1 ARCHITECTURAL DESIGN AND STUDIES OF REQUIREMENTS IN METROPOLITAN AREA NETWORKS Richard Rabba and Edmond Kayi Lee Tel: Fax: Fujisu Laboraories of America, Inc. 595 Lawrence Expressway, Sunnyvale, CA Tel: Fax: Massachuses Insiue of Technology 77 Massachuses Avenue, Cambridge, MA Absrac: Key words: We sudy he offline raffic pah assignmen in mero ring neworks ha use Wavelengh Division Muliplexing (WDM) ransmission. We describe in his paper he issues relaing o capaciy, conneciviy and flexibiliy requiremens for supporing daa and voice raffic in he Meropolian Area Nework. We define an opimizaion problem and presen a novel Mixed Ineger Program o solve he opimizaion problem. The novely lies in he abiliy of he program o make recommendaions for he ypes of nodes on he ring and o carry as well as no carry a proporion of raffic. We conduc several sudies for simulaed scenarios of raffic growh ha allow us o make nework design and node placemen decisions. In addiion, we sudy failures, opology changes and raffic upsurge due o special evens and heir effec on he nework. Resuls of hese sudies allow us o make choices on raffic served and ypes of nodes used in order o allow resiliency o failure and rerouing of raffic based on a new opimal soluion. nework design opimizaion, mixed ineger programming, meropolian area neworks.
2 2 Richard Rabba and Edmond Kayi Lee 1. INTRODUCTION Wavelengh Division Muliplexing (WDM) has experienced widespread adopion in a relaively shor period of ime, moving from he core nework o he bandwidh-sarved mero nework. Today s mero neworks have been buil up wih lile consideraion o he raffic hey are supposed o carry. They essenially deploy SONET/SDH service ha offers Time-Division Muliplexing (TDM) circuis. We describe in his paper he capaciy, conneciviy and flexibiliy requiremens for supporing daa and voice raffic in he Meropolian Area Nework. The mero nework remains o his dae he larges underdeveloped par of he nework. We define an opimizaion problem and presen a novel Mixed Ineger Program ha balances conflicing requiremens of carrying subsanial raffic while remaining cos-effecive. 1.1 Previous Work Several works have addressed he opic of efficien rouing and wavelengh assignmen. Oher works have looked a circui selecion in deail. Reference  discusses dynamic rouing of resorable bandwidh guaraneed pahs, where wo pahs are se up o allow for resoraion, wih no a priori knowledge of fuure raffic. Reference  discusses he use of a conneciviy marix, a concep ha we use in his paper. I also refers o he issue of node configuraion o opimize he size of such node, given he cos srucure of swiches. This is discussed furher in , where he auhors describe a mehodology for choosing among some ypes of cross-connecs. References  and  presen algorihms o solve he classic problem of Rouing and Wavelengh Assignmen while  and  focus on he use of wavelengh converers. Reference  invesigaes differen mehods of solving he problem of designing and opimizing he physical opology of all-opical neworks wih and wihou he use of wavelengh conversion. Finally,  proposes he use of ring neworks wih limied wavelengh conversion capabiliy ha can suppor a large number of lighpah requess. 1.2 Focus of Meropolian Nework Sudy While hese papers address imporan problems of opical neworking, hey do no consider a soluion o he problem of changing raffic mix and requiremens, increasing volume and dealing wih special evens including a shor-lived even such as a naural disaser and long-lived evens such as a killer applicaion. This paper addresses he problems relaed o he efficien allocaion of bandwidh (in he form of circuis) in a ring nework, by aking
3 Archiecural Design and Sudies of Requiremens in Meropolian Area Neworks 3 ino consideraion he relaive cos of supporing each ype of raffic, he cos of deploying equipmen (nodes and fibers) and he opporuniy cos of no supporing some raffic. Secion 2 presens a formal formulaion of he problem, inroduces our assumpions of he raffic and nework model and discusses he auxiliary graph ha we consruc from he original nework graph as well as simplifying assumpions needed in solving he problem. Secion 3 solves he problem as a Mixed Ineger Program. Inpus, noaions and variables are presened in his secion. Secion 4 presens he simulaion sudy done using our raffic assumpions wih experimens dealing wih increasing bandwidh demand, special evens, opology augmenaion and componen failures. Secion 5 presens analyical resuls for simplifying assumpions ha we use in our simulaion. Secion 6 concludes his paper. 2. PROBLEM FORMULATION We formulae he problem as a nework design and raffic assignmen problem. The nework design problem finds opimal node funcionaliy while he raffic assignmen problem opimizes he circui selecion. The inpu of he problem is he physical opology of he nework, raffic demand marix and he available node ypes for differen nodes in he nework. Each node ype is associaed wih an equipmen cos and each uni of raffic is associaed wih an opporuniy cos which represens he revenue derived from he raffic demand. Also, an operaional cos can be used o model he uilizaion of nework resources. The objecive is o selec he node ype for each node and assign he raffic in order o minimize he cos funcion. 2.1 Modeling Traffic Inpu We assume here are muliple classes of raffic, where each class of raffic has a differen cos as well as differen bandwidh requiremens. Each raffic class k is represened by an N N raffic marix D k, where each elemen specifies he number of connecion requess from node i o node j. We assume all connecion requess have he same bandwidh demand, which is normalized o Nework Model The nework is modeled as a direced graph. Each link in he graph represens a unidirecional fiber link in he nework. Each fiber carries W wavelengh channels, and each channel has G unis of capaciy. For
4 4 Richard Rabba and Edmond Kayi Lee example, if he connecion requess are STS-1 circuis and he wavelenghs are OC-48 channels, we se G o 48. Each node is associaed wih a se of available node ypes, each of which has a cerain cos and funcionaliy. The funcionaliy of a node ype is modeled by a pair (g, M ). The elemen g is he swiching granulariy of he node ype, which specifies he number of individual circuis he node ype processes as a single uni. If he node ype is a packe swich, we assume g = (since a packe could be very small, his is a fair assumpion). Oherwise we assume ha g divides G, and each wavelengh channel is divided ino G/g imeslos, where each imeslo can carry g unis of raffic. The marix M is he conneciviy marix ha describes he por-o-por conneciviy of he node ype a he wavelengh level. The marix elemen is se o 1 if raffic from he i h inpu wavelengh-por can be swiched o he j h oupu wavelengh-por and 0 oherwise. Given he inpus, we firs consruc an auxiliary graph so ha he rouing, wavelengh and TDM channel assignmen of he connecion requess can be modeled simulaneously. The auxiliary graph presens a more deailed model for he nodes and fibers in he physical nework. Each node in he nework is modeled by a sub-graph, and such sub-graphs are conneced by parallel edges represening he fibers. This exends work done by Cinkler e al  o solve he problem of node selecion. In , he node ype a each locaion is fixed. In our approach, variable node ypes are candidaes for each locaion; so insead of modeling he node archiecures as differen subgraphs, we capure hem in he MIP formulaion by assigning each edge a capaciy. The assignmen of roue and wavelengh channel for a connecion reques can be deermined by a corresponding pah in he auxiliary graph, which fully specifies he roue, wavelengh channels and he ype of swiching (size of channel) aken a each node. To avoid confusion beween he physical graph and auxiliary graph, in his paper we use he erms nodes and links when we refer o he objecs in he physical graph, and he erms sub-nodes and sub-links when referring o objecs in he auxiliary graph. 2.3 The Auxiliary Graph A sub-graph is consruced for each node in he nework o model in he swiching and add/drop capabiliy of he node. Each node has an inpu erminal por and an oupu erminal por, where raffic is added and dropped respecively, as well as a number of inpu pors and oupu pors, which receive or deliver raffic from an upsream or downsream node. In he subgraph, we creae a sub-node for each wavelengh for each of wo pors. In addiion, a pair of inpu/oupu represenaive sub-nodes is creaed for each
5 Archiecural Design and Sudies of Requiremens in Meropolian Area Neworks 5 node o model he enry and exi poins for he raffic sourced/desined a he node. The inpu sub-nodes are conneced o he oupu sub-nodes by direced sub-links o model possible raffic flow pahs. The represenaive sub-nodes are conneced o he sub-nodes ha belong o he inpu/oupu erminal pors. Represenaive sub-node Sub-node for erminal por Sub-node for inpu por Sub-node for oupu por Figure 1. An auxiliary graph for wo conneced nodes The sub-graphs of he nodes are inerconneced by sub-links according o heir physical opology. Figure 1 shows he auxiliary graph modeling wo conneced nodes, each wih one inpu por, one oupu por, a pair of inpu/oupu erminal pors and wo wavelenghs. 2.4 Simplifying Assumpions In his paper we model he deails of he node on he wavelengh level. Wihin he same wavelengh channel, we assume ha raffic in a imeslo can be swiched o a differen imeslo. This may no be rue for equipmen such as SONET Add-Drop Muliplexers (ADM s), in which raffic says in he same imeslo unless i is added or dropped. An alernaive approach is o model he nodes a he imeslo level. From an absrac poin of view a imeslo is equivalen o a wavelengh channel, which maps o a physical wavelengh service. In ha case, we can model he sysem using he same idea of he auxiliary graph, in which a sub-node represens he uni of a imeslo raher han a wavelengh. However, such a model would increase he number of sub-links by a facor of O(G 2 ) in he auxiliary graph. Therefore, we consider he model a he wavelengh level as
6 6 Richard Rabba and Edmond Kayi Lee an approximaion of he more precise model. The inaccuracies inroduced will be discussed in secion A MATHEMATICAL FORMULATION The node selecion and raffic assignmen problem can be formulaed as a Mixed Ineger Program (MIP). Basically, for each node i and node ype, here is a variable w i indicaing wheher ype is chosen for node i. In he auxiliary graph, here is a capaciy u pq for each sub-link (p, q) in he subgraph represening a node, limiing he maximum amoun of raffic ha can be assigned o he sub-link. The capaciy for each sub-link depends on he node ype seleced for he node. For example, if ype is seleced for he node and ype canno swich raffic from wavelengh-por p o wavelenghpor q, hen he capaciy u pq is 0. Oherwise, he capaciy u pq is G. k,ij We denoe by x pq he flow variable for each sub-link specifies he amoun of class k raffic, sourced a node i and desined o node j, assigned o he sub-link (p, q). The amoun of assigned raffic canno exceed he capaciy u pq of he sub-link. 3.1 Inpus, Noaions and Variables We define he noaions used in our MIP formulaion. In he res of he paper, we use indices i and j o refer o he nodes in he nework, and indices p and q o refer o sub-nodes of he auxiliary graph. Table 1 describes he lis of inpu parameers; able 2 describes some addiional noaions used in he MIP formulaion; able 3 describes variables used in he MIP. Parameer V G T H i M pq g K k D ij α β k γ pq Table 1. Descripion of Inpu Parameers Descripion Nodes in he physical opology Grooming granulariy Se of node ypes Enry in he node ype availabiliy marix. Se o 1 if ype available a node i's locaion and 0 oherwise Enry in he conneciviy marix of node ype. Se o 1 if raffic can be forwarded from wavelengh-por p o wavelengh-por q and 0 oherwise Grooming granulariy for node ype Se of raffic classes Traffic demand from node i o node j for class k raffic Equipmen cos for node ype Opporuniy cos of each uni of class k raffic if uni of demand no saisfied by nework Operaional cos for using a uni of resource on sub-link (p,q)
7 Archiecural Design and Sudies of Requiremens in Meropolian Area Neworks 7 Noaion V A E A I(p) O(p) s(i) E(i) R Table 2. Descripion of addiional noaions Descripion Sub-nodes in he auxiliary graph Sub-links in he auxiliary graph Se of sub-nodes conneced o sub-node p in he auxiliary graph Se of sub-nodes ha sub-node p is conneced o in he auxiliary graph Represenaive verex in he auxiliary graph for node i Edges connecing he wavelengh-pors in he sub-graph in he auxiliary graph represening node i Se of represenaive sub-nodes in auxiliary graph Table 3. Descripion of variables used in formulaion Variable Descripion w i Variable indicaing wheher ype node is seleced for node i u pq Capaciy of sub-link (p, q) in he auxiliary graph i, v pq For node ype a node i, number of ime-slos assigned for swiching from wavelengh por p o wavelengh por q k,ij x pq Amoun of class k raffic from node i o node j assigned o sub-link (p, q) k y ij Amoun of class k raffic saisfied from node i o node j z pq Toal amoun of raffic assigned o sub-link (p, q) Le us describe he MIP formulaion. We explain he objecive funcion and he consrains used in he program. 3.2 MIP Formulaion Minimize Subjec o: k k k + ( ) + α w β D y γ z i ij ij pq pq i V T k K i, j V A ( pq, ) E wi = 1 i V (1) T wi Hi i V, T (2) u, A pq G i V q V \ S (3) p I( q) u, A pq G i V p V \ S (4) q O( p) g v GM w i V; T;( p, q) E( i) (5) i, pq pq i u g v i V;( p, q) E( i) pq k ij i, pq T k, ij s ( i) q, ; (7) q O( s( i)) y = x i j V k K (6)
8 8 Richard Rabba and Edmond Kayi Lee pq kij, A pq (, ) (8) k Ki, j V z = x p q E, ; (9) A x = x i, j V; p V \ S; k K (10) kij, k y Dij i j V k K kij, kij, qp pq q I( p) q O( p) z pq upq i V;( p, q) E( i) (11) 0 wi 1 T, i V (12), 0 v i G T; i V;( p, q) E( i) (13) pq kij, xpq 0 i, j V;( p, q) E ; k K (14) wi Z T, i V (15) kij, A xpq Z i, j V;( p, q) E ; k E (16) i, v Z if g (17) pq The objecive funcion consiss of hree cos componens. The firs erm is he equipmen cos, which akes ino accoun he ype of equipmen seleced for each node. The second componen is he opporuniy cos, defined o be he loss of revenue for he raffic ha he nework is unable o suppor; his is calculaed as he oal raffic demand minus he amoun of raffic saisfied. The hird componen is he operaional cos, which is he variable cos incurred by uilizing nework resources for carrying exising raffic. The goal is o minimize he oal cos defined here. Consrain (1) requires ha exacly one ype of equipmen be seleced for each node. Consrain (2) saes ha he node ype can be chosen for node i only if he node ype is available a node i. Consrains (3) and (4) se he capaciy of each wavelengh por o G. Consrain (5) saes ha raffic can be swiched from wavelengh por p o wavelengh por q using node ype if and only if node ype is available and seleced for node i, and he number of imeslos assigned is limied by he capaciy of he wavelengh. Consrain (6) compues he capaciy assigned o sub-link (p, q). Consrain (7) simply compues he oal raffic demand saisfied for he class k raffic from node i o node j. Consrain (8) compues he oal raffic assigned o sub-link (p, q). Consrain (9) saes ha he oal raffic saisfied for each class beween each node pair is limied by he raffic demand. Consrain (10) is he flow conservaion consrain. Consrain (11) saes ha he amoun of raffic assigned o sub-link (p, q) canno exceed he capaciy of he sub-link. Consrains (12), (13) and (14) specify he variable ranges. Consrains (15), (16) and (17) are ineger consrains. In paricular, for consrain (17), if he node ype is a packe swich, ha is, g = 0, hen he capaciy assigned o sub-link (p, q) can be a real number.
9 Archiecural Design and Sudies of Requiremens in Meropolian Area Neworks 9 4. SIMULATION We have implemened a ool solving he node selecion and raffic assignmen problem. To solve he MIP, he simulaion ool calls he opimizaion library provided by CPLEX. The ime needed o solve he simulaed problems varied from several seconds o a few hours. 4.1 Traffic Inpu In our simulaion, we use raffic daa expecaions based on raffic sudies conduced by Andrew Odlyzko  and Lawrence Robers . In paricular, we assume ha he oal raffic demand is a combinaion of raffic saying in he mero (defined as local raffic) and raffic going o/from he core (defined as hubbed raffic). The simulaed nework uses 1/50 h of he oal bandwidh. We reproduce inpu raffic values in able 4. Table 4. Traffic volume expecaions in represenaive mero nework [ ] Year Toal raffic demand (Gbps) The oal raffic demand includes boh he local raffic (raffic ha is sen beween nodes of he mero ring) and core raffic (raffic coming from or going o he core nework hrough he gaeway or hub node). We assume he proporion of local raffic increases linearly from 30% in 2002 o 50% in This would reflec more communiy-based daa exchanges. The local raffic is assumed o be disribued uniformly among he nodes on he Mero nework. 4.2 Simulaion Experimens We carried ou four differen experimens in order o sudy he effec on nework performance under changes in differen aspecs, including: Archiecures wih differen node ypes and capaciies Abiliy o suppor exra raffic in he case of special evens Changes in physical opology Componen Failures.
10 10 Richard Rabba and Edmond Kayi Lee Effec of Differen Node Types The nework under sudy has 6 nodes, conneced in a ring opology bidirecionally. We assume ha a de-muliplexer and a muliplexer funcion are used for all he simulaions. We sudy four differen archiecures: a) ADM s and Opical Bypass (ADM/OB): Each node erminaes some of he wavelenghs while ohers are opically bypassed. b) ADM/OB wih Full Cross Connec a Hub (ADM/OB/FCCH): I is similar o ADM/OB, wih a full cross-connec a he hub node. c) Sacked Ring (SR): We use his erminology o indicae ha all wavelenghs are erminaed a every node using ADM s. In ha respec, raffic on one wavelengh canno be swiched o a differen wavelengh. d) Sacked Ring wih Full Cross Connec a Hub (SR/FCCH): I is similar o SR wih a full cross-connec a he hub node. Archiecure A Archiecure B Number of Circuis Saisfied Toal W=4, G=48 W=8, G=48 W=4, G=192 Number of Circuis Saisfied W=4, G=48 W=8, G=48 W=4, G=192 Toal Archiecures C and D Comparison Char for OC-192, W= Number of Circuis Saisfied W=4, G=48 W=8, G=48 W=4, G=192 Toal Traffic Number of Circuis Saisfied ADM/OB ADM/OB/FCCH SR SR/FCCH Figure 2. Simulaion resuls for differen archiecures For each of he above archiecures, we evaluae he performance on hree differen ses of channel capaciies: four OC-48 channels, eigh OC-48 channels, and four OC-192 channels. In he archiecures wih opical bypass, we assume he node erminaes half of he wavelenghs. However, we le he solver selec he opimal se of how i places he wavelengh bypasses a each node.
11 Archiecural Design and Sudies of Requiremens in Meropolian Area Neworks 11 Figure 2 shows he performance of differen nework archiecures wih differen capaciies. I also gives a comparison among differen nework archiecures for a four-wavelengh, OC-192 sysem. The performance measure here is he number of circuis saisfied by he nework. Simulaion resuls show ha he differen nework archiecures will come o sauraion in years 2004 o 2007, depending on he node ypes in he nework, number of wavelenghs as well as he bandwidh capaciy of each wavelengh channel. A four-wavelengh OC-48 sysem sars o saurae in 2004 or 2005, whereas a four-wavelengh OC-192 sysem can susain he increasing raffic demand hrough Among differen archiecures, he neworks wih opical bypass suppor fewer circui requess han heir opaque counerpars: raffic canno be added or dropped a he opically bypassing wavelenghs, hereby reducing flexibiliy. Using a full cross-connec a he hub improves performance, bu sill underperforms a sacked ring for he projeced fuure raffic. One ineresing observaion is ha he performance of SR and SR/FCCH seems idenical. Tha means he abiliy o cross-connec raffic among differen wavelenghs does no help improve performance significanly. We will address his observaion in secion 5, where we conduc a mahemaical analysis comparing archiecures using ADM s and cross-connecs Suppor of Exra Traffic In his secion, we sudy he abiliy of he nework o handle special evens ha lead o a sudden surge in raffic demand. Such exra raffic is suppored by he available bandwidh of he nework, afer saisfying he projeced normal raffic. Hub node Non-hub node (a) (b) Figure 3. Exra raffic in special even, (a) o he hub, and (b) o a local node
12 12 Richard Rabba and Edmond Kayi Lee We look a wo simple models of special evens where all he exra raffic is direced o he hub in one case, and all he exra raffic is direced o a local node in he oher. Figure 3 illusraes he wo scenarios. Figure 4 shows he number of exra circuis ha can be suppored in he wo scenarios. As normal raffic demand grows over he years, he amoun of bandwidh available for exra raffic decreases. In addiion, more exra raffic can be suppored if he raffic is direced o a local node. This is due o he fac ha he link conneced o he hub is he mos highly uilized link. (a) Exra Circuis Direced o Hub 3500 W=4, G= W=8, G=48 W=4, G= (b) Exra Circuis Direced o Local Node W=4, G=48 W=8, G= W=4, G= Figure 4. Exra circuis ha can be suppored on a 6-node ring of sacked ADM s Topology Augmenaion We now sudy he effec of differen opologies on he performance of he nework. Specifically, we sudy he effec of adding one or muliple fibers. The opologies we consider are depiced in figure 5. Figure 5 also shows he number of circuis saisfied for a four-wavelengh OC-192 nework. T0 T1 T T3 T4 T5 Number of Circuis Saisfied T0 T1 T2 T3 T4 T5 Toal Figure 5. Topology augmenaion scenarios and simulaion resuls
13 Archiecural Design and Sudies of Requiremens in Meropolian Area Neworks 13 Alhough adding an exra fiber helps o suppor more raffic, he exen of improvemen varies depending on which fiber is added. The improvemen is more significan when a new fiber is added o connec he hub node o a local node since he links conneced o he hub are he boleneck. Topology T0 suppors a smaller amoun of he oal projeced raffic afer he year 2004 while opology T5 suppors he oal raffic up o This indicaes ha he ring opologies impose sringen resricions on abiliy o carry raffic Failure Sudies We look a he effec of failures on he nework performance. The failures considered include fiber failure and ransponder failure. The scenarios sudied are depiced in figure 6. F1 and F2 depic a fiber failure, a he hub and a non-hub node respecively, while F3 and F4 show a ransponder failure affecing hub and non-hub nodes as well. F1 F F3 F4 Number of Circuis Saisfied Iniial F1 F2 F3 F Effecs of failures for OC-48 Figure 6. Differen failure scenarios and simulaion resuls The resuls for four-wavelengh OC-48 are shown. As expeced, a fiber failure shows more significan impac on performance han a ransponder failure. There is service disrupion for par of he saisfied raffic for boh fiber failures and ransponder failures. The locaion where he failure akes place also affecs performance. A failure closer o he hub node leads o larger disrupion han a failure away from he hub node, especially in he earlier years. The difference dwindles gradually as he raffic is more localized as more unsaisfied raffic accumulaes.
14 14 Richard Rabba and Edmond Kayi Lee 5. ANALYTICAL RESULTS In his secion, we compare he performance of he sacked ring archiecure versus a ring wih full cross-connecs. We analyze he number of wavelenghs required for he wo nework archiecures in order o saisfy all he connecion requess for a given class of raffic. The resul can be inerpreed in wo ways. Firs, his would give us insigh abou he benefi wih full cross-conneciviy. Specifically, his gives us an idea how many exra wavelenghs is sufficien o make up for he inabiliy o swich raffic among differen wavelenghs. Second, his gives us a bound on inaccuracies inroduced by he simplifying assumpion discussed in Secion 2.4, in which we assume raffic can always be swiched among differen TDM imeslos wihin he same wavelengh. From an absrac poin of view, a TDM imeslo is funcionally equivalen o a wavelengh channel. Therefore a W- wavelengh, g-imeslo sysem can be considered as a Wg-wavelengh sysem wih 1 imeslo. The abiliy o swich among imeslos can be considered as having muliple parial cross-connecs inside he swich. The performance of such an archiecure will lie somewhere beween he one wihou cross-connecs and one wih full cross-connec capabiliy. In he following analysis, we assume he ring nework has N nodes and each connecion requess a lighpah (i.e., enire wavelengh) beween a wo nodes in he nework. The raffic demand is represened by a raffic marix M. Given he marix M, le W SR (N,M) be he minimum number of wavelenghs required for an N-node sacked ring o saisfy all raffic specified by M. Similarly, le W CC (N,M) be he minimum number of wavelenghs required for a ring wih full cross-connecs. The performance raio: R(N,M) = W SR (N,M) / W CC (N,M) will be used for he comparison. The raio is a leas 1 because a nework wih full crossconnecs will never require more wavelenghs han a sacked ring. Theorem 1: For arbirary M, 1 RNM (, ) 2 Proof: Direc resul from , p.372. Theorem 2: If raffic marix M is a mix of hubbed raffic and uniform local raffic, where he proporion of local raffic is p. Then: 1 RNM (, ) 1+ p pn + (1 p)( N 1) 2 Proof: Suppose we decompose M ino M l and M h, where M l is he raffic marix for local raffic and M h is he marix for he hub raffic. Le s(m) be
15 Archiecural Design and Sudies of Requiremens in Meropolian Area Neworks 15 he sum of enries in he marix M. Then s(m l ) is he oal local raffic reques, and s(m h ) is he oal hub raffic reques. We have he equaion: s( Ml) s( Mh) = p 1 p (18) For hubbed raffic, since every connecion reques has o go hrough one of he four fibers, for any rouing here is a fiber wih load a leas s(m h )/4. Le L h be he load of he highes loaded link. Then we have: s( M h) Lh (19) 4 For an N-node ring, each enry in M l has value d, where d = sm ( l ) N( N 1) (20) Equaion (8.1) in , p.360 saes ha he load on each link conribued by he local raffic is: 2 dn L l = (21) 8 From (19) and (21), here is a fiber wih load a leas L h + L i ; herefore: 2 CC dn W ( N, M) Lh + (22) 8 Equaion (8.8) in , p.362 implies ha local raffic specified in M l can be suppored wih W l wavelenghs wihou wavelengh conversion, where 2 N N Wl d (23) Hubbed raffic can be suppored by L h wavelenghs. For a sacked ring he oal number of wavelenghs required W SR (N,M) is bounded as follows:
16 16 Richard Rabba and Edmond Kayi Lee W ( N, M) W + W SR SR SR h l 2 N N Lh + d( + ). 8 4 (24) As a resul, he raio is given by (22) and (24): 2 N N Lh + d dn SR + W ( N, M) 8 4 =1+ 4 CC W ( N, M) dn dn Lh + Lh dn 1 4 p + =1+ 2 sm ( h ) dn pn + (1 p)( N 1) (25) If all raffic is hubbed, i.e. p=0, he performance raio is 1. Tha means a ring wih full cross-connecs will no have any advanage over a sacked ring. On he oher hand, if he raffic is local and uniformly disribued, i.e. p=1, he raio becomes 1+2/N. Full cross-connecs become more beneficial as raffic disribuion becomes more local. When p=0.5, he performance raio would be a mos 9/8 for a 6-node nework and he resuls of he simulaion. This also gives a bound on he inaccuracies inroduced by assuming ha raffic can be swiched beween differen imeslos. 6. CONCLUSION We have sudied he issue of raffic and node placemen in he opical mero nework. We have presened a Mixed Ineger Program ha solves boh problems. In ha respec, we have assumed a cerain raffic growh in he meropolian area nework and have presened he changes and increased conneciviy needs o accompany ha raffic growh. We have also sudied he effec of dramaic change on he nework including conneciviy change (increased conneciviy and componen failures) and special evens. We have shown he advanage of increasing he number of wavelenghs and increasing he node conneciviy on he nework uilizaion. Since his sudy focuses on off-line compuaion, issues such as compuaional speed and finding a polynomial-ime algorihm ha approximaes he resuls of he MIP were no considered. Fuure work will
17 Archiecural Design and Sudies of Requiremens in Meropolian Area Neworks 17 address online compuaion sudies o deal wih dynamic bandwidh changes using heurisic algorihms. We are currenly assessing he performance of several heurisics o solve he problems in polynomial-order ime. We would also like o explore more dynamic nework changes using predicion mehods based on connecion reques arrival imes o solve he online problem efficienly. REFERENCES  M. Kodialam and T.V. Lakshman, Dynamic Rouing of Locally Resorable Bandwidh Guaraneed Tunnels using Aggregaed Nework Resource Usage Informaion, INFOCOM  Y.-W. Leung, Design of Node Configuraion for All-Opical Muli-Fiber Neworks, IEEE/ACM Transacions on Neworking, Vol. 50, No. 1, Jan  G. Jeong and E. Ayanoglu, Comparison of Wavelengh-Inerchanging and Wavelengh- Selecive Cross-Connecs in Muli-Wavelengh All-Opical Neworks, INFOCOM  D. Cavendish and B. Sengupa, Rouing and Wavelengh Assignmen in WDM Rings wih Heerogeneous Wavelengh Conversion Capabiliies, INFOCOM  B. Ramamurhy and B. Mukherjee, Wavelengh Conversion in WDM Neworking, IEEE Journal on Seleced Areas of Communicaion, vol. 16, no. 9, Sep  G. Xiao, Two-Sage Cu Sauraion Algorihm for Designing All-Opical Neworks, IEEE Transacions on Communicaions, Vol. 49, No. 6, Jun  S. Subramanian, M. Azizoglu and A. Somani, All-opical neworks wih sparse wavelengh conversion, IEEE/ACM Transacions on Neworking, Vol. 4, No.4, Aug  G. Li and R. Simha, On he Wavelengh Assignmen Problem in Mulifiber WDM Sar and Ring Neworks, INFOCOM  R. Ramaswami and G. Sasaki, Muliwavelengh Opical Neworks wih Limied Wavelengh Conversion, IEEE/ACM Transacions on Neworking,  Coffman, K. G., and A. M. Odlyzko, Inerne Growh: Is There a Moore s Law for Daa Traffic?, Handbook of Massive Daa Ses, J. Abello, P. M. Pardalos, and M. G. C. Resende, eds., Kluwer,  Robers, Lawrence, Inerne Traffic Measuremen 2001 and 2002, Caspian Neworks, January 2002, Whie Paper.  Tibor Cinkler, Dániel Marx, Claus Popp Larsen, Dániel Fogaras, Heurisic Algorihms for Join Configuraion of he Opical and Elecrical Layer in Muli-Hop Wavelengh Rouing Neworks, Proceedings of INFOCOM  R. Ramaswami and K. Sivarajan, Opical Neworks: A Pracical Perspecive, Morgan Kaufmann, 1998.