Non-Cooperative Design of Translucent Networks

Transcription

1 Non-Cooperatve Desgn of Translucent Networs Benoît Châtelan, She Mannor, Franços Gagnon, Davd V. Plant McGll Unversty, Electrcal and Computer Engneerng, 3480 Unversty, Montreal, Canada, H3A A7 École de technologe supéreure, Département de géne électrue, 00 Notre-Dame Ouest, Montreal, Canada, H3C K3 Abstract-Ths paper ntroduces a new game theoretc formulaton for the desgn and routng of reslent and translucent networs. An nteger lnear programmng (ILP) modelng s also presented and used as a reference to evaluate the game theoretc algorthm performances. Both formulatons nclude prmary and ln-dsjont protecton paths pre-calculaton and tae nto account the system maxmal optcal reach dstance. Numercal results show that the game theoretc formulaton consderably decreases the optmzaton tme and provdes near optmal solutons, n term of reured number of regenerator nodes. I. INTRODUCTION Although t s foreseen that future optcal wde area networs (WAN) wll rely on all-optcal technology, current mplementatons are rather based on electronc swtchng. In today s WAN networs, most of the ntermedate nodes are opaue. For now, optcal-electrcal-optcal (OEO) converson s the only vable alternatve for 3R regeneraton (reamplfcaton, reshapng and retmng). Translucent networs are a combnaton of all-optcal and opaue technologes. As long as the dstance traveled by the optcal sgnal s wthn the optcal reach lmt, all-optcal swtchng s used. Otherwse, when the sgnal transmt dstance exceeds the optcal reach lmt, t has to be regenerated by an OEO node. Wth the ncreasng avalablty of optcal swtchng technology, mproved savngs can be acheved by replacng some of the opaue nodes by all-optcal swtchng nodes. Translucent networs are a brdge between the alloptcal and all-opaue approaches and are strong canddates for the next generaton of optcal networs. Game theory has been largely appled n the study of economcs. It now fnds applcatons n poltcs, psychology, bology, socology and engneerng. In networ desgn, control and optmzaton, game theory has many appealng propertes. It s smple to mplement and computatonally frendly ; t s well adapted to dynamcs and real-tme modelng; t can be easly adapted for dstrbuted control and t represents well the nteracton between users or servce provders n a networ, each havng ther own need and selfsh objectves. In ths paper, an nteger lnear programmng (ILP) and a game theoretc model are proposed for the desgn of translucent optcal networs. The maxmal reach dstance of the optcal sgnal and the networ reslence are taen nto consderaton. The paper shows how to tacle the problem of locatng OEO nodes usng the two approaches. More specfcally, the paper s organzed as follow. In secton II, the lterature on translucent networs and game theory applcatons s revewed. In secton III, the desgn model s presented and n secton IV, the ILP and game theoretc formulatons of the desgn problem are ntroduced. Fnally, n secton V, a numercal analyss of the optmzaton procedures s provded. II. LITERATURE REVIEW Translucent networs are a relatvely recent research topc. They were frst proposed and studed n 999 by Ramamurthy and al. []. In ther defnton of translucdty, they state that a sgnal from the source travels through the networ as far as possble before ts ualty degrades, thereby reurng t to be regenerated at an ntermedate node. The same sgnal could be regenerated several tmes n the networ before t reaches the destnaton. In ths study, ths defnton s appled. There are two man approaches for the desgn of WAN translucent networs. The frst allows the sgnal regeneraton to tae place at any nodes n the networ whle the second mples the creaton of transparency domans, nterconnected by OEO nodes. As noted n [] and [3], sparse placement of OEO nodes usually reures less regeneraton nodes and s more cost effectve than the domans or slands of transparency approach. Reslence n translucent networs has been nvestgated n [4-7] for statc traffc and n [8] for the dynamc case. In [4] a + protecton scheme s proposed, whle n [5, 7-8] varants of shared path protecton, allowng OEO sharng, are studed. Protecton n [6] s envsaged n the context of generalzed multprotocol label swtchng (GMPLS) operaton. Noncooperatve path protecton have been studed n [9] wth the prmary paths already defned and fxed. The only actons or strateges left for the players to decde n ths case are the bacup paths. Although ths approach reduces the search space, t s too restrctve when seeng an optmal soluton. Networ desgn can be acheved by usng exact algorthms such as ILP or by the use of heurstcs. The problem assocated wth ILP models s that for even medum networ szes (8 to 4 nodes), they ether become untractable or very tme consumng. These are the man reasons why heurstcs are developed. For WAN translucent networs, both ILP formulaton [5-7, 0-] and heurstc optmzaton [4-8, 0, 3-4] were proposed. These formulatons tae nto consderaton a large dversty of physcal mparments such as chromatc dsperson, polarzaton mode dsperson, attenuaton, amplfed spontaneous emsson and crosstal. Although there exsts a rch lterature on game theoretc applcatons n networ desgn [5], to the best of the authors X/07/$ IEEE

2 nowledge no game theory model has been developed for translucent networs optmzaton and desgn. In ths study, a non-cooperatve game theoretc model based on the so-called congeston games [6] s presented. An exact potental functon [6] s also derved and used to descrbe the evoluton of the game as an optmzaton procedure. III. DESIGN MODEL Throughout ths wor, t s assumed that dense wavelength dvson multplexng (DWDM) s used and that the number of avalable wavelengths s greater than the overall traffc demand n the networ. No attempt wll be made to reduce the number of wavelengths used n the networ. The only desgn objectve s to mnmze the number of OEO nodes n a translucent networ, whch are usually the most lmted and expensve resources n a DWDM optcal networ. It s also taen for granted that the chosen lght transmsson paths are properly power managed by perodcally placed optcal amplfers. In ths study, only the optcal reach dstance wll be consdered whch encompasses for most of the physcal mparments. Real-world networ plannng s often accomplshed wthout precse nowledge of the fber plant and also reles on dstance-based regeneraton [7]. Gven that chromatc dsperson can be compensated by the use of electronc sgnal processng algorthms [8], the man lmtaton n the transmsson of optcal sgnals s the attenuaton caused by fber loss. The optcal reach (O R ) s defned as the maxmal dstance that the sgnal can traverse wthout amplfcaton. The maxmal number of spans (N S ) refers to the maxmum number of amplfed fber spans above whch sgnal regeneraton s necessary. The reach lmt (R L ) corresponds to the maxmal dstance that a sgnal can travel: RL = NS OR. () Transparent nodes use optcal cross-connects (OXC) or wavelength selectve swtches (WSS) to commute the sgnal from one fber to another. Opaue nodes, on the other hand, use regenerators and electronc swtchng. In translucent networs, opaue nodes can be all-opaue or partly opaue. In the all-opaue confguraton, a transcever par s used for all ncomng and outgong wavelengths. In the partly opaue confguraton, only the wavelengths that need to be regenerated are converted to an electronc sgnal, the others are processed n the optcal doman. Snce the optmzaton goal n the present wor s to mnmze the number of OEO stes, no dstncton s made between all-opaue or partly-opaue nodes, the two beng consdered as OEO nodes. IV. FORMULATION OF THE DESIGN PROBLEM The desgn problem can be formulated as follow: gven the networ topology, a set of pont-to-pont demands (connectons) along wth canddate paths for the prmary routes and canddate ln dsjont paths for the protecton routes, fnd the mnmal number of OEO eupped nodes satsfyng the worng and protecton paths constrants. The developed algorthms are only for regenerator placement and routng, smlarly to the wor of [6] and [4]. Whle selectng the regenerator nodes, no traffc matrx or ln capacty constrants are consdered. The objectve s to place a mnmum number of regenerators to satsfy the reurement that there s at least two ln-dsjont paths between any par of nodes, satsfyng the maxmal reach dstance constrant. Ths formulaton s the euvalent of havng a full demand matrx, such that a prmary and secondary ln dsjont path must exst between every possble connecton. Wavelength assgnment s not covered n our framewor but, as shown n [7], separatng the routng and wavelength assgnment process s a common practce and yelds n effcent networ desgn. Two dfferent approaches are used to solve the problem. The frst one nvolves ILP and the other, a game theoretc model. Both models rely on the pre-calculaton of worng and protecton paths. For each path, the postons of OEO nodes are also pre-determned, based on the gven reach lmt. The prmary paths are calculated usng Yen s algorthm [9] that computes the -shortest acyclc paths between a source and a destnaton. For each prmary path, -protecton lndsjont paths are computed by removng from the orgnal networ the lns used n the prmary path of nterest. The OEO placement s done for each prmary and protecton path. Whenever the accumulated dstance s greater than the reach lmt, an OEO node s added. A. ILP Formulaton ILP formulatons for all-optcal or translucent networs are usually ntractable or very tme consumng. By applyng relaxaton technues, they can provde bounds on the performances of optmzaton heurstcs. In ths wor, an ILP formulaton based on pre-calculated path s proposed. Not only s t easer to solve but, gven that enough paths are consdered, t s optmal or very close of beng so. The results gven by the ILP optmzaton wll be used to compare the effcency and accuracy of the game theoretc approach. The followng notaton s used: G = (V, E): An undrected graph wth the set of networ nodes V and the set of optcal lns E., j: Orgnatng and termnatng nodes of a lghtpath. Values gven: N: The number of nodes n the networ. P: The networ physcal topology matrx. If P =, there exst a physcal ln between and j. T: The networ traffc matrx, n terms of wavelengths reured between and j. In ths wor, the traffc matrx s full; ndcatng that a wavelength demand exsts for all possble connectons n the networ. c: The number of connectons n the networ, gven by the number of nonzero elements n the matrx T. x: The number of prmary paths for each connecton. y: The number of protecton paths for each prmary path. A: The set of prmary paths for each connecton. A contans c x paths X/07/$ IEEE

3 B: The set of protecton paths for each prmary path. B contans c x y paths. C: The set of reured regenerator nodes for every paths of A. C s the same sze as A. D: The set of reured regenerator nodes for every paths of B. D s the same sze as B. Varables: X: A bnary vector of N elements representng the presence () or the absence (0) of regenerators at each node. a : A bnary varable ndcatng, when t euals to, that among the x possble prmary paths between nodes and j, the th path s selected. r b : A bnary varable ndcatng, when t euals to, that among the y possble protecton paths of the prmary path between nodes and j, the r th path s selected. The objectve s to mnmze the number of OEO nodes n the networ. Ths can be wrtten as: N Mnmze X. () n n= The constrants on the prmary paths are as follow: x a = T > 0, (3) = T > 0, C > 0 C a c, =,,, x l = d x+, d =,,, c l. (4) The constrant represented by (3) s necessary so that at least one path per connecton s selected. The constrant represented by (4) s used to assocate wth a gven path the OEO nodes that are needed. For example, f the nd possble path between nodes and 3 s chosen, then a 3 =. If connecton -3 ( =, j = 3) corresponds to connecton number (d = ) and f 3 prmary paths are pre-calculated (x = 3), then = 5. Therefore, all the non-zero elements of matrx C ffth row, whch ndcate the reured regeneraton nodes for ths path, must be superor or eual to a 3. If n ths case the ffth row of matrx C euals to [ ], then nodes and 4 must be OEO nodes. To represent the reslence reurement, the followng condtons are added: y r b = a T > 0, (5) r = T > 0, Dgh > 0 r Dgh b c, r =,,..., y g = d y+ r, d =,,, c. (6) The constrant expressed by (5) represents the dependence of the protecton path n the prmary path: a secondary path can only exst f the assocated prmary path s selected. The constrant denoted by (6) s to assocate the OEO nodes reurement wth the pre-calculated protecton paths, smlarly to constrant (4). B. Game Theoretc Formulaton A game s essentally descrbed by the players that tae part n t, by ther possble actons or strateges and by ther utlty functons or payoffs. Non-cooperatve players are selfsh and care only about maxmzng ther own utlty. They act ndependently. The non-cooperatve approach was chosen prmarly for ts low mplementaton complexty and secondarly for ts flexblty: n the non-cooperatve framewor, t s very easy to change the utlty functon whle retanng the same convergence propertes. In ths wor, the game theoretc formulaton s used as an optmzaton procedure. The fnal goal s not to precsely model the players behavor, but to create a framewor that facltates smpler optmzaton setup. The game s defned usng the followng notaton: G = (V, E): An undrected graph wth the set of networ nodes V and the set of optcal lns E. n: The number of players. There are as many players as reuested connectons n the networ. A connecton s defned as a reuest for assgnment of a wavelength from a source node to a destnaton node. For a gven connecton, a player must decde on dfferent actons. The frst represents the prmary the prmary path selecton whle the second determnes the choce of the secondary path. A path s formed of multple lns, between a source node and a destnaton node n the graph G. s : The chosen acton of player, representng the prmary path of connecton. s : The chosen acton of player, representng the secondary path of connecton. S: S = s x s x x s n : the set of actons of the game. u : : u S, the utlty functon of player. R: The set of resources. In ths game, the resources are the OEO regenerators and the overall goal s to mnmze the number of resources. The maxmal number of resources eual to the number of nodes n the networ, or graph G. n : The number of players who select resource e on ther prmary and secondary path. c : The cost assocated wth resource e, functon of n. The utlty functon s computed n the followng way: where: and u u u = u + u, (7) ( ) f s not a drect path c n s s = s 0 f s s a drect path ( ) (8) c f s not a drect path n s s = s. (9) 0 f s s a drect path X/07/$ IEEE

4 A drect path s an all optcal path that does not reure OEO, the total dstance between the source and destnaton beng nferor to the reach lmt. In ths game, players should share as much OEO as they possbly can, n order to reduce the overall number of OEO n the networ. The goal of each player s to mnmze hs utlty. The cost functon assocated wth a resource s therefore gven by: ( ) c n s = n s. (0) A Nash eulbrum (NE) s a partcular set of actons s* of the game where players have no ncentve to devate, that s, to choose another acton. A NE s reached f: * (, * * ) (, ) u s s u s s s S, () where s - can be nterpreted as the actons of all other players, but s: s = s,..., s, s+,..., sn. () The game descrbed by (7) falls n the category of congeston games. As such, t can be transformed n a potental game [6]. The potental functon Φ( s): S descrbes the change n the objectve functon when one player modfes hs acton whle the others eep ther strateges. For ths partcular game, the potental functon s gven by: n ( s) Φ s. (3) = c ( j) The frst summaton represents the sum of costs of each player usng the resource whle the outer sum s over all resources used. Euaton (3) s an exact potental functon snce: j= u xs, u zs, =Φ xs, Φ zs, xz, S, (4) meanng that a change n the utlty of a player s exactly reflected n the potental functon value. Snce the number of strategy confguratons s fnte, one of them must lead to the mnmzaton of the potental functon. When ths s so, no player can further decrease hs utlty and conseuently, a NE s reached. The exstence of a NE s therefore assured and a steady state, n whch no player devates, can be attaned. In other words, the convergence of the optmzaton algorthm s guaranteed. In ths game setup, players update ther acton followng a smple rule referred to n the game theory lterature as the best response dynamc. At every stage, or teraton of the game, a player chooses the acton that mnmzes hs utlty functon, n lght of the actons of other players that too place n the prevous stage. Wth a t the acton of player n stage t, the best response dynamc can be formulated as: a t t a = BR a. (5) The detaled game descrpton s gven below: Networ ntalzaton and -shortest paths computaton. Specfy the optmzaton parameters (number of prmary and secondary paths, reach lmt);. Load the networ parameters (traffc, dstance and adjacency matrces, nodes poston); 3. For every connecton, compute the x shortest prmary paths usng Yen s algorthm; 4. For every prmary paths, compute the y shortest ln dsjont protecton paths usng Yen s algorthm; 5. If for any prmary path, the number of possble protecton paths n the networ s nferor to the number of specfed protected paths, reduce the number of possble actons of the player representng the protecton path of ths connecton; 6. For every prmary path, determne the reured OEO nodes; 7. For every secondary path, determne the reured OEO nodes; Strateges ntalzaton 8. Assgn a random acton to every player representng the prmary paths of all the connectons n the networ; 9. For every prmary path, assgn a random acton to the player representng the protecton path of the correspondng connecton; Start of the game 0. For every connecton : a. Compute u, the possble utltes of the player representng the prmary path. The number of possble utltes depends on the specfed number of prmary paths; b. Compute u, the possble utltes of the player representng the secondary path. The number of possble utltes depends on the specfed number of secondary paths; c. Compute u, the global utltes, the sum of utltes calculated n (a) and (b); d. Choose the actons of the player that mnmze the player s utlty (Best Response Dynamc); e. Compute the potental functon. V. NUMERICAL EXAMPLES The proposed ILP and game theoretc problem formulatons were valdated on two European WAN networs. The ILP soluton for the German networ presented n Fg. (a) s 5 OEO nodes, represented by the bold crcular marers. Wth a specfed number of prmary and protecton paths of, and the reach dstance set at 600 m, the problem formulaton generates 33 varables and 3 37 constrants. Usng ILOG s CPLEX 9.0 optmzer, the soluton s obtaned n 7 mnutes on a Pentum D 3.4 GHz computer. The mnmum number of prmary and protecton paths that can be specfed to reach the optmal soluton s 8, resultng n an optmzaton tme of 4 mnutes, for a total number of varables and constrants of 89 and 4 977, respectvely. For the same German networ, the game theoretc optmzaton procedure results n a mean number of 5.05 OEO X/07/$ IEEE

5 nodes (average performance over 40 runs), very close to the socal optmum of 5 nodes. Ths result s obtaned by specfyng 8 prmary and protecton paths. The game duraton (the optmzaton tme) s 7.4 seconds; 8.6 seconds are dedcated for the paths calculaton and 8.8 seconds for the actual game consstng n the best response dynamc. The algorthm was mplemented wth Matlab, an nterpreted language. The optmzaton tme could be further reduced by usng a complable language such as C++. When specfyng 4 prmary and protecton paths and a reach lmt of 600 m, the ILP problem formulaton for the Italan WAN networ presented n Fg. (b) generates 44 varables and constrants. Ths problem cannot be solved by the CPLEX verson that was used. However, a soluton s found by the game theoretc algorthm n 3.3 mnutes, resultng n 6.00 OEO nodes (average performance over 40 runs) (a) (b) Fgure. ILP and game theoretc results optmzaton on German (a) and Italan (b) WAN networs VI CONCLUSION The man contrbutons of ths wor were to propose a scalable ILP formulaton and a non-cooperatve game theory framewor for the desgn of reslent, WAN translucent networs. To the authors nowledge, the present wor s the frst attempt at modelng translucent networs as games. As wth other heurstcs such as genetc algorthms, tabu search and smulated annealng, a mnmal nowledge of the algorthm behavor for dfferent networ szes s reured when choosng the parameters (number of prmary and protecton paths). Ths nconvenence s however mtgated by the fact that the convergence of the algorthm s fast. It can thus be run repeatedly wth dfferent parameters values, untl a good soluton s obtaned. The advantages of the game theoretc approach for the desgn of reslent translucent networ can be summarzed as follow: t s fast and smple to mplement, t s guaranteed to converge, and on average, t provdes a soluton very close to optmalty. Future wor wll nclude the evaluaton of the proposed game model for the desgn of larger networs and the modfcaton of the utlty functon to consder not only the sgnal attenuaton but also other physcal mparments. In ths wor, only one possblty of regenerator nodes postonng s computed per path. To further reduce the overall number of OEO stes, more than one regenerator confguraton for each path should be taen nto account. VII. ACKNOWLEDGMENT The authors wsh to than Danel O Bren for provdng a Matlab verson of Yen s -shortest path algorthm. 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