Link Dimensioning and LSP Optimization for MPLS Networks Supporting DiffServ EF and BE traffic casses 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 Capacity panning is indispensabe for future Internet providing QoS. Accurate dimensioning is especiay important when no per-fow signaing or contro exists. In this paper, we address the probem of ink dimensioning and Labe Switching Path (LSP) optimization for MPLS networks supporting DiffServ EF and BE traffic casses. The probem is formuated as an optimization probem, where the goa is to minimize the noninear tota ink cost, subject to the performance constraints of both expedited forward (EF) and best effort (BE) traffic casses. The variabes to be determined are the routing of LSPs carrying both EF and BE traffic, and the discrete capacities of the inks. We show that Lagrangean Reaxation and subgradient optimization methods can be used to effectivey sove this difficut probem. Computationa resuts show that the soution quaity is verifiaby good, whie the running time remains reasonabe on practica-sized networks. This is the first work on capacity panning for MPLS networks supporting mutipe DiffServ service casses. 1. Introduction Capacity panning is the process of designing and dimensioning networks to meet the expected demands on them. The nature of offering ony best effort (BE) service makes capacity panning a straightforward matter[1] in the current Internet. With the popuarization of e-commerce and new vaue-added services over IP, Quaity of service (QoS), the abiity of a network eement to have some eve of assurance that its traffic and service requirements can be satisfied, has become a must. Capacity panning wi be an imperative part of IP network management to support various quaities of service. Mutiprotoco abe switching (MPLS) [2] [3] and Differentiated services (DiffServ) [4] [5] are regarded as two key components for providing QoS in the Internet. MPLS uses a short, fixed-ength, ocay significant abe in the packet header to switch the packets. The initia abe is chosen and inserted by the ingress node of a MPLS domain, based on the information in the IP header, associated QoS, or any other poicies in effect. The intermediate nodes use the abe as an index to find the next hop and the corresponding new abe. A abe distribution protoco (LDP) propagates abe bindings among the nodes in order to estabish and tear down the abe switched path (LSP). The power of MPLS ies in the fact that the mappings between the packet fows and the LDPs are fexibe, which enabes IP network to be traffic engineered. Packets with the same source and destination addresses, which wi inevitaby foow the same path in the traditiona IP network, can be assigned different abes This work is supported by DARPA and AFOSR (under grants F30602-99-1-0540 and F49620-99-1-0264). The authors wish to thank Dr. Coe Smith and Dr. Micha Pioro for their hepfu comments and suggestions.
and subsequenty be sent to separate LSPs. LSPs can be setup expicity to optimize the resource utiization. The use of MPLS abes may aso provide faster switching than the norma IP forwarding agorithm. The essence of DiffServ is prioritization. In traditiona IP networks, the DiffServ Code Point (DSCP) fied in the headers of IP packets is marked at the edge of the network. Routers within the core of the network forward packets using different predefined per-hop behaviors (PHBs), according to their DSCP fied. In MPLS networks, the DSCP fied is not visibe to the core LSRs. The abe vaue and EXP fied (3bit experimenta fied) are used instead to determine the PHB scheduing cass (PSC) associated with packets [6]. Note that even with MPLS, the signaing and traffic contro are sti at the eve of fow aggregation. Since there is no per-fow signaing or contro, accurate dimensioning of the network is particuary important for achieving performance guarantees. To prepare for the depoyment of DiffServ in MPLS network, it is necessary to study the capacity panning probem in the context of mutipe cass-of-service networks. The IETF DiffServ working group has standardized two PHBs: Expedited Forwarding (EF) and Assured Forwarding (AF). The EF PHB [7] is defined as being such that the EF packets are guaranteed to receive service at or above a configured rate. The EF PHB can be used to buid a ow oss, ow atency, ow jitter, assured bandwidth, end-to-end service, through a DiffServ Domain. As has been discussed in [8], three expected major initia appications of QoS in the IP network are: 1) to distinguish mission critica or preferred customers; 2) to provide voice over IP service; 3) to enabe services competitive with eased ines. It can be easiy seen that the services based on the EF PHB are idea for a three of these appications. Because of its great vaue, the EF PHB is very ikey to be the first PHB to be put into action. The priority queue is widey considered to be the canonica way to impement the EF PHB, due to its abiity to offer a tighter deay bound and smoother service over reativey short time scaes [7]. AF PHBs are designed to reaize different forwarding assurances, or dropping preferences, for IP packets. AF PHBs are considered usefu to differentiate TCP traffic, where the performance is sensitive to the packet oss rate. However, simuations showed that the standard traffic contro methods of routers, such as RED(Random Eary Detection), do not satisfactoriy differentiate between AF PHBs and best effort traffic [9]. 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. In this paper we address the probem of ink dimensioning and path optimization for MPLS networks providing DiffServ EF and BE traffic 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. The goa is to minimize the tota ink cost, subject to the performance 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 [7], 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. Whie the performance constraint of EF traffic is ony represented by a bandwidth requirement, the performance constraint of the BE cass is characterized by the average deay in each ink. Queueing is modeed as M/G/1 strict priority queues. Our intention is to not ony define the capacity panning probem for MPLS network supporting DiffServ and discose feasibe soutions, but aso provide hepfu insights for capacity panning of other QoS architectures. Athough there is no previous work specificay targeting the dimensioning and routing probems for MPLS networks supporting Diffserv, there are many papers addressing the issues of QoS routing in genera. An extensive survey can be found in [10]. But because the routing and ink dimensioning probems are cosey reated to each other, it is inappropriate to separate them. There are extensive works deaing with traffic engineering issues in MPLS network, such
as [11] [12], or LSP setup and dimensioning probem [13]. However, in those works the ink capacity is fixed and not subject to be optimized. Papers where the routing and capacity assignment probems are treated simutaneousy incude [14] [15] [16][17] [18] [19] [20] [21]. Gera and Keinrock [14] presented heuristic methods based on the fow deviation agorithm [22]. Gavish and Neuman [15] formuated the probem as a non-inear integer programming probem, and proposed a Lagrangean reaxation based approach. [21] studied the network with eastic traffic and approximated the non-inear cost function to a piece-wise inear function. The networks studied in [14] [15] [21] ony incude one traffic cass, though. Medhi and Tipper [19] proposed four approaches for reconfigurabe ATM networks based on the Virtua Path concept. Even though ATM networks incude mutipe traffic casses, Medhi proposed a mode that assumes the deterministic mutipexing of different virtua paths, which resuts in inear performance constraints. We studied the probem of capacity panning for DiffServ networks in a previous work [23], where the MPLS protoco is not supported. Because of the absence of MPLS in [23], traffic demands with the same origin and destination wi be constrained to foow the same path. [23] ony considers the routing of EF demand pairs, whie the routing of both EF and BE casses are optimized in this paper. A noninear cost function is assumed in this paper, whie [23] uses a inear cost function. The nove aspect of our capacity panning probem is the fact that two traffic casses, EF and BE, with independent behaviors and performance requirements, share the same capacity resource, which resuts in a compex non-inear performance constraint. In addition to the noninear performance constraint, non-bifurcated routing and discrete ink capacity constraints dramaticay increase the degree of difficuty, and significanty imit the viabe soution approaches. 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 Section 6. 2. Notation and Modes The foowing notation wi be used throughout the paper. K set of (both EF and BE) Origin-Destination (O-D) pairs 3 M k set of EF demands for O-D pair k, k K L set of inks in the network J k set of possibe candidate LSP paths for the O-D pair k, k K δj ink-path indicator; 1 if path j uses ink, j J k,k K, L, 0 otherwise α ef average arriva rate of an EF traffic demand m, m M k, k K ρ ef requested bandwidth of an EF traffic demand m, m M k, k K j EF path routing variabe; 1 if EF demand m, m M k, k K uses path j J k,0 otherwise. η tota requested bandwidth of EF demand on ink L average arriva rate of tota EF traffic demand on ink L kj BE path routing variabe: the portion of BE demand k uses candidate path j, j J k. kj can be any rea vaue between 0 and 1 αk be average arriva rate of a BE traffic demand, k K γ average arriva rate of extra BE traffic demand on ink, L average arriva rate of tota BE traffic demand on ink L β be
d d max g T u t ψ t average deay experienced by BE traffic on ink L maximum vaue of d aowed for ink L BE deay bound factor index of avaiabe ink types for ink L ink type decision variabe; 1 if ink type t is used for ink L, 0 other wise. size of the capacity of ink type t, t T ψ tota capacity of ink, L C t cost of the ink type t, t T, in ink C tota cost of ink, L ỹ,ỹ2 the first and second moment of packet size, (units: bits & bits 2 ) Link based formuation is used in this paper. The network is defined by (L,K,J k ). For an EF demand m, m M k, we differentiate between the average arriva rate, α ef, and the requested bandwidth, ρ ef. ρef is usuay a vaue between the average arriva rate and the peak rate. It is noted in [7] 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. = α ef j δ j (1) k K m M k η = Γ({ρ ef }, {δ j}, { j }) ρ ef k K m M k j δ j (2) where Γ() is the EF demand mutipexing function (discussed further beow). The inequaity becomes equaity ony when there is no mutipexing gain. For each O-D pair k, ony one BE demand pair is defined. We aow an arbitrary portion of the BE demand to route through any candidate LSP. Therefore the aggregation of BE traffic woud potentiay improve the effectiveness of traffic engineering. Because of the connectioness nature of IP traffic, it is unikey that a the BE demand can be ceary mapped to specific O-D pairs. We introduce another variabe γ, which is the average arriva rate of extra BE traffic in ink besides that from the BE demand pairs {k : k K}. Thus the tota BE oad (average arriva rate) on ink is: β be = γ + k K α be k kj δj The capacity and the cost of ink, ψ and C respectivey, are: (3) ψ = t T u t ψ t, C = t T u t C t (4) There is no inear reationship assumed between C t and ψ t, therefore C is not necessariy a inear function of ψ. There are many discussions about the origina EF PHB [24] concerning the imits on EF utiization. Charny reported in [25] that the worst case deay jitter can be made arbitrariy arge using a FIFO queue uness the utiization of EF traffic was imited to a factor smaer than 1/(H 1), where H is the number of hops in the ongest path of the network. Other
impementations of packet scheduing may improve the upper bound on the EF utiization. The revised EF PHB [7], RFC 3246, 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 departure time of the same packet. It is not immediatey cear whether this revision totay eiminates the constraint on the EF utiization, or simpy aows a trade-off between the EF utiization and the deay jitter. In this paper, we assume that the projected EF user demand η is much ess than the capacity of the ink, so there is no concern about this imit on the EF utiization, and the exact form of the mutipexing function Γ() does not have any impact on the fina soutions. How to specify the performance of BE traffic in the service eve agreement (SLA) is sti an active research topic. [26] suggests using the atency averaged over a arge time scae as the primary criteria for the performance of BE traffic in IP network service eve agreements (SLAs). We pick the average deay as the soe performance measurement for BE traffic in this paper. 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 ỹ deay bound. Let d max = g, where g ψ 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. Every router is modeed as a M/G/1 system with Poisson packet arrivas and an arbitrary packet ength distribution. Whie it has been suggested that the Internet traffic is ong-range dependent [27] and thus bursty, a recent work [28] shows that the network traffic can be smooth and Poisson-ike. [29] concudes, through both simuation and anaytic study, that even though the traffic exhibits bursty behavior at certain time scaes, the variance-mean reation is approximatey inear 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 [30], we obtain the performance constraint for BE traffic: ỹ ψ + ỹ2 2ỹ ( ψ + β be )( ψ β be ) g ỹ (5) ψ In order to have a meaningfu soution for constraint (5), ψ > With some rearrangement, (5) yieds ψ f( ), where f( ) = + βbe 3. Probem Formuation 2 + ỹ2 ( + β be ) 4(ỹ) 2 (g 1) + 1 2 (2 + β be + + β be is required. ỹ 2 ( + β be ) 2(ỹ) 2 (g 1) )2 4β be (β be + ) The forma probem definition is presented beow. Given: K, M k,l,j k,α ef,ρef,αbe k,γ,δj,t,ψ t,c t Variabe: j,xbe kj,u t Goa: min C L Subject to: ψ f(βef) j =0/1, j =1 (6) (7)
kj =1 u t =0/1, t T u t =1 (8) (9) Constraint (6) ensures the performance of BE traffic. (9) imposes a discrete constraint on the ink capacities. (7) ensures that a traffic from one EF O-D pair wi foow one singe path. Because C is a non-decreasing function of and β be, this probem can be reformuated as: min C (10) L Subject to (6) (7) (8) (9) and: k K β be γ + k K αbe k m M k α ef j δ j (11) kj δ j (12) We refer to the probem defined by (10, 6, 7, 8, 9, 11, 12) 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 very difficut to optimize in genera. 4. Soution Method 4.1. Lagrangean Reaxation Lagrangean Reaxation is a common technique for muticommodity fow probems [31]. It has been successfuy appied to the capacity panning and routing probems [32] [15] [20] [18] [19]. We describe its use for our probem in this section. Using Lagrangean Reaxation, reax (11) and (12), and we have the Lagrangean as: L( j,xbe kj,u t,,λ be ) = C The Lagrangean dua probem (D) is then: max h(,λ be ),λ be 0 where: h(,λ be ) = min j,xbe kj,u t L( j,xbe ( k K λ be (β be γ k K kj,u t, Since,β be and j,xbe kj are independent variabes, min L = min = + min k K + k K α be k [ C λ be (β be γ )] + min k K α be k α ef j δ j) m M k kj δ j) (13) (14),λ be ) (15) α ef m M k j δ j λ be kjδ j (16) min[ C λ be (β be γ )] + k K min(α be k m M k min(α ef j δ j) λ be kjδ j) (17)
4.2. Soving the Subprobems Equation (17) shows that the probem (15) can be separated into the foowing three subprobems: Subprobem (i): min[ C λ be (β be Subject to: f(,β be γ )] (18), ψ ) d max (19) Subprobem (i) can be soved by the gradient projection method [33]. Subprobem (ii): min(α ef j δ j) (20) This is simpy a shortest path probem where the cost of ink is set to. The soution is to et j =1for j satisfying: where P (j) = P (j ) = min j (P (j)) (21) j δ j Subprobem (iii): min(α be k λ be kjδ j) (22) Simiar to Subprobem (ii), the soution is to set kj to 1 for j satisfying: Q(j ) = min(q(j)) j where Q(j) = λ be kjδ j (23) 4.3. Subgradient Method The subgradient method is used to update and λ be. Due to space imitations, the reader is referred to the standard reference [31], or to our earier work [23], for a detaied description of the procedure and the choices of parameters. At each iteration, the soution of j and kj for the prima probem (P) can be obtained from the soution of subprobem (ii) and(iii). The ink capacity ψ can be computed according to (6). 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 [23][31]s. 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) [34]. The ocations of origins and destinations are randomy seected. For each O-D pair, 10 candidate paths are cacuated using Yen s K-shortest path agorithm [35]. If not specified, EF and BE demand pairs are randomy generated with a uniform distribution from 0 Mbps to 10 Mbps, whie the average BE traffic oad of each ink is aso uniformy distributed from 30 Mbps to 100 Mbps. The number of EF demands for the same O-D pair is uniformy distributed from 1 to 10. The number of candidate ink types 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 [36]. In practice, the BE deay bound factor, g, shoud be carefuy chosen to refect the actua traffic pattern, the desired BE performance, and the expected ink utiization. g is set to 2 for a inks in our experiments. The average ink utiization is about 60% when g equas 2. The agorithm was tested on 8 different sizes of networks, ranging from 10 nodes to 1000 nodes. Some detais of the network topoogies are isted in Tabe 1. Note that the O-D demand number shown in Tabe 1 incudes both EF and BE demands. To obtain confidence intervas, we generate 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 (24) 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 1 shows the running time and Duaity Gap (where a vaue of 0 means optima quaity) of various network sizes, expressed in terms of the 95% confidence intervas. In a 240 test cases, the agorithm converges without difficuty. It is easy to see from the tabe that the Lagrangean Tabe 1 Network topoogy information and experimenta resuts Node Number Link Number O-D Demand Number Duaity Gap (%) Running Time (sec) 10 25 30 (0.15, 2.24) (3.79, 5.20) 20 50 90 (0.01, 3.96) (15.84, 22.68) 50 125 350 (0, 2.78) (75.40, 98.38) 100 250 1000 (0.16, 3.51) (207.55, 241.49) 200 500 3000 (0, 2.15) (1430.79, 1935.30) 500 1250 12000 (0, 3.02) (8895.36, 12237.93) 700 1750 20000 (0, 2.41) (18470.24, 23918.52) 1000 2500 40000 (0, 2.93) (34807.64, 44630.93)
Reaxation together with the subgradient method produces reasonabe resuts as the duaity gap is bounded by no more than 4%. 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 99% 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. Because capacity panning is usuay performed on the time scae of weeks to months, the running time of the agorithm is not the most critica factor. But it is sti desirabe to know how the running time scaes up with respect to the network size. 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 capacity panning. It is fair to predict that the running time of the agorithm wi stay reasonabe for practica sized networks. The other observation is that over 93% of the time, demands with the same O-D use ony singe candidate path. It means that when designing a east cost network, traffic engineering capabiity enabed by the MPLS protoco does not have a significant impact on the optima resut at the stage of network panning. This is understandabe, since traffic engineering is most usefu when the rea traffic fuctuates in an operationa network. Further investigation is required to have a better understanding of the effect. 6. Concusions and Future Directions In this paper, we addressed the probem of ink dimensioning and routing for 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. The performance guarantee of BE traffic resuts in noninear constraints. The variabe here is the non-bifurcated (singe-path) routing of EF demands, mutipath routing of BE demands, and the discrete ink capacities. We presented a Lagrangean Reaxation-based method to effectivey decompose the origina probem. A subgradient method is used to find the optima Lagrangean 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. This paper presents a preiminary investigation of the capacity panning issue for MPLS networks supporting DiffServ. The novety of the probem presented in this paper is that it invoves two traffic casses, EF and BE, which have totay different forms of performance requirements. The probem formuation and soution approaches may be appied to other traffic casses and simiar network architectures. There are opportunities to extend this work in severa directions. We are working on a method where an empirica performance mode may be used, and thus AF traffic casses can be incorporated into the probem. We are aso investigating the adaptation of this technique to other type of networks, when there are mutipe casses of service. REFERENCES 1. S. Keshav, An Engineering Approach to Computer Networking, Addison-Wesey, 1997. 2. D. A. et a., Requirement for traffic engineering over MPLS, IETF RFC 2702. 3. E. Rosen, A. Viswanathan, R. Caon, Mutiprotoco abe switching architecture, IETF RFC 3031. 4. S. Bake, et a., An Architecture for Differentiated Service, IETF RFC 2475.
5. K. Nichos, V. Jacobson, L. Zhang, A Two-bit Differentiated Services Architecture for the Internet, IETF RFC 2638. 6. L. Wu, B. David, etc, Muti-Protoco abe switching MPLS support of differentiated services, IETF RFC 3270. 7. B. David, A. Charny, et A., An Expedited Forwarding PHB, IETF RFC 3246. 8. B. E. Carpenter, K. Nichos, Differentiated services in the internet, Proceedings of the IEEE 90 (9) (2002) 1479 1494. 9. S. Ano, T. Hasegawa, N. Decre, Experimenta TCP performance evauation on diffserv AF PHBs over ATM SBR service, Teecommunication Systems 19 (3-4) (2002) 425 441. 10. S. Chen, K. Nahrstedt, An overview of quaity-of-service routing for the next generation high-speed networks: Probems and soutions, IEEE Network Magazine 12 (1998) 64 79. 11. A. Ewaid, C. Jin, S. Low, I. Widjaja, MATE: MPLS adaptive traffic engineering (2001), in: IEEE Infocom 01, Vo. 3, IEEE, Anchorage, Aaska, 2001, pp. 1300 1309. 12. M. Kodiaam, T. Lakshman, Minimum interference routing with appications to MPLS traffic engineering, in: IEEE Infocom 00, Vo. 2, IEEE, Te Aviv, Israe, 2000, pp. 884 893. 13. I. Akyidiz, T. Anjai, L. Chen, J. de Oiveira, C. Scogio, A. Sciuto, J. Smith, G. Uh, A new traffic engineering manager for DiffServ/MPLS networks: Design and impementation on an IP QoS testbed, Computer Communications 26 (2003) 388 403. 14. M. Gera, L. Keinrock, On the topoogica design of distributed computer networks, IEEE Transactions on Communications COM-25 (1) (1977) 48 60. 15. B. Gavish, I. Neuman, A System for Routing and Capacity Assignment in Computer Networks, IEEE Transaction on Communications 37 (4) (1989) 360 366. 16. H. Gersht, R. Weihmayer, Joint Optimization of Data Network Design and Faciity Location, IEEE Journa on Seected Areas in Communications 8 (9) (1990) 149 152. 17. T. Ng, D. Hoang, Joint Optimization of Capacity and Fow Assignment in a Packet-switched Communication Network, IEEE Transaction on Communications COM-35 (2) (1987) 202 209. 18. D. Medhi, Muti-Hour, Muti-Traffic Cass Network Design for Virtua Path-based Dynamicay Reconfigurabe Wide-Area ATM Networks, IEEE/ACM Transactions on Networking 3 (6) (1995) 809 818. 19. D. Medhi, D. Tipper, Some Approaches to Soving a Mutihour Broadband Network Capacity Design Probem with Singe-path Routing, Teecommunication Systems 13 (2-4) (2000) 269 291. 20. A. Amiri, A System for the Design of Packet-Switched Communication Networks with Economic Tradeoffs, Computer Communications 21 (18) (1998) 1670 1680. 21. G. Maicsko, G. Fodor, M. Pioro, Link capacity dimensioning and path optimization for networks supporting eastic services, in: IEEE ICC 2002, Vo. 4, 2002, pp. 2304 2311. 22. L. Fratta, M. Gera, L. Keinrock, The Fow Deviation Method: An Approach to Store-and-Forward Communication Network Design, Networks 3 (2) (1973) 97 133. 23. K. Wu, D. S. Reeves, Capacity panning of DiffServ networks with Best-Effort and Expedited Forwarding traffic, in: To appear in IEEE 2003 Internationa Conference on Communications, Anchorage, Aaska, 2003. 24. V.Jacobson, K.Nichos, K.Poduri, An Expedited Forwarding PHB, IETF RFC 2598. 25. A. Charny, J.-Y. L. Boudec, Deay Bounds in Network with Aggregate Scheduing, in: First Internationa Workshop on Quaity of Future Internet Service, Berin, Germany, 2000, pp. 1 13. 26. J. Martin, A. Nisson, On service eve agree ments for IP networks, in: IEEE Infocom 02, IEEE, New York, 2002. 27. V. Paxson, S. Foyd, Wide-area traffic: The faiure of poisson modeing, in: ACM SIGCOMM 94, London, 1994. 28. R. Morris, D. Lin, Variance of aggregated web traffic, in: IEEE Infocom 00, IEEE, Te Aviv, Israe, 2000, pp. 360 366. 29. X. Tian, J. Wu, C. Ji, A unified framework for understanding network traffic using independent waveet modes, in: IEEE Infocom 02, Vo. 1, IEEE, 2002, pp. 446 454. 30. L. Keinrock, Queueing Systems, Voume II: Computer Appication, Wiey Interscience, 1976. 31. R. Ahuja, T. Magnanti, J. Orin, Network Fows, Theory, Agorithms and Appications, Prentice Ha, New Jersey, 1993. 32. B. Gavish, I. Neuman, Capacity and Fow Assignment in Large Computer Networks, in: IEEE Infocom 86, IEEE, 1986, pp. 275 284. 33. D. Bertsekas, R. Gaager, Data Networks, Prentice Ha, Engewood Ciffs, NJ, 1987. 34. E. Zegura, K. Cavert, M. J. Donahoo, A Quantitative Comparison of Graph-based Modes for Internet Topoogy, IEEE/ACM Transactions on Networks 5 (6) (1997) 770 783. 35. J. Yen, Finding the K Shortest Loopess Paths in a Network, Management Science 17 (11) (1971) 712 716. 36. NLANR passive measurement and anaysis project, http://pma.nanr.net (2001).