In the smulaton, we compare the blockng probabltes of the Route-Frst Algorthm and the Wavelength-Scan Algorthm. For each of the algorthms, we run the smulaton IEEE Communcatons Socety Globecom 004 0 0-7803-8794-5/04/$0.00 004 IEEE
for an extended perod of tme, under varous traffc loads, and compare ther blockng probabltes. The results are obtaned wth confdence level between 90% to 95% and confdence nterval around 5%. The results are depcted n Fg.6. protecton. One possble area of future work would be to further mprove the performance of the Wavelength-Scan Algorthm at hgher load. In addton to traffc balancng, we may also adust the lnk costs based on other factors such as the number of free wavelengths on a lnk that are reachable to the destnaton. Ths type of adustments may have a postve mpact on the algorthm s performance. Blockng Probabltes Fgure 5. 6-node SFET backbone network..00e-0.00e-0.00e-03.00e-04.00e-05.00e-06.00e-07.00e-08 Traffc Load (Erlang) 3 4 5 6 8 0 Route-Frst Algorthm Route-Frst Algorthm wth Shared Protecton Wavelength-Scan Algorthm Wavelength-Scan Algorthm wth Shared Protecton Fgure 6. Blockng probablty versus load. The trends contnue for traffc load hgher than 0 Erlangs. The smulaton results confrm our analyss n Secton III. From the smulaton, we frst observe that shared protecton sgnfcantly mproves blockng probablty, regardless of the traffc load. Secondly, when the traffc load s very low, the Route-Frst Algorthms perform better than the Wavelength- Scan Algorthms. Whle the traffc load ncreases, the Wavelength-Scan Algorthms become better than the Route- Frst Algorthms. V. COCLUSIO In ths paper we consdered the problem of dynamc lghtpath protecton n optcal networks under the wavelength contnuty constrant. We proved that the problem s Pcomplete. We then developed an ILP formulaton and heurstc algorthms to solve the problem. We conducted computer smulatons to evaluate the heurstc algorthms and compared ther blockng probabltes under varous traffc loads. The smulaton reveals that, when network load s low, the Route-Frst Algorthm performs better. When network load s hgher, the Wavelength-Scan Algorthm performs better. The smulatons also confrmed that shared protecton sgnfcantly mproves blockng probablty over dedcated REFERECES [] B. Mukheree, Optcal Communcaton etworks. ew York: McGraw Hll, 997. [] R. Ramaswam and K.. Svaraan, Optcal etworks: A Practcal Perspectve. San Francsco, Morgan Kaufmann, 998. [3] O. Gerstel, Opportuntes for optcal protecton and restoraton, OFC, Techncal Dgest, pp. 69-70, February 998. [4] T. Wu, Fber etwork Survvablty. MA: Artech House, 99. [5] S. Ramamurthy and B. Mukheree, Survvable WDM mesh networks, part I protecton, Proceedngs, IEEE IFOCOM, vol., pp. 744-75, March 999. [6] A. Fumagall, et al. Survvable networks based on optmal routng and WDM self-healng rngs, Proceedngs, IEEE IFOCOM, vol., pp. 76-733, March 999. [7] S. Ramamurthy and B. Mukheree, Survvable WDM mesh networks, part II restoraton, Proceedngs, IEEE ICC, vol. 3, pp. 03-030, June 999. [8] R. R. Iraschko and W. D. Grover, A hghly effcent pathrestoraton protocol for management of optcal network transport ntegrty, IEEE Journal on Selected Areas n Communcatons, vol. 8, no. 5, pp. 779-794, May 000. [9] R. M. Karp, On the computatonal complexty of combnatoral problems, etworks, vol. 5, pp. 45-68, 975. [0] http://www.log.com/products/cplex [] Andrea Dacomo, et al. Desgn of statc reslent WDM mesh networks wth multple heurstc crtera, Proceedngs, IEEE IFOCOM, vol. 3, June 00. [] Hu Zang, et al., Path-protecton routng and wavelength assgnment (RWA) n WDM mesh networks under duct-layer constrants, IEEE/ACM Transacton on etworkng, vol., no., Aprl 003. [3] R. A. Barry and P. A. Humblet, Models of blockng probablty n all-optcal networks wth and wthout wavelength changers, IEEE Journal on Selected Areas n Communcatons, vol. 4, no. 5, pp. 858-867, June 996. [4] S. Subramanam, et al. All-optcal network wth sparse wavelength converson, IEEE/ACM Transactons on etworkng, vol. 4, no. 4, pp. 544-557, August 996. [5] J. W. Suurballe and R. E. Taran, A quck method for fndng shortest pars of dsont paths, etworks, vol. 4, pp. 35-336, 984. [6] Remesh Bhandar, Survvable etworks Algorthms for Dverse Routng. Kluwer Academc Publshers. [7] Wefa Lang, Robust routng n wde-area WDM networks, Parallel and Dstrbuted Processng Symposum, Proceedngs, 5th Internatonal, Aprl 00. [8] G Mohan, et al. Effcent algorthms for routng dependable connectons n WDM optcal networks, etworkng, IEEE/ACM Transactons on, vol. 9, no. 5, October 00. [9] V. Anand and C. Qao, Dynamc establshment of protecton paths n WDM networks. Part I, Proceedngs, 9th Intl. Conference on Computer Communcatons and etworks, pp. 98-04, October 000. [0] M. R. Garey and D. S. Johnson, Computers and ntractablty: A gude to the theory of P-Completeness. ew York: W. H. Freeman Company, 979. [] S. Yuan and J. P. Jue, Dynamc path protecton n WDM mesh networks under wavelength-contnuty and rsk-dsont constrants, Techncal Report, no. UTDCS-6-04, Unversty of Texas at Dallas. IEEE Communcatons Socety Globecom 004 03 0-7803-8794-5/04/$0.00 004 IEEE