Performance Evaluation of Wavelength Band Switching in Multi-fiber All-Optical Networks

Transcription

1 Performace Evaluatio of Wavelegth Bad Switchig i Multi-fiber All-Optical Networks Xiaoju Cao Vishal Aad Yizhi Xiog Chumig Qiao Departmet of Computer Sciece ad Egieerig State Uiversity of New York at Buffalo {xcao2, vaad, yxiog, qiao}@cse.buffalo.edu Abstract Wavelegth bad switchig (WBS) has oly recetly attracted attetio from the optical etworkig idustry for its practical importace i reducig the cotrol complexity ad cost of optical cross-coects (OXCs). However, WBS-related problems of theoretical iterest have ot bee addressed thoroughly by the research commuity, ad may issues are still wide ope. I particular, WBS is differet from wavelegth routig, ad thus techiques developed for wavelegth-routed etworks (icludig e.g., those for traffic groomig) caot be directly applied to effectively address WBS-related problems. I this paper, we first propose a ew multi-graular OXC (MG-OXC) architecture for WBS, which is more flexible tha ay existig WBS ode architectures. We also adopt the most powerful wavebad assigmet strategy, ad develop a efficiet heuristic algorithm called Balaced Path routig with Heavy- Traffic first (BPHT). To verify its ear-optimality, we also develop a iteger liear programmig (ILP) model. Both the ILP ad the BPHT algorithms ca hadle the case with multiple fibers per lik ad hece are more geeral tha our previous sigle-fiber solutios [1]. We coduct a comprehesive evaluatio of the beefits of WBS through detailed aalysis ad simulatios. We show that the proposed heuristic BPHT ca perform much better tha a heuristic which applies the optimal routig ad wavelegth assigmet (RWA) method. We also show that WBS usig BPHT is eve more beeficial i multi-fiber etworks tha i sigle-fiber etworks i terms of reducig the port cout. Our aalytical ad simulatio results also provide valuable isights ito the effect of wavelegth bad graularity, as well as the trade-offs betwee the wavelegth-hop ad the port cout required i WBS etworks. I. INTRODUCTION All-optical etworks usig wavelegth-divisiomultiplexig (WDM) have become promisig cadidates for the Iteret backboe. The WDM techology divides the eormous fiber badwidth ito a large umber of wavelegths ad with curret techologies, each fiber ca have 100 or more wavelegths (each operatig at 2.5Gbps or higher). However, such advaces i the trasmissio techology also brig about tremedous icrease i the umber of ports (ad thus the cost) at optical cross-coects (OXCs) as well as the complexity ad difficulty associated with cotrol ad maagemet of such large scale OXCs. Recetly, the cocept of Wavelegth Bad Switchig (WBS) or simply wavebadig has bee proposed by several optical etworkig compaies to maitai this complexity at a reasoable level. The mai idea of WBS is to group several wavelegths together as a bad ad switch the bad usig a sigle port wheever possible. Systems havig a high wavelegth cout ca use a hierarchical or multi-graular OXC (MG-OXC) such as the oe proposed i [2] [4]. Merits of the MG-OXC such as small-scale modularity, cross-talk ad complexity reductio were summarized i [2]. The cocept of WBS was applied to WDM rig etworks i [5] [7], ad a three-layer MG-OXC architecture for mesh etwork ad its applicatio to metro-area etworks were described i [3], [8] while a sigle-layer architecture was proposed i [9]. Research i efficiet WBS algorithm desig ad their performace evaluatio, as well as other problems of more theoretical iterest has oly begu i academia [1], [10], ad may WBS-related problems are still wide ope. I this paper, we first propose a ew MG-OXC architecture (show i Figure 1),which is more flexible tha ay previously proposed architectures icludig those i [1], [3]. This architecture allows dyamic cofiguratio of the add, drop ad bypass ports, while the earlier architectures oly allow fixed add, drop ad bypass. The ratioal behid usig such a MG- OXC to reduce the port cout is that a fiber is demultiplexed ito bads if ad oly if ecessary (for example, it carries at least oe lightpath which eeds to be dropped or added). Similarly, a bad is demultiplexed ito idividual wavelegths if ad oly if ecessary. O the other had, the MG-OXC with oly a wavelegth cross-coect (WXC) layer becomes what will be called a ordiary-oxc (sigle-graular). WBS differs from covetioal wavelegth routig i several ways, for example, each has differet objectives. Accordigly, techiques developed for wavelegth-routed etworks (icludig e.g., those for traffic groomig) caot be directly applied to effectively address WBS-related problems. More specifically, i etworks employig ordiary-oxc, the routig ad wavelegth assigmet (RWA) problem is to fid a route for a lightpath ad assig a wavelegth to it. Oe of the key objectives of the traditioal RWA algorithms is to miimize the total umber of wavelegth-hop (WH) or the maximum umber of wavelegths required to satisfy a give set of lightpath requests, which is kow to be NP-complete [11] [13]. I this paper, we study the optimal WBS problem, with its mai objective beig to route lightpaths ad assig appropriate wavelegths to them so as to miimize the total umber of ports required by the MG-OXCs. As to be show, eve though traditioal RWA is still a importat compoet of WBS, ew wavebad assigmet algorithms eed to be

2 developed i order to effectively achieve the objective. The above optimal WBS problem is still a ope ad challegig problem. Recetly, a iteger liear programmig (ILP) model was proposed i [10], but the model is restrictive i that it tries to bad or group lightpaths with the same destiatio oly. Further, it requires wavelegth coversio capability at WXC layer. I this paper, we adopt a more geeral, ad i fact the most powerful wavebad assigmet strategy that ca group lightpaths with differet sources ad differet destiatios whe developig a ILP model. Sice the optimal WBS problem cotais a istace of RWA, which is NP-complete, it ca be solved for a small problem size (e.g. etwork size) oly. Accordigly, we also develop ad compare heuristic algorithms for large problem sizes. We show that for small etworks, the proposed wavebad assigmet heuristic, called Balaced Path routig with Heavy-Traffic first (or BPHT), ca achieve results that are close to the optimal results, ad ca sigificatly outperform a heuristic based o the optimal RWA algorithm. We develop a aalytical model to calculate the umber of ports eeded ad i particular, aalyze the performace of the BPHT heuristic, which is verified through extesive simulatio. To the best of our kowledge, this is the first comprehesive performace evaluatio study o optimal WBS ad efficiet WBS heuristics for mesh etworks. Note that the ILP formulatio, ad the BPHT heuristic developed i this paper are applicable to multi-fiber etworks, ad hece are more geeral tha our previous sigle-fiber solutios preseted i [1]. The study o multi-fiber etworks is motivated by the work i [14] [17], which has show that the performace improvemet i terms of reduced blockig ad better fault tolerace ca be obtaied by usig multifiber etworks. Ref. [14] showed that doublig the umber of fibers per lik is aki to doublig the umber of wavelegths per lik with the additioal advatage of simulatig a partial wavelegth coversio capability. I this paper, we aalyze the performace advatages of usig multi-fiber liks over siglefiber liks i WBS etworks ad show that sigificat savigs i umber of ports ca be achieved i multi-fiber etworks. Sice the objective of WBS is to miimize the umber of ports i MG-OXCs, rather tha to miimize WHs as i traditioal RWA algorithm, WBS may require more WHs tha that eeded by optimal RWA to satisfy a give set of lightpaths requests. I other words, there is a trade-off betwee miimizig the umber of WHs versus miimizig the umber of ports i WBS etworks. Our results idicate that our heuristic BPHT achieves a large reductio i the umber of required ports with oly a small icrease i WHs. This paper is orgaized as follows. I Sectio II, We describe the proposed WBS ode architecture ad explai how WBS is performed usig this flexible MG-OXC. Sectio III presets our ILP model while Sectio IV presets heuristic algorithms icludig the proposed BPHT heuristic. I Sectio V, we aalyze the performace of BPHT. Numerical results from our ILP model ad heuristic algorithms are preseted i Sectio VI. Fially, Sectio VII cocludes the paper with a summary of its major cotributios. B drop F add W add BTW BTF Fig. 1. WXC BXC FXC W drop WTB FTB Architecture of a MG-OXC II. WAVELENGTH BAND SWITCHING B add :;& /D\HU %;& /D\HU F drop );& /D\HU I this sectio, we describe the proposed, flexible MG- OXC architecture show i Figure 1, which icludes the FXC, BXC ad WXC layers. As show i the figure, the WXC ad BXC layers cosist of cross-coect(s) ad multiplexer(s)/demultiplexer(s). The WXC layer icludes a wavelegth cross-coect (WXC) switch that is used to bypass/add/drop lightpaths at this layer, bad-to-wavelegth (BTW) demultiplexers, ad wavelegth-to-bad (WTB) multiplexers. The BTW demultiplexers are used to demultiplex bads ito wavelegths, while the WTB multiplexers are used to multiplex wavelegths ito bads. At the BXC layer, the wavebad cross-coect (BXC) switch is used to bypass/add/drop wavebads. The BXC layer also icludes the fiber-to-bad (FTB) demultiplexers ad bad-to-fiber (BTF) multiplexers. Similarly, fiber cross-coect (FXC) switch is used to bypass/add/drop fibers at the FXC layer. The MG-OXC architecture proposed i this paper is differet ad more flexible tha ay existig architectures icludig those cosidered i [1], [3], as it allows for dyamic itercoectio or cofiguratio of the MG-OXC. For example, at the FXC layer, as log as there is a free FTB port, ay fiber ca be demultiplexed ito bads. Similarly, at the BXC layer ay bad ca be demultiplexed to wavelegths usig a free BTW port by appropriately cofigurig the FXC, BXC crosscoects ad associated demultiplexers. O the other had, i the earlier proposed architectures, these cofiguratios are fixed, i that oly certai fixed fibers (bads) ca be demultiplexed. Due to the differece betwee this ew architecture ad the oe cosidered i [1], the way to cout the umber of ports are also differet.

3 More specifically, whe coutig the umber of ports, we will oly focus o the iput-side of the MG-OXC 1. We defie the iput-side of a MG-OXC to cosist of locally added traffic ad traffic comig ito the MG-OXC ode from all other odes (which cosists of bypass traffic ad locally dropped traffic). I order to reduce the umber of ports, the MG- OXC switches a fiber usig oe port (space switchig) at the FXC cross-coect if oe of its wavelegths is used to add or drop a lightpath. Otherwise, it will demultiplex the fiber ito bads, ad switch a etire bad usig oe port at the BXC cross-coect if oe of its wavelegths is used to add or drop a lightpath. I other words, oly the bad(s) whose idividual wavelegths eed to be added or dropped will be demultiplexed, ad oly the wavelegths i those bads that carry bypass traffic eed to be switched usig the WXC. This is i cotrast to the ordiary-oxcs, which eeds to switch every wavelegth idividually usig oe port. For example, assume there are 10 fibers, each havig 100 wavelegths, ad oe wavelegth eeds to be dropped ad oe to be added at a ode. The total umber of ports required at the ode whe usig a ordiary-oxc is 1000 for icomig wavelegths (icludig 999 for bypass ad 1 for drop wavelegth), plus 1 for add wavelegth for a total of However, if the 100 wavelegths i each fiber are grouped ito 20 bads, each havig 5 wavelegths, the usig a MG-OXC, oly oe fiber eeds to be demultiplexed ito 20 bads (usig a 11-port FXC). The, oly oe of these 20 bads eeds to be demultiplexed ito 5 wavelegths (usig a 21-port BXC). Fially, oe wavelegth is dropped ad added (usig a 6-port WXC). Accordigly, the MG-OXC has oly = 38 ports, a almost 30 times reductio. Hereafter, we cocetrate o oe of the proposed WBS schemes i [1], wherei each fiber has a fixed umber (B) bads ad each bad has a fixed umber (W) aswellasa fixed set of wavelegths. Note that the ILP model ad heuristic algorithms developed i this paper ca be exteded to the other WBS schemes (e.g., allowig variable umber of bads per fiber) as well. With the curret state of art, wavelegth coversio techology is still too immature (ad expesive), hece, i the followig, we assume that there is o wavelegth coversio i our model. The case with wavelegth coversio ad other variatios of the WBS scheme will be studied i the future. III. ILP MODEL FOR WBS This sectio formulates the WBS scheme usig iteger liear programmig (ILP). The ILP formulatio is for multifiber etworks, ad is more geeral tha existig ILP models, icludig that for the sigle-fiber case i [1]. 1 Due to the symmetry of the MG-OXC architecture, the umber of ports o the iput-side ad output-side are equal. A. Notatios I : I f,m: Set of iput fibers at ode (excludig those for local add); Iput fiber f at ode, coected to ode m. So I = I,m; f m,f O : Set of output fibers at ode (excludig those for local drop); O,m: f Output fiber f at ode, coected to ode m. So O = O,m; f m,f A : Set of local add fibers at ode, icludig those used at the ports of WXC, BXC ad FXC layer; D : Set of local drop fibers at ode, icludig those used at the ports of WXC, BXC ad FXC layer; IA : I A. This set icludes the set of all icomig fibers (local ad o-local) at ode ; OD : O D. This set icludes the set of all outgoig fibers (local ad o-local) at ode ; b : Set of wavelegths i bad b; F: Number of fibers per lik that ca be used for each directio; K: Number of wavelegths per fiber; B: Number of wavelegth bads per fiber; W: Number of wavelegths per wavelegth bad (K = B W); P: Set of ode pairs havig o-zero traffic demad. Each ode pair ca be deoted by p = (p.src, p.dest), where p.src ad p.dest represet the source ad destiatio odes of oe or more request lightpaths, respectively; T[p]: Traffic matrix whose elemet t p is a iteger, represetig the traffic demad (i.e. umber of lightpaths) of the ode pair p. B. ILP Variables To facilitate presetatio ad uderstadig of our ILP model, we defie variables to describe the properties of a ode (istead of a lik as i other ILP formulatios for RWA). More specifically, to obtai ad represet the detailed iformatio of the routig ad wavelegth assigmet, we itroduce the followig biary variables to be used i the ILP formulatio. Note that the traffic at a ode ca be drop traffic, bypass traffic or add traffic. The followig four variables: i,o,p, W,w i,o, B,b i,o ad F i,o are used for describig the lightpaths, each of which ca represet bypass traffic whe i I,o O ; add traffic whe i A,o O or drop traffic whe i I,o D. A icomig (or outgoig) fiber refers to either a iput (or output) fiber from (or to) a eighborig ode or a fiber coectig the local ode to ay add (or drop) port at the WXC, BXC ad FXC layer (as metioed earlier i Sectio III-A).

4 i,o,p : 1 if ode has a lightpath for ode pair p = (p.src, p.dest) o wavelegth w from icomig fiber i to outgoig fiber o, ad 0 otherwise; W,w i,o B,b i,o : F i,o : : 1 if ode has a wavelegth w bypass/add/drop at the WXC layer from icomig fiber i to outgoig fiber o, ad 0 otherwise; 1 if ode has a wavebad b (b [1, 2,...,B]) bypass/add/drop at the BXC layer from icomig fiber i to outgoig fiber o, ad 0 otherwise; 1 if ode has a icomig fiber i bypass/add/drop to outgoig fiber o at the FXC layer, ad 0 otherwise; Four additioal variables used for describig multiplexig/demultiplexig are also defied. FTBi : 1 if iput fiber i (i I ) eeds to be demultiplexed ito bads at ode, ad 0 otherwise; BTW,b i : 1 if bad b o iput fiber i (i I ) eeds to be demultiplexed ito wavelegths at ode, ad 0 otherwise; BTFo : 1 if a bad eeds to be multiplexed oto a output fiber o (o O ) at ode, ad 0 otherwise; WTBo,b : 1 if a wavelegth eeds to be multiplexed o to bad b of a output fiber o (o O ) at ode, ad 0 otherwise;, λ1 V i 1, o 2, p1 ) ) ) Q λ 2 λ 1 E E BTW i Fig. 2., λ2,λ1 Va 0, o2, p2 1, o2, b1 1 BTFo 2 :;& %;& );& W i WTB o 2 FTB i 1 λ 2 λ 1,b1 Wavebad at ode As a cosequece of multiplexig/demultiplexig, we eed to use multiplexer/demultiplexer port(s) at the respective layers. Figure 2 shows oe such example ivolvig two lightpaths, oe for ode pair p 1 usig λ 1 o iput fiber 1 ad the other for ode pair p 2 usig λ 2 to be added locally. Usig the MG- OXC, the two lightpaths are grouped together i the same bad of the same output fiber (e.g. fiber 2). By defiitio, we have V,λ1 i 1,o 2,p 1 = Va,λ2 0,o 2,p 2 =1. For this, iput fiber 1 (cotaiig the lightpath for p 1 ) has to be demultiplexed ito bad b 1 (ad other bads) usig a FTB demultiplexer (hece, FTBi 1 =1). E E ) ) ) Q Bad b 1 the has to be further demultiplexed ito λ 1 ad other wavelegths (hece, BTW,b1 i 1 =1) to switch the lightpath for p 1 (hece, W,λ1 i 1,o 2 =1). The secod lightpath for p 2 is added ito bad b 1 usig a WTB multiplexer (hece, WTBo,b1 2 =1). Now that the two lightpaths are i the same bad, the bad is multiplexed oto a fiber usig a BTF multiplexer (hece, BTFo 2 =1), ad the trasmitted oto output fiber 2. C. Objective Fuctio Let WXC, BXC ad FXC be the umber of ports at WXC,BXC ad FXC layers at ode, respectively. There are two reasoable objectives. The first is to miimize the total cost associated with the MG-OXC ports i the etwork, that is, miimize [α WXC + β BXC + γ FXC ] (1) where α, β ad γ are the coefficiets or weights correspodig to the cost of each port at the WXC,BXC ad FXC layer, respectively. Whe α = β = γ =1, the objective becomes to miimize the total umber of MG-OXC ports i the etwork, which is the sum of the port cout at FXC, BXC ad WXC layers respectively. The secod objective is to miimize the maximum cost at each ode over all odes. This ca be formulated as: miimize max(α WXC + β BXC + γ FXC ) (2) Whe α = β = γ =1, this becomes equal to miimizig the maximum port cout (ode size) over all the odes i the etwork. D. Costraits For Routig ad Wavelegth Assigmet, the followig costraits o traffic flows, wavelegth-capacity ad wavelegth-cotiuity are similar to those i the traditioal RWA ILP formulatios. i A,o O w,i A,o O w,i I,o D i IAm,o O f m, i,o,p = i,o,p =0 p.src, p.dest, w (i) i,o,p i I,o D = tp = p.src, (ii) i,o,p = tp = p.dest, (iii) (3) p,o OD p,i IA V m,w i,o,p i,o,p 1 w, i I; (4) i,o,p 1 w, o O; (5) i I f,m,o OD i,o,p =0 m,, p, w, f; (6) Equatio (3) is the traffic flow costrait; Equatios (4) ad (5) are the wavelegth capacity costrait; Equatio (6) is the wavelegth cotiuity costrait.

5 For Wavebad Switchig, we eed the followig additioal costraits. 1 F i,o + B,b i,o + W,w i,o 1 F i,o + 1 B,b i,o + p,o 1 o p,o 1 o p i,o 1,p, i,o 1,p, i,o,p w b,i IA,o OD ; (7) 1 F i,o + 1 B,b i,o + p,i 1 i p,i 1 i i 1,o,p w, i, o; (8) i 1,o,p i, o, w b; (9) Costraits (7), (8) ad (9) esure that if a lightpath uses wavelegth w belogig to bad b of icomig fiber i ad outgoig fiber o (i.e. i,o,p =1), the at ode, p - exactly oe of FXC, BXC ad WXC cross-coect port will be used for switchig this lightpath whe it is a bypass (i.e. i I,o O )or - exactly oe of F add, B add ad W add port will be used for addig this lightpath whe it is added (i.e. i A,o O )or - exactly oe of F drop, B drop ad W drop port will be used for droppig this lightpath whe it is dropped (i.e. i I,o D ). BTF o WTB,b o W,w i,o w b,o O,i IA ; (10) The above costrait esures that a wavelegth w at ode switched or added at the WXC layer has to pass a WTB multiplexer to the BXC layer. At the same time, every bad from a WTB multiplexer has to pass a BTF multiplexer before it ca leave ode. Similarly, Equatio (11) below specifies that a wavelegth w switched or dropped at the WXC layer has to come from BXC layer usig a BTW demultiplexer, ad i additio every bad demultiplexed by BTW ca oly come from a FTB demultiplexer. FTB i,b BTWi W,w i,o w b,o OD,i I ; (11) Fially, ay bypass or add bads should pass a BTF multiplexer as specified i equatio (12) ad similarly, ay drop or bypass bad ca oly come from a FTB demultiplexer as specified i Equatio (13). BTF o FTB i B,b i,o B,b i,o o O,i IA; (12) o OD,i I; (13) For Port Numbers, the followig costraits specify the miimum umber of ports required at each layer of the MG- OXC. WXC = W,w i,o ; (14) i IA,o OD,w BXC = B,b i,o + WTB,b o + i IA,o OD,b o O,b FXC = i IA,o OD BTW,b i ; i I,b (15) F i,o + BTF o + FTB i ; (16) o O i I For the WXC layer, the umber of iput-side ports iclude the bypass, add/drop lightpaths as specified i (14). The umber of iput-side ports eeded at the BXC layer is the (BXC cross-coect ad add/drop/bypass bads) ad the umber of wavebads from the WTB/BTW multiplexers/demultiplexers as i (15). Similarly, Equatio (16) ca be used to determie the umber of ports at the FXC layer. sum of the umber of wavebads B,b i,o I short, our ILP model (ad heuristics to be described ext) cosiders the desig of MG-OXC odes (i.e. the umber of ports allocated at each of the layers) with the objective to miimize either the total port cout or the maximum port cout over all MG-OXC odes i the etwork give a set of traffic demads to be satisfied o a give etwork topology, wherei each lik i the etwork may have sigle or multiple fibers. Note that if we elimiate the FXC ad BXC layers (i.e. set correspodig variables to 0) from the MG-OXC, the above ILP formulatio with Objective (1) will miimize the total umber of ports, which is equivalet to miimizig WHs usig ILP for optimal RWA. As such ILP formulatios developed ca oly be solved for small systems with a few odes ad a few wavelegths o each fiber, we eed to develop efficiet heuristic-based approaches for large systems. IV. HEURISTIC ALGORITHMS FOR WAVEBAND SWITCHING I this sectio, we describe the heuristic algorithms developed for WBS. There are several wavebad assigmet strategies i WBS etworks, icludig: (1) groupig the lightpaths with the same source-destiatio pair oly; (2) groupig the lightpaths from the same source oly; (3) groupig the lightpaths with same destiatio oly; (4) groupig the lightpaths with commo itermediate liks (from ay source to ay destiatio). The authors i [10] oly cosidered the Strategy 3 ad sigle-fiber etworks, while our ILP formulatio covers the fourth strategy, which is the most geeric ad flexible i multi-fiber etworks. Below, we describe a heuristic that takes ito cosideratio lightpath routig as well as wavebad assigmet Strategy 4 i multi-fiber etworks. A. Wavebad Oblivious Optimal RWA (WBO-RWA) To study the relatioship betwee WBS ad traditioal RWA, we use ILP formulatios for RWA [12] that miimize the total umber of used WHs. The we try to group the assiged wavelegths ito bads ad calculate the umber of required ports. Note that the heuristic is completely oblivious to the existece of wavebads. From ow o, we refer WBO- RWA as gettig wavelegth assigmet from the optimal RWA ILP formulatio ad the groupig them [1].

6 B. Balaced Path routig with Heavy-Traffic first wavebad assigmet (BPHT) Ituitively, to maitai wavelegth-cotiuity i wavelegth routed optical etworks without wavelegth coversio, it is better to assig wavelegths to loger paths (i terms of hops) first. Further, to reduce the umber of ports i MG-OXC, it is better to assig paths that have maximum umber of liks i commo, wavelegths i the same fiber (ad bad), thus icreasig the probability of switchig the whole fiber (ad bad) by just usig a sigle FXC (ad BXC) port. The followig is our three-stage heuristic algorithm called Balaced Path routig with Heavy-Traffic (BPHT)first wavebad assigmet, which tries to maximize the reductio i the MG-OXC size usig the above ideas. Stage 1: Balaced Path Routig I this stage, we use the followig steps to achieve load balaced routig. - Fid K-shortest routes for every ode pair (s, d) with o-zero traffic demad as i [18], ad order them from the shortest to the logest (i terms of hop umber) as Ps,d 1,P2 s,d,,pk s,d. Let the umber of hops of the shortest route be H s,d. -Defie the load o every lik l to be the umber of routes already usig lik l (iitially, this is 0). Let C be the maximum lik load over all the liks. - Use C to achieve load balaced routig, startig with the ode pair (s, d) with the largest H s,d value over all ode pairs, to determie the route for each ode pair. More specifically, for the K-shortest routes Ps,d i of the selected ode pair (s, d), where i =1, 2,,k, we compute the C ad pick oe of the routes that miimizes C. If more tha oe routes, say Ps,d i ad P j s,d, have the same miimum C, the shortest oe (i.e. Ps,d i,ifi<j) will be used as the route for (s, d). That is, all the lightpaths from s to d will take this route. After the route for (s, d) is chose, the process cotiues to choose oe route for each of the remaiig ode pairs, startig with the oe havig the largest umber of hops alog the shortest path, util every ode pair with o-zero traffic demad is assiged a route. Stage 2: Wavelegth Assigmet Based o the observatio that bypass traffic, which goes through two or more hops accouts for 60% 80% of the total traffic i the backboe, we assig the wavelegths to those bypass lightpaths first. At the same time, we also wat to give preferece to the lightpaths that overlap with may other (shorter) lightpaths i order to maximize the advatage of wavebadig. The followig steps are used to assig wavelegths to all the lightpath demads oce the routig is doe i Stage 1. To maximize the beefit of WBS i multi-fiber etworks, we itroduce a ew wavebad assigmet algorithm, called wavebad assigmet for multi-fiber WBS (WA-MF-WBS,see Step (D) below). (A) For every ode pair (s, d), whose route is determied as s = s 0 s 1 s 2...s 1 s = d i Stage 1, defie a set Q s d, which icludes all ode pairs (s i,s j ), whose route is s i,s i+1,...,s j, as determied i Stage 1, where 0 i 2, ad i +2 j. Note that it is possible that the route chose for (s i,s j ) i Stage 1 is ot a sub-path of the route chose for (s, d), i which case, (s i,s j ) will ot belog to Q s d. (B) Calculate the weight (similar to the cocept of wavelegth-hops) for each set Q s d as W sd = h p t p, where p = (s i,s j ) Q s d, h p is the p Q s d umber of hops ad t p is the required umber of lightpaths from s i to s j ; (C) Fid the set Q s d with the largest W sd. (D) Call set Q s d as L, ad assig wavelegths to L as follows. i.) Suppose that the logest path i L is as follows: s 0 s 1 s 2...s 1 s.lets = s 0 ad d = s (which is the case iitially based o the defiitio of Q s d ). Assig wavelegths to the requested lightpaths for the ode pair (s, d) by tryig to group them ito the same fiber, ad withi each fiber, ito the same bad(s). More specifically, for each fiber, let 0 w K 1 ad 0 b B 1be the idex of wavelegth ad bad respectively, startig from which, a available wavelegth ad bad will be searched i order to fulfill ew lightpath requests; I additio, let 0 f F 1 be the idex of the fiber curretly uder cosideratio (i.e., whose wavelegths may be used for ew lightpaths). Iitially, f = 0 ad w = b = 0 for all fibers. The followig algorithm WA-MF-WBS assigs wavelegths to the lightpaths for a specified ode pair p. Algorithm: WA-MF-WBS while t p > W do Fid a fiber startig from idex f that has as may free bads as possible (say a tp W ) { Call the foud fiber g, where g may or may ot be the same as f; Assig the bads i fiber g to the a W lightpaths for p; t p = t p a W; Set f = g, ad update w ad b for fiber g accordigly; } ed while while t p > 0 do Fid a fiber (g), startig from idex f, that has at least oe free wavelegth; Assig a free wavelegth (x), startig from idex w, to a lightpath for p, where x is most likely to be w; t p = t p 1; Set f = g, ad w = x +1. Also, update b for fiber g accordigly; ed while ii.) Use WA-MF-WBS to assig wavelegths to the

7 requested lightpaths for (s,s j ) startig with the largest j (i.e. j = 1, 2,...,2). iii.) Use WA-MF-WBS to assig wavelegths to the requested lightpaths for (s i, d) startig with the smallest i (i.e. i =1, 2,..., 2). iv.) If there are still ode pairs (s i,s j ) Q s d that have ot bee cosidered, repeat from Step (D) by treatig s i with the smallest i as s, ad s j with the largest j as d. Otherwise go to Step (E). (E) Recompute the weight for those ode pairs whose routes use ay part of the route used by ode pair (s, d). For each fiber, re-adjust b ad w to be the ext wavebad ad the first wavelegth i the ext wavebad, respectively, so as to prevet the lightpaths of the ext ode pair set (e.g. Q s d ) from usig the same bads as the lightpaths of Q s d (thus reducig the eed to demultiplex ad multiplex the lightpaths belogig to these two sets whe they merge ad diverge). More specifically, set b =(b +1)mod B, ad w = b W, ad the go to step (C). Repeat util all the bypass (multi-hop) lightpath demads are satisfied. W S 0 S 1 3 S S 3 S 4 s = S d = S 4 sd Fig. 3. = h p Q s d p t p = = 16 A example illustratig the Steps (C) ad (D) i Stage 2 of BPHT For example, suppose we are cosiderig a ode pair set Q 0 4 i Figure 3, where t p = 1 for ay p Q 0 4. Assumig that the lightpaths umbered from 2 to 6 will be routed alog the same route as the lightpath 1 ( i.e. s 0 s 1 s 2 s 3 s 4 ) as dictated by the load balaced routig algorithm. The, the weight of the ode pair set is 16 as show, ad i additio, the order i which these lightpaths will be assiged wavelegths accordig to Steps (C) ad (D) is from 1 to 6. (F) Fially, assig wavelegths to lightpaths betwee two odes separated by oly oe hop, startig with the ode pair havig the largest lightpath demad. Stage 3: Wavebad Switchig Oce the wavelegth assigmet is doe, WBS ca be performed i a fairly straight-forward way. Basically, we switch as may fibers usig FXCs as possible; ad the as may wavebads usig BXCs as possible. The remaiig lightpaths are the idividually switched at the WXC layer. The total umber of ports used at a give ode ca the be determied as discussed at the Sectio III-D. Ideally, BPHT will group traffic from the same source to the same destiatio, ad most of the traffic that has commo itermediate liks. Oe of the variatios of BPHT (i Stage 1) is to balace the amout of traffic (i terms of the actual umber of lightpaths istead of just oe route for each ode pair) o every lik. Aother variatio is to assig wavelegths to lightpaths with the largest hop cout or those for ode pairs with the largest weighted traffic demad (i.e. h p t p ) first (assumig e.g., shortest-path routig) i Stage 2. I our experimets, we have compared may heuristics ad foud that the overall performace of BPHT is the best. Due to space limitatio, we will oly compare the results of BPHT, with that of WBO-RWA ad previous ILP model. V. PERFORMANCE ANALYSIS To simplify the aalysis of the performace of our heuristic algorithm BPHT, we assume a radom etwork with N odes ad 2L uidirectioal liks ad a uiform traffic model where the traffic demad of every ode pair p is t (i.e. t p = t). Let δ = 2L N be the average ode degree, H(p) be the shortest path t H(p) p P legth for ode pair p, ad G = 2L be the average umber of lightpaths o a lik. Note that the miimum amout of total traffic goig ito the iput-side of all the odes is MP = t H(p)+N(N 1) t, where N(N 1) t p P is total umber of added lightpaths. Further, ote that MP is also equal to the miimum umber of ports i ordiary-oxc etworks. Give that the proposed heuristic BPHT uses load balaced routig, it is reasoable to assume that all added (or dropped, bypass) traffic, measured i terms of the umber of lightpaths, is evely distributed amog all the output (or iput) liks. This implies that the umber of dropped (D) or added (A) lightpaths o a lik at a ode is A = D = (N 1) t δ ad the umber of bypassig lightpaths o a lik is I = G D. Case 1: Traffic demad (t) is ot a multiple of the wavebad graularity (W) If the traffic demad (t) is ot a multiple of the wavebad graularity, each of the three layers cotributes to the total port cout of the MG-OXC. Below, we calculate the ports at each of the layers at a ode startig at the FXC layer. We ote that, for a give lik, F a = F d = D K is the umber of full fibers that ca be added or dropped. O the other had, F b = I K is the umber of full fibers that will bypass. The remaiig umber of lightpaths o the lik is thus λ FTB = G F d K F b K. Hece, FTB = λftb K is the umber of FXC layer ports eeded for FTB demultiplexers. Due to the symmetry i added ad dropped traffic (i.e. uiform traffic), we have FTB = BTF (ad F a = F d, as above). Accordigly, at the FXC layer of ode (havig δ liks), the umber of required ports iclude F i = G K ports for iput fibers (from other odes), F a ports for locally added fibers ad BTF ports for lightpaths from BTF multiplexers o each

8 coected lik. Thus, we have: FXC =[F i + F a + BTF] δ (17) From the above aalysis, we kow that there are D = D F d K remaiig lightpaths (per lik) that eed to be dropped (through the BXC ad WXC layers) ad I = I F b K remaiig bypass lightpaths. Hece, B d = D W is the umber of ports for bads to be dropped locally, B b = I W is the umber bypass bads ad BTW = λbt W W is the umber of ports for BTW demultiplexers where λ BTW = λ FTB (B d + B b ) W. Similarly, the umber of ports required at the BXC layer of ode icludes B i = I +D W ports for iput bads, B a (=B d ) ports for locally added bads ad WTB (=BTW) ports for bads from WTB multiplexers o each coected lik. Thus, we have: BXC =[B i + B a + WTB] δ (18) Fially, for the etire etwork, the remaiig MP (F b + F d + F a ) K δ N (B b + B d + B a ) W δ N lightpaths will go through the WXC layer. Hece, the umber of ports required at the WXC layer is: WXC = MP (F b +F d +F a) K δ N (B b +B d +B a) W δ N (19) Therefore, from equatios (17) - (19), we ca obtai the total umber of ports i the etire etwork as: Total =(FXC + BXC ) N + WXC (20) Case 2: Traffic demad (t) is a multiple of the wavebad graularity (W) If the traffic demad per ode pair is a multiple of W, themg- OXC will add/drop/bypass bads, ot idividual wavelegths i order to reduce the umber of ports. Thus, all traffic is switched usig the FXC ad BXC layers oly, ad the WXC layer is ot eeded. Accordigly, at the BXC layer, this is similar to havig a ordiary-oxc, wherei each port switches a bad of wavelegths. Hece, i this case, we obtai the total umber of ports at the iput-side as follows. Equatio (17) gives us the umber of ports required at FXC layer at each ode. After switchig at the FXC layer, the umber of remaiig lightpaths goig through the BXC layer is MP (F b + F d + F a ) K δ N Thus, we have: Total = FXC N + MP (F b + F d + F a) K δ N W Upper ad Lower bouds Note that, i the best case, all the G + A lightpaths ca be added/switched oly at the FXC layer (i.e., o eed for WXC/BXC), ad hece the lower boud o the umber of ports eeded at each ode is ( A K + G K ) δ FXC ports. O the other had, i the worst case, all these lightpaths will have to be added/switched at the WXC layer, ad thus the maximum WXC is (G + A) δ. At the FXC layer, the maximum umber of ports eeded at each ode should also be bouded by (G + A) δ. I additio, i the worst case, all the iput fibers may go through the FTB demultiplexers. Thus, BTF = FTB F δ, ad the, FXC F δ 2. I other words, the maximum FXC is mi[(g+a) δ, F δ 2]. Similarly, the maximum BXC is mi[(g + A) δ, F B δ 2]. Thus, the bouds for the total umber of ports i the etwork are as follows: LowerBoud = ( A K + G ) δ N (22) K UpperBoud = {mi[(a + G) δ, F δ 2] + mi[(a + G) (21) δ, F B δ 2] + δ (A + G)} N (23) VI. NUMERICAL RESULTS AND DISCUSSIONS I this sectio, we focus o simulatio to compare WBS algorithms based o the ILP model, WBO-RWA ad BPHT. We will first preset the results for ILP, WBO-RWA ad BPHT for a radom 6-ode etwork, as our experimets show that the optimal WBS based o ILP formulatio is feasible oly for such a small etwork. We the compare the heuristic algorithms such as BPHT, ad WBO-RWA oly, for the larger 14-ode NSF etwork. We defie the followig three performace-metrics. Each metric is a fuctio of a WBS algorithm. Total port umber ratio T(a): Total(FXC +BXC +WXC ) used by W BS algorithm a Total(OXC ) of ordiary OXC Max port umber ratio M(a): max(fxc +BXC +WXC ) used by W BS algorithm a max(oxc ) of ordiary OXC Used wavelegth-hop ratio W(a): 2 wavelegth hops used by W BS algorithm a wavelegth hops used by RW A without W BS The results preseted below are obtaied via extesive simulatio, ad each poit is the averaged result over a large umber of simulatio rus with differet radom traffic patters. A. Six-ode Network For the six-ode etwork, a traffic matrix is radomly geerated, such that the umber of lightpaths requested by a (s, d) pair is i the rage of 0 4. Whe usig ILP formulatio, we set α = β = γ =1i the objective equatio (1) ad use CPLEX to obtai the optimal results. For three differet represetative radom traffic patters where the total lightpaths (i.e. t p ) is 25, 31 ad 53 respectively, Table I shows the umber of ports ad performace ratios for optimal WBS (based o ILP), WBO-RWA ad BPHT. As the basis for the compariso, the last row (OXC) idicates the miimum total umber of ports required whe ordiary OXCs without WBS are used. The rows T(a), M(a) ad W(a) represet the performace ratios. TABLE I RESULTS FOR THE SIX-NODE NETWORK (F =2, B=2,W=2) Optimal WBS WBO-RWA BPHT tp T(a) M(a) W(a) OXC 71 ( tp =25); 83 ( tp =31); 142 ( tp =53) From the table, we see that the performace of BPHT is close to that of the ILP model (Optimal WBS) ad much better tha that of WBO-RWA, i particular, BPHT ca save about 50% of the total ports tha usig just ordiary OXCs. I additio, i the process of tryig to reduce the total umber of ports, both our ILP solutio ad heuristic (BPHT) have 2 Note that by defiitio W(WBO-RWA)=1.

9 W(a)>1, that is, use more wavelegth-hop (WH) tha the ILP solutio for RWA (i.e. WBO-RWA). This ca be explaied as follows: sometimes, to reduce port cout, a loger path that utilizes a wavelegth i a bad may be chose eve though a shorter path (that caot be packed ito a bad) exists. I other words, miimizig the umber of ports at MG-OXC does ot ecessarily imply miimizig the umber of WHs (eve though miimizig WHs i etworks without MG-OXC is equivalet to miimizig the umber of ports). I fact, there is a trade-off betwee the required umber of WHs ad ports. Though the results are ot show, we ote that i multi-fiber etworks there is a slight improvemet i the performace i terms of T(a), M(a) ad W(a) (over sigle-fiber etworks), which we attribute to the iheret partial wavelegth coversio capability. Additioal results for BPHT ad WBO-RWA i sigle ad multiple fiber etworks will be show i Sectio VI-C. For a large etwork such as the NSF etwork, the ILP becomes itractable, hece we will oly study the previously described heuristic algorithms i the followig sectios. B. NSF etwork - Uiform Traffic I this sectio, we focus o the validatio of our aalysis of BPHT (Sectio V). Figure 4 shows the umber of ports of wavelegths i a bad. This ca be explaied as i Figure 5(a), whe the traffic demad (umber of lightpaths) per ode pair is a multiple of the umber of wavelegths i a bad (i.e. W), usig algorithm BPHT will eable MG-OXCs to add/drop/bypass traffic at the bad graularity, rather tha the wavelegth graularity. This is similar to havig a ordiary- OXC, wherei each port switches a bad of wavelegths, thus reducig the umber of ports. Further, ote that whe t p is such that Mod(t p, 4) = 2 (e.g. t p is 2, 6, 10, 14 etc.), the performace of BPHT is agai close to its aalytical value, this is due to the high probability of groupig bypass traffic as i Figure 5(b). S 0 S 1 S 2 S 3 (a)uiform traffic (t p =8) Fig. 5. b 2 b 3 b 0 b 1 b 0 λ λ 2 3 λ 0 λ 1 S 0 S 1 S 2 S 3 (b)uiform traffic (t p =2) A example of bad groupig by BPHT (W=4) We ca also see that the performace aalysis is accurate, as verified by our simulatio results, especially at the poits where the traffic demad is a multiple of the wavebad graularity. At other poits, the assumptio i the aalysis such as every add/drop traffic is evely distributed o all the output liks causes the deviatio i the performace of BPHT from the aalytical results. C. NSF Network - No-uiform Traffic I our experimets with multi-fiber etworks, we oce agai fid that BPHT performs better tha its variatios, ad hece oly show the compariso of algorithms BPHT ad WBO-RWA, for the sake of cociseess. Fig. 4. T(a) i NSF etwork (F=2). obtaied from our aalysis with simulatio results for the NSF etwork assumig uiform traffic. Curve Aalysis is from Equatio (20) ad (21), curve LowerBoud is from Equatio (22) ad curve UpperBoud is from Equatio (23). The figure illustrate how T(a) chages with the traffic itesity t p (traffic demad per ode pair), assumig that each fiber has 30 bads, ad each bad has 4 wavelegths (so the total umber of wavelegths, is F B W= 240). We see that whe usig heuristic algorithm BPHT, the umber of MG-OXC ports drops sigificatly whe t p is a multiple of W =4(e.g. t p is 4, 8, 12, 16 etc.), the umber Fig. 6. Ratio T(a) (Fixed Load)

10 ports whe compared to havig oly a sigle fiber. O the other had, havig too may fibers causes a icrease i port cout. The ratio W (BPHT), does ot vary much with a chage i the umber of fibers, but remais larger tha 1 by a small amout idicatig the trade-off betwee the umber of WHs ad ports. Fig. 7. Ratio M(a) (Fixed Load) 1) Fixed Load: Figures 6 ad 7 illustrate how the ratios T (a) ad M(a) vary with chagig wavebad graularity (i.e. umber of wavelegths i a bad) ad umber of fibers but a fixed umber of total wavelegths per lik (i.e. F B W= 240) ad a fixed traffic load (i.e. the total traffic does ot chage with F or B or W) but a radom patter. From the figures, we otice that the total umber of ports i the etwork ad the maximum umber of ports at a ode amog all odes by usig BPHT is much less tha those from WBO-RWA, ad heuristic WBO-RWA requires more ports at MG-OXC tha usig ordiary OXCs (as T(WBO-RWA) > 1). Iterestigly, the curves for BPHT i Figures 6 ad 7 also idicate that with a appropriate wavebad graularity (W 6), BPHT performs the best i terms of both T (a) ad M(a), achievig a savigs of early 70% i umber of ports whe usig MG- OXCs istead of ordiary OXCs. More specifically, we otice that multi-fiber MG-OXC etworks perform better tha sigle-fiber MG-OXC etworks, as they ca achieve a larger reductio i port cout whe usig BPHT. This is because with multiple fibers (e.g. F = 4) there is a higher probability to switch lightpaths as a group (whole fiber or bad). I sigle-fiber etworks the advatage of havig a FXC layer ad fiber switchig is ot evidet [1]. O the other had, the situatio is slightly reversed for WBO-RWA, sice WBO-RWA does ot appropriately cosider bad or fiber switchig, the wavelegth assigmet is doe i maer usuitable for reducig port cout. Hece the beefit of multifiber i reducig port cout does ot show up i WBO-RWA algorithm. I additio, Figure 8 shows how the ratios T (a),w(a) ad M(a) vary with chagig umber of fibers per lik (i.e. F)but a fixed umber of wavelegths per lik (i.e. F B W = 240), a fixed wavebad graularity (i.e. W = 4) ad fixed traffic load. From the figure we ca see that with appropriate umber of fibers, havig multiple fibers per lik reduces ecessary Fig. 8. Effect of multiple fibers per lik (Fixed Load) 2) Proportioal Load: Figures 9-10 depict the variatio i total port umber ad maximum port umber with chagig wavebad graularity ad fiber umber but with a fixed umber of wavelegths per fiber (i.e. W B= 60). Fig. 9. Total port cout (Proportioal Load) The traffic demad is directly proportioal to the umber of fibers (i.e. if F is doubled, the traffic demad T [p] is also doubled). Although the results of W(a) are ot show here due

11 of WBS etworks. I particular, with appropriate wavebad graularity, usig MG-OXCs i multi-fiber etworks ca save up to 70% ports compared to usig ordiary-oxcs. While our ILP formulatios ad heuristics are especially useful for the efficiet desig of MG-OXC odes (i.e. the dimesioig of the switchig matrices) for a give set of traffic demads, they ca also be used to miimize the umber of used active ports i a existig etwork, ad thus lower etwork operatig costs, ad reduce blockig probability of future requests. Fig. 10. Maximum umber of ports (Proportioal Load) to space limitatio, the WHs icrease almost proportioally to the umber of fibers (ad traffic load) for BPHT. O the other had, we otice that the total umber of ports ad maximum port umber icrease sub-liearly with a icrease i the umber of fibers (ad traffic load) for BPHT. This is because there is a higher probability for wavelegth groupig (ito fibers ad bads) i multi-fiber etworks usig the BPHT algorithm. However, we fid that the total port umber ad maximum port umber for WBO-RWA icrease rapidly with a icrease i the umber of fibers, further idicatig the effectiveess of our proposed BPHT algorithm. VII. CONCLUSION I this work, we have studied the problem of optimal WBS i multi-graular all-optical mesh etworks. We have proposed a ew MG-OXC architecture which is more flexible tha ay existig architectures. We have also adopted the most powerful wavebad assigmet strategy ad developed a correspodig ILP ad a efficiet heuristic algorithm called BPHT. The ILP model ad the heuristic algorithm (which uses a ew multifiber wavebad assigmet algorithm) ca hadle the case with multiple fibers per lik, ad hece are also more geeral tha our previously proposed sigle-fiber solutios. We have verified that the proposed BPHT heuristic ca achieve ear-optimal results by comparig its performace with that of the ILP formulatio (which is feasible for small etworks oly). I additio, the performace of BPHT has bee aalyzed ad the results from aalysis are verified with simulatio. We have also compared the performace of BPHT with that of a heuristic that uses ILP to perform optimal RWA via extesive simulatios for varyig etwork topologies ad traffic patters, ad show that BPHT is sigificatly better. Our performace evaluatio has also show that WBS is eve more beeficial i multi-fiber etworks. I additio, the wavebad graularity has a large effect o the performace REFERENCES [1] X. Cao, Y. Xiog, V. Aad, ad C. Qiao, Wavelegth bad switchig i multi-graular all-optical etworks, i SPIE s Proc. vol. 4874, OptiComm 02, Bosto Massachusetts, 2002, pp [2] K. Harada, K. Shimizu, T. Kudou, ad T. Ozeki, Hierarchical optical path cross-coect systems for large scale WDM etworks, i Proceedigs - OFC, 1999, pp. WM55 3. [3] L. Noirie, M. Vigoureux, ad E. Dotaro, Impact of itermediate groupig o the dimesioig of multi-graularity optical etworks, i Proceedigs - OFC, 2001, pp. TuG3 3. 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