Capacity Planning of Survivable MPLS Networks Supporting DiffServ

Capacity Panning of Survivabe MPLS Networks Supporting DiffServ Kehang Wu and Dougas S. Reeves Departments of Eectrica and Computer Engineering and Computer Science North Caroina State University kwu@unity.ncsu.edu, reeves@eos.ncsu.edu Abstract It is essentia for ISPs to offer both performance and survivabiity guarantees at the IP/MPLS ayer. Capacity panning is needed to ensure there wi be sufficient resource avaiabiity and quaity of service under norma circumstances, and during network faiures. In this paper we study the issue of capacity panning for survivabe MPLS networks providing DiffServ EF and BE traffic casses. Our goa is to minimize the tota ink cost, subject to the performance and survivabiity constraints of both EF and BE casses. The probem is formuated as an optimization probem, where we jointy seect the routes for edge to edge EF and BE user demand pairs, and assign a discrete capacity vaue for each ink. Capacity for the EF cass is fuy restorabe under the singe ink faiure mode, whie restorabiity of the BE cass can be adjusted by the network operator. We propose an efficient soution approach based on the Lagrangian Reaxation and subgradient methods. Computationa resuts show that the soution quaity is within a few percent of optima, whie the running time remains reasonabe for networks with 1000 nodes, 2500 inks, and 40,000 demand pairs. This represents the first work on capacity panning for survivabe mutipe-cass-of-service IP/MPLS networks with non-inear performance constraints. 1 Introduction The ast few years witnessed the exposive growth of mission-critica data traffic carried by the Internet, and the emergence of genera societa dependencies on the Internet. We are now amost as dependent on This work is supported by DARPA and AFOSR (under grants F30602-99-1-0540 and F49620-99-1-0264). 1

the avaiabiity of the communication networks as on other basic infrastructures, such as roads, water, and power. With up to 100 terabits per second of data traffic possiby carried in a singe fiber with DWDM, faiure of a singe fiber coud be catastrophic. Despite efforts to physicay protect the fiber optic cabes, cabe cuts are surprisingy frequent according to the FCC. Survivabe network design, which refers to the incorporation of survivabiity strategies to mitigate the impact of the given faiure scenarios, has become a vita part of network design, not an afterthought. The current network architecture is moving toward a two ayers structure where IP/MPLS [15] [29] is on top of optica WDM networks. Traditiony, network resiience primariy reies upon the functionaities provided by SONET and ATM [35][31]. As mentioned in [18], most ISPs rey on the IP ayer when faiures occur. But IP-based faiure recovery schemes suffer from ong deay and/or routing instabiity [27]. MPLS-based recovery has become a viabe soution because of the faster restoration time than IP rerouting. There are many works that investigated recovery schemes in MPLS enabed networks [14][20]. With MPLS ayer protection, the faiure is detected either by the MPLS ayer detection mechanism (such as exchange of Heo message ), or the signaing message propagated from the ower ayer [14]. Once a faiure is detected, protection switching based recovery mechanisms are used. A LSP is protected by a pre-estabished recovery path or path segment. The resource aocated to the recovery path, the socaed spare capacity, may be fuy avaiabe to preemptibe ow priority traffic. The spare capacity can be shared by mutipe recovery paths corresponding to different faiure scenarios, to further reduce the resource redundancy [19]. With MPLS-based recovery, it is possibe to differentiate the eve of protection for different casses of service [5]. For exampe, Expedited Forwarding (EF) based service, such as virtua ease ine service, can be supported by ink protection with a reserve path that offers 100% restorabiity. Best effort (BE) traffic, on the other hand, may not need to be fuy restorabe, or may simpy rey on IP rerouting without being assigned any spare capacity in the network. The cost of redundancy for survivabiity can be very high, compared to a corresponding network 2

designed ony to serve the working demands under norma conditions. One of the major interests in survivabe network design has been providing cost-efficient resource aocation. Most of the work on survivabe network design has been concentrated on the cassic spare capacity aocation probem (SCA) [13][23][22]. The SCA probem assumes that information about the operationa network is given; the focus is ony on the cacuation of backup paths and spare capacities, not the primary path and tota capacity. It has been shown in [19], however, that even though optimizing the working path and the backup path together is computationay more expensive, it offers significant advantage for the tota capacity savings. Since capacity panning is usuay performed infrequenty (on the time scae of weeks to months), we can afford the extra running time in exchange for a ower network cost. With the popuarization of e-commerce and new vaue-added services over IP, it is desirabe for the next generation Internet to offer Quaity of Service (QoS), the abiity to support mutipe traffic casses with different eves of performance guarantees. MPLS and Differentiated Service [8][28] are regarded as two key components of QoS. There has been work on survivabe network design touching on the issues of muticass traffic, such as the architectura aspects of differentiated survivabiity [5]. However, no quantitative treatment of the subject has been done so far. In this paper, we study the issue of capacity panning of survivabe MPLS networks. The targeted MPLS networks wi be providing DiffServ EF and BE traffic casses [34][11]. Since our approach requires a precise performance mode for optimization, we do not incude the AF traffic casses in this paper, due to the ack of a consensus on the impementation of the AF PHB [4]. We focus on the probem of resource dimensioning and traffic routing, and assume that the network topoogy is given. The probem is formuated as an optimization probem, where we jointy seect the primary and backup LSPs for edge to edge EF and BE user demand pairs 1, and assign a discrete capacity vaue for each ink. The goa is 1 The term demand pair and demand are used interchangeaby through out the paper. Depending on the goas of the network designer, a demand pair may correspond to a singe fow, or an aggregation of fows sharing the same QoS requirements, source and destination, and path through the network. 3

to minimize the tota ink cost, subject to the performance and survivabiity constraints of both EF and BE casses. The non-bifurcated (i.e., singe-path) routing mode is used for the EF cass as required by [11], so that the traffic from a singe EF demand pair wi foow the same LSP between the origin and the destination. Traffic in the BE cass is aowed to be spit across mutipe LSPs. The performance constraint of EF traffic is ony represented by a bandwidth requirement, as specified in [11]. Most of iterature on the design of survivabe network ony consider the case of singe ink faiures [14]. Whie there are recent works investigating the issues of doube-ink faiure [10], the existing methods for protection switching are designed for singe ink faiures. We assume there is ony singe ink faiure in this paper. We assume that each EF user demand is assigned a primary LSP as we as a backup LSP with enough preemptibe bandwidth; therefore it wi not be affected by any singe ink faiure. The performance as we as the survivabiity constraints of the BE cass are characterized by the upper imit on the average deay in each ink, assuming no more than a singe ink faiure. Queueing is modeed as M/G/1 strict priority queues. The nove aspect of our probem is the fact that two traffic casses, EF and BE, with independent performance and survivabiity requirements, share the same capacity resource, which resuts in a compex non-inear performance constraint. In addition to the non-inear performance constraint, non-bifurcated routing and discrete ink capacity constraints dramaticay increase the degree of difficuty, and significanty imit the viabe soution approaches. There are reated papers on capacity panning that do not dea with survivabiity issues. A survey of those works can be found in [32]. The remainder of this paper is organized as foows. In Section 2, notation and detaied assumptions and modes are presented. The probem definition is given in Section 3. Section 4 shows a Lagrangean reaxation of the origina probem, and describes the subgradient procedure to sove the resuting dua probem. Section 5 presents some numerica resuts on the use of the method. The paper is concuded in 4

Section 6. 2 Notation and Assumptions The issue of capacity panning of reiabe MPLS networks supporting DiffServ EF and BE traffic casses is studied in this paper. The network is considered reiabe as ong as the given performance constraints are satisfied under any singe ink faiure scenarios. A ink based formuation is used throughout this paper. Our notation is defined in Tabe 1. Compared to our previous work [34], B kj,δb, ˆxef jkb, ˆxbe jkb, and r are new variabes introduced to refect the survivabiity constraints. Pease refer to [32] for more compete discussion on the notations and assumptions. The network topoogy is assumed to be given, as we as the set of candidate LSPs J k for each O-D (Origin-Destination) pairs. As mentioned, we ony study the faiure scenario where there is at most a singe ink faiure. The given networks must be two-ink-connected [12], which impies that there are at east two ink-disjoint paths for any O-D pair. For every candidate path j, j J k, there is a set of set of candidate backup paths B kj, which consists of LSPs that are ink-disjoint from the primary LSP j, to ensure the avaiabiity of the backup path regardess of the faiure scenarios. δb defines the route of backup path b. The backup paths for the EF and BE user demands on path j, wi be specified by the backup path routing variabes ˆx ef jkb, ˆxbe jkb respectivey. T denotes the index of avaiabe ink types for ink. Ony one ink type can be used for each individua ink. The capacity and the cost of ink, ψ and C respectivey, are: ψ = t T u t ψ t, C = t T u t C t (1) There is no inear reationship assumed between C t and ψ t, therefore the ink cost is not necessariy a inear function of the ink capacity. For an EF demand m, m M k, we differentiate between the average arriva rate, α ef km, and the requested bandwidth, ρef km. ρef km is usuay a vaue between the average arriva 5

Network Candidate paths and routing L set of inks in the network J k set of possibe candidate LSP paths for the O-D pair k, k K K set of (both EF and BE) Origin- δj ink-path indicator; 1 if path j uses ink, Destination (O-D) pairs j J k,k K, L, 0 otherwise T index of avaiabe ink types for ink L B kj set of candidate backup paths, which are ink disjoint from j, j J k u t ink type decision variabe; 1 if ink type δb ink-path indicator; 1 if backup path b t is used for ink L, 0 other wise uses ink, b B kj,j J k,k K, L, ψ t size of the capacity of ink type t, t T x ef kmj 0 otherwise EF primary path routing variabe; 1 if EF demand m, m M k, k K uses path ψ tota capacity of ink, L j J k, 0 otherwise x be kj BE primary path routing variabe: the C t cost of the ink type t, t T, L portion of BE demand k that uses candidate path j, j J k. x be kj can be any rea vaue C tota cost of ink, L between 0 and 1 jkbbackup path routing variabe, 1 if backup path b B kj is used by path j J k EF demand reated BE demand reated M k set of EF demands for O-D pair k, k K αk be average arriva rate of a BE traffic demand, k K α ef km average arriva rate of an EF traffic de- β be average arriva rate of tota BE traffic demand on ink L mand m, m M k, k K d average deay experienced by BE traffic ρ ef km η ˆx ef jkb, ˆxbe d max on ink L maximum vaue of d aowed for ink L requested bandwidth of EF traffic demand m, m M k, k K tota requested bandwidth of EF demand r BE traffic restoration eve on ink L β ef average arriva rate of tota EF traffic de- g BE deay bound factor mand on ink L ỹ,ỹ2 the first and second moment of packet size, (units: bits & bits 2 ) Tabe 1: Notation rate and the peak rate. It is noted in [11] that the packets of the EF traffic cass beonging to the same fow shoud not be reordered. Consequenty, traffic from the same EF demand can not be separated into different LSPs. For the same reason, ony one backup path wi be used per EF user demand. Given a faiure of ink F, F L, the tota amount of EF user demand on ink is: η = k K ρ ef km (x ef m M k j J k kmj δ j + x ef kmj δf j ˆx ef jkb δ b ) (2) where (x ef kmj δ j ) equas 1 if the EF demand m uses LSP j as its working path and the path j incudes ink, 6

and 0 otherwise, whie (x ef kmj δf j ˆxef jkb δ b ) equas 1 ony if the working path of the EF demand m is interrupted by the ink faiure F and the backup path b uses ink. Simiary, average arriva rate of tota EF traffic on ink can be cacuated according to the foowing equation: β ef = α ef km (x ef kmj δ j + x ef kmj δf j ˆx ef jkb δ b ) (3) k K m M k j J k For each O-D pair k, ony one BE demand pair is defined, which can be ooked as the aggregation of mutipe BE demands with the same O-D. Unike EF traffic, we aow the traffic within a singe BE demand to spit arbitrariy across any number of candidate LSPs, therefore the aggregation of BE traffic woud potentiay improve the effectiveness of traffic engineering. However, we assume ony one backup path wi be used for traffic on the same working path. We assume that the BE traffic may not be fuy restorabe in case of ink faiure. BE traffic restoration eve r, a vaue between [0,1], represents the proportion of BE traffic on each ink that is being protected. The abiity to set the restoration eve of BE traffic and accordingy differentiate the resiience eve provides a powerfu too to the service provider. It aows the ISP to ceary demonstrate the advantage of premium service based on EF cass, and aso aows the ISP to fine tune the redundancy eve and tota network cost. The tota BE oad (average arriva rate) on ink is: β be = k K α be k j J k (x be kj δ j + r x be kj δ F j ˆx be jkb δ b ), F L (4) It is assumed that the requested bandwidth of a EF user demand is defined in a way such that the performance of an EF demand wi be guaranteed if it is given the requested bandwidth. η ψ, L (5) There are many discussions regarding the imits on the ink utiization of EF user demands [9]. The revised EF PHB, RFC3246 [11], introduces an error term E a for the treatment of the EF aggregate, which represents the aowed worst case deviation between the actua EF packet departure time and the idea 7

departure time of the same packet. This revision makes it possibe for the EF utiization to go up as high as 100% [37]. In this paper, we assume that there are no additiona constraints on the EF utiization. We pick the average deay as the soe performance measurement for BE traffic in this paper as suggested by [24]. We evauate the performance of BE traffic on a per-ink basis (i.e., not end-to-end). The vaue ỹ ψ stands for the average transmission deay of packets. We use ỹ ψ as the basis for the deay bound. Let d max = g ỹ ψ, where g is a parameter defined by the network designer. The arger the vaue of g, the more bandwidth is required for ink, therefore the ower the ink utiization. We assume that the performance of BE traffic is satisfactory if d d max. Note that the deay bound d max remains the same under the norma condition and the faiure scenarios. Every router is modeed as a M/G/1 system with Poisson packet arrivas and an arbitrary packet ength distribution. [30] concudes, through both simuation and anaytic study, that even though the traffic exhibits bursty behavior at certain time scaes, the reationship between the variance and the mean is approximatey inear, or Poisson-ike, if they are measured over arger time scaes, where the traffic can be treated as if it were smooth. Our choice of the Poisson arriva mode is justified because we are more concerned about the average BE performance over a arge time scae for capacity panning purposes. From the average queueing deay formua of the priority queue [21], we obtain the performance constraint for BE traffic: d = ỹ ỹ + 2 ψ 2ỹ With some rearrangement, (6) yieds ( ψ β ef β ef + β be )( ψ β ef β be ) g ỹ ψ (6) ψ f(β ef,β be ) (7) where f(β ef,β be ) = β ef + βbe 2 + ỹ2 (β ef + β be ) 4(ỹ) 2 (g 1) + 1 2 (2β ef + β be + ỹ 2 (β ef + β be ) 2(ỹ) 2 (g 1) )2 4β be (β be + β ef ). 8

In order to have a meaningfu soution for constraint (7), ψ >β ef + β be is required. 3 Probem Formuation The forma probem definition is presented beow. Given: L, K, T,ψ t,c t,j k,δ j,b kj,δ b,m k,α ef km,ρef km,αbe k,d max,r,g, ỹ, ỹ2 Variabe: u t,x ef kmj,xbe kj, ˆxef jbk, ˆxbe jbk Goa: min L C t Subject to: u t =0/1, t T u t = 1 (8) x ef kmj =0/1, x ef kmj = 1 (9) j J k x be kj = 1 (10) j J k ˆx ef jkb =0/1, ˆx ef jkb = 1 (11) ˆx be jkb =0/1, ˆx be jkb = 1 (12) ψ η (13) ψ f(β ef,β be ) (14) (8) imposes a discrete constraint on the ink capacities. Constraint (9) ensures that a traffic from one EF O-D pair wi foow one singe path. (11)(12) denotes that ony one backup path wi be used for each working path. Constraint (13)(14) ensures the performance of EF and BE traffic respectivey. Because C is non-decreasing with respect to β ef, η, and β be, this probem can be reformuated as: min L C (15) 9

Subject to (8)(9)(10)(11)(12) (13)(14) and: β ef η k K k K β be k K ρ ef km (x ef m M k j J k α ef km (x ef m M k j J k α be k j J k (x be kj δ j + r kmj δ j + kmj δ j + x be x ef kmj δf j ˆx ef x ef kmj δf j ˆx ef jkb δ b ) (16) jkb δ b ) (17) kj δj F ˆx be jkb δ b ) (18) We refer to the probem defined by (15,8,9,10,11,12,13,14,16, 17, 18) as probem (P) in the rest of this paper. As can be seen from the above probem formuation, probem (P) is a non-inear integer programming probem, which is considered to be at east NP hard [2]. 4 Soution Method 4.1 Lagrangean Reaxation Lagrangean Reaxation is a common technique for muticommodity fow probems [2]. It has been successfuy appied to the capacity panning and routing probems [16] [17] [3] [25] [26], and our previous work [33][34]. We describe its use for our probem in this section. Pease refer to [32] for more detai. Using Lagrangean Reaxation, reax (16)(17) and (18), and we have the Lagrangean as: = L(x ef kmj,xbe kj, ˆxef jbk, ˆxbe jbk,u t,λ ef ( C λ η η λ ef β ef λ be β be ) + k K + k K m M k j J k x be x ef kmj j J k kj L The Lagrangean dua probem (D) is then:,λ η,λbe ) (λ η ρef kj + λef L α ef km )(δ j + δ F j ˆx ef jkb δ b ) λ be α ef k (δ j + r δ F j ˆx be jkb δ b ) (19) λ η,λef max,λ be 0 h(λ η,λef,λ be ) (20) 10

where: h(λ η,λef,λ be ) = min x ef kmj,xbe kj,ˆxef jbk,ˆxbe jbk,u t L(x ef kmj,xbe kj, ˆxef jbk, ˆxbe Since η,β ef,β be and x ef kmj,xbe kj, ˆxef jkb, ˆxbe jkb are independent variabes, min L = min( C λ η η λ ef β ef λ be β be ) + min[ x ef kmj (λ η ρef k K m M k j J k L + min[ x be kj k K j J k L 4.2 Soving the Subprobems kj + λef jbk,u t,λ ef,λ η,λbe ) (21) α ef km )(δ j + δ F j ˆx ef jkb δ b )] λ be α ef k (δ j + r δj F ˆx be jkb δb )] (22) Equation (22) shows that the probem (21) can be separated into the foowing three set of subprobems: Subprobem (i): min( C λ η η λ ef β ef λ be β be ) (23) Subject to: (8)(13)(14) (24) Subprobem (i) can be soved by the gradient projection method [7]. Subprobem (ii): min[ j J k x ef kmj (λ η ρef kj + λef L α ef km )(δ j + δj F ˆx ef jkb δ b )] (25) Since we have a sma set of candidate path for each O-D, we can simpy set the ink cost to (λ η ρef kj + λ ef α ef km ), and then enumerate the choice of working paths and backup paths, picking the combination of working and backup paths with the east tota cost. The resuting choice of working path and backup path is cached. In the subsequent iteration, ony paths whose cost is changed wi be compared to the previous soution, and hence save the running time. Subprobem (iii): min[ j J k x be kj L λ be α ef k (δ j + r δj F ˆx be jkb δb )] (26) 11

Simiar to Subprobem (ii), the soution is to enumerate the choice of working path and backup path, then choose the one with the east combined cost. The resuts are simiary cached to speed up the program. 4.3 Subgradient Method The subgradient method is used to update λ η, λef and λ be. For an in depth description of the subgradient method and its variants, pease refer to [6]. Given the initia vaues of λ η, λef and λ be, once we sove the probem (D), subgradients for the three set of Lagrangeans are computed as foows: ω η = η k K ω ef = β ef k K ω be = β be k K ρ ef km (x ef m M k j J k α ef km (x ef m M k j J k α be k j J k (x be kmj δ j + kj δ j + r x ef kmj δf j ˆx ef kmj δ j + x be x ef kmj δf j ˆx ef jkb δ b ) (27) jkb δ b ) (28) kj δj F ˆx be jkb δ b ) (29) To improve the convergence speed, the surrogate subgradient method is used, where ony one of the three subprobems is soved at a time. The subsequent Lagrangean mutipiers are updated using the atest vaue. λ η min(0,λ η + tωη ), L (30) λ ef min(0,λ ef + tω ef ), L (31) λ be min(0,λ be + tω be ), L (32) where the step size, t, is defined by: t = φ ω eta h h 2 + ω ef (33) 2 + ω be 2 where h is the vaue of the best feasibe soution found so far, and φ is a scaar between 0 and 2. φ is set to 2 initiay in our study and is haved if the soution does not improve in 10 iterations. 12

At each iteration, the soution of x ef kmj,x be kj, ˆx ef jkb, and ˆxef jkb for the prima probem (P) can be obtained from the soution of subprobem (ii) and(iii). The ink capacity ψ is computed according to (13). Consequenty, the primary objective function can be derived. As the iteration proceeds, we store the best soution found so far for the prima probem (P). In this way, we are aways abe to obtain a feasibe soution. The maximum number of iterations is set to 400 in the impementation [33][2]. The soution of the dua probem provides a ower bound for the prima probem. Therefore, the soution quaity can be assessed by the duaity gap, which is the difference between the soutions of probem (P) and probem (D). Note that because the duaity gap is aways no smaer than the actua difference between the obtained feasibe soution and the optima soution, it is a conservative estimate of the soution quaity. 5 Computationa Resuts In this section, we present numerica resuts based on experimentation. The objective of our experiment is to evauate the soution quaity and running time of the agorithm. The program is impemented in C and the computation is performed on a Pentium IV 2.4GHz PC with 512M memory, running Redhat Linux 7.2. The network topoogies are generated using the Georgia Tech Internetwork Topoogy Modes (GT-ITM) [38]. The ocations of origins and destinations are randomy seected. For each O-D pair, 10 candidate working paths are cacuated using Yen s K-shortest path agorithm [36]. 10 candidate backup paths are aso generated for each candidate working path. If not specified, EF demand pairs are randomy generated with the average rate uniformy distributed from 0 Mbps to 10 Mbps. The requested bandwidth of a EF demand pair is a vaue between the average rate and the peak rate. It is randomy specified to be 150%-300% of the average rate in our experiments. The average rate of BE demand is uniformy distributed from 10Mbps to 50Mbps. The number of EF demands for the same O-D pair is uniformy distributed from 1 to 10. The number of candidate ink types 13

for ink is uniformy distributed from 5 to 10. The capacities of ink types are set to be mutipes of 45Mbps, whie the costs of the ink types for ink are randomy generated in such a way that the cost goes higher and the unit cost per Mbps goes down as the ink capacity increases. We use average packet size ỹ = 4396 bits and second moment of packet size ỹ2 = 22790170 bits 2 for a the test cases. They are cacuated based on a traffic trace (AIX-1014985286-1) from the NLANR Passive Measurement and Anaysis project [1]. The restoration eve of BE traffic is determined by the network designer, depending on the budget and the desired BE cass resiience. It is set to 0.5 in a of the experiments. The BE deay bound factor, g, is set to 2 for a inks in our experiments. In practice, g shoud be carefuy chosen to refect the desired BE performance and the expected ink utiization. Figure (1) shows the figure of f(β ef ) with respect to β ef for various vaues of g. As can be seen from the figure, the ink utiization varies significanty as g changes. The average ink utiization, the vaue of β ef +β be, is about f(βef ) 60% when g equas 2. 14 x 107 bps 13 12 g = 1.5 11 f ( β ef ) 10 Utiization = 0.6 9 8 7 6 g = 2 g = 4 g = 8 5 1 2 3 4 5 6 7 8 9 10 x 10 6 bps β ef Figure 1: f(β ef ) vs. β ef, (β be =5 107 bps) Two additiona greedy type heuristics were impemented for the purpose of comparison: 14

1. The first heuristic method starts with one randomy chosen O-D pair k, finds out both the best working path j and backup path b for both EF and BE demand among a candidate working and backup paths that woud resut in the east tota cost, and then repeats the process for a k K. 2. The second heuristic is an iterative method inspired by the heuristic used in [23]. The primary difference here is that [23] deas ony with the spare capacity panning probem, whie we have to find out the working path first. In the first iteration, for each O-D pair k, the agorithm finds out ony the best working paths j for both EF and BE demands among candidate working paths that woud resut in the east tota cost. After a the working paths have been determined, the agorithm ooks for the best backup paths b for both EF and BE demands among candidate backup paths. In the subsequent iteration, the agorithm goes through each O-D pair to see whether any change of working path or backup path wi ower the tota cost. The iteration wi continue as ong as there is at east one adjustment of path in the current iteration. We wi ca these two methods (H) and (S), respectivey. The agorithms were tested on 8 different sizes of networks, ranging from 10 nodes to 1000 nodes. Some detais of the network topoogies are isted in Tabe 2. Note that the O-D demand number shown in Tabe 2 incudes both EF and BE demands. To obtain confidence intervas, we generated 30 different topoogies for each network size, with the same number of nodes, inks, and O-D pairs. Duaity Gap = s p s d s d (34) The soution quaity is represented by the duaity gap, which is the percentage difference between the soution of the prima probem and the dua probem. s p and s d are the soutions of prima probem and dua probem respectivey. A vaue cose to zero means the soution is very cose to optima. Tabe 2 shows the Duaity Gaps of the three soution methods, expressed in terms of the 95% confidence intervas. It is easy to see from the tabe that the Lagrangian Reaxation together with the subgradient 15

Node Link O-D Duaity (%) Number Number Number LR H S 10 25 30 0.12-3.24 3.82-21.89 0-3.40 20 50 90 0.01-2.96 0.2-15.21 0-2.3 50 125 350 0.17-1.78 1.7-13.70 0-1.87 100 250 1000 0.16-1.90 0-8.36 0-1.4 200 500 3000 0-1.64 0-10.21 0-1.55 500 1250 12000 0-2.59 0-11.47 700 1750 20000 0.2.82 0-9.35 1000 2500 40000 0-1.97 0-15.09 Tabe 2: Network topoogy information and experimenta resuts method produces reasonabe resuts as the duaity gap is bounded by no more than 3.3%. Note the prima probem itsef is approximated when reducing the size of candidate path set for a possibe path set. But according to our experimenta resuts, more than 97% of the time, the fina soution is chosen among the 5 shortest candidate paths. Therefore, 10 candidate paths are considered adequate. Having more than 10 candidate paths wi have minima impact on the soution quaity, whie significanty increasing the running time. Given the arge number of networks being tested, we are confident that the soution shoud have good quaity for other sizes of networks. Heuristic (S) achieves comparabe soution quaity in networks sized up to 200 nodes, whie method (H) yieds soutions with a duaity gap of more than 20%, which is not very desirabe. Figure 2 shows the running time with respect to the network size. In a 240 test cases of Lagrangian Reaxation based method, the agorithm converges without difficuty. As can be seen from the figure, heuristic (S) takes much onger time than the other two methods. From the experiments shown above, Lagrangian Reaxation based approach demonstrates a good tradeoff between the soution quaity and the running time. The size of the argest network evauated in this paper is representative of a arge network, and is much arger than the test cases used in most work on network design. It is fair to predict that the running time of the agorithm wi stay reasonabe for practica sized networks. 16

9 x 10 4 8 H LR S 9 x 104 8 H LR S 7 7 Running time (sec) 6 5 4 Running time (sec) 6 5 4 3 3 2 2 1 1 0 0 200 400 600 800 1000 1200 Node number 0-1000 0 1000 2000 3000 4000 5000 O-D pair number Figure 2: Running Time vs. Network Size The fow aggregation based agrangian Reaxation method proposed in this paper can be appied to more genera noninear integer muticommodity fow probems. We are seeking the opptunitues to extend the method to other types of network design probems, such as design of optica networks and overay networks. 6 Concusions and Future Directions In this paper, we addressed the probem of ink dimensioning and routing for survivabe MPLS networks supporting DiffServ EF and BE traffic. We formuate the probem as an optimization probem, where the tota ink cost is minimized, subject to the performance constraints of both EF and BE casses under norma circumstance and singe ink faiures. The variabe here is the routing of working and backup LSP of both EF and BE user demands, and the discrete ink capacities. This paper presents a preiminary investigation of the capacity panning issue for survivabe muticass MPLS networks. The novety of the probem presented in this paper is that it invoves two traffic casses, 17

EF and BE, which have totay different forms of performance requirements and resuts in non-inear performance constraint. Unike SCA probems, this paper dea with the the working and backup paths at the same time, which is considered to be much more compex. The combination of targeting mutipe traffic casses with non-inear performance constraints, optimizing working path and backup path simutaneousy, and the use of non-inear cost function produces a probem more difficut than most reated works. We presented a Lagrangian Reaxation-based method to effectivey decompose the origina probem. A subgradient method is used to find the optima Lagrangian mutipier. We investigated experimentay the soution quaity and running time of this approach. The resuts from our experiments indicate that our method produces soutions that are within a few percent of the optima soution, whie the running time stays reasonabe for practica sized networks. The probem formuation and soution approaches may be appied to other traffic casses, such as the Assured Forwarding (AF) cass. The technique of reaxing fow baance equations enabes the Lagrangian Reaxation techniques to be appied to difficut integer non-inear programming probems, which is not possibe otherwise. We are investigating the adaptation of this technique to the genera non-inear muticommodity fow probems. References [1] NLANR passive measurement and anaysis project, http://pma.nanr.net, 2001. [2] R.K. Ahuja, T.L. Magnanti, and J.B. Orin. Network Fows, Theory, Agorithms and Appications. Prentice Ha, New Jersey, 1993. [3] A. Amiri. A System for the Design of Packet-Switched Communication Networks with Economic Tradeoffs. Computer Communications, 21(18):1670 1680, Dec. 1998. [4] Shigehiro Ano, Toru Hasegawa, and Nicoas Decre. Experimenta TCP performance evauation on diffserv AF PHBs over ATM SBR service. Teecommunication Systems, 19(3-4):425 441, Apr. 2002. [5] Achim Authenrieth and Andreas Kirstadter. Components of MPLS recovery supporting differentiated resiience requirements. In IFIP Workshop on IP and ATM Traffic Management WATM 2001, Paris, France, September 2001. [6] D.P. Bertsekas. Noninear Programming. Athena Scientific, Bemont, MA, 1995. [7] D.P. Bertsekas and R.G. Gaager. Data Networks. Prentice Ha, Engewood Ciffs, NJ, 1987. [8] S. Bake and et a. An Architecture for Differentiated Service, IETF RFC 2475. Dec. 1998. 18

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