Fault-Tolerant Design of Wavelength-Routed Optical. Networks. S. Ramamurthy and Biswanath Mukherjee

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1 DIMACS Series in Discrete Mathematics and Theoretical Computer Science Fault-Tolerant Design of Wavelength-Routed Optical Networks S. Ramamurthy and Biswanath Mukherjee Abstract. This paper considers optical networks which employ wavelengthrouting switches that enable the establishment of wavelength-division-multiplexed (WDM) channels, called lightpaths, between node-pairs. In such networks, the failure of a single ber link may cause the failure of several optical channels. Given a set of lightpaths, that may comprise of a logical topology, and alternate routing tables at each node, this paper examines the following problem: how to allocate capacity in an optimal fashion and assign routes and wavelengths to each lightpath, such that, upon any link failure, lightpaths that are aected by the failure can be reestablished without any blocking. A simple approach, called one-on-one protection, protects against link failures by reserving for each lightpath, a link-disjoint backup lightpath. In this paper, we present an alternate protection scheme called multiplexed-spare-capacity protection, that multiplexes the spare capacity among all lightpaths in the logical topology in an optimal fashion. This paper examines the performance (in terms of capacity utilization) of this approach on representative network topologies, and shows that this approach is superior to one-on-one protection. 1. Introduction Wavelength-division multiplexing (WDM) [1] divides the tremendous bandwidth of a ber (potentially a few tens of terabits per second) into many nonoverlapping wavelengths (WDM channels). Each channel can be operated asynchronously and in parallel at any desirable speed, e.g., peak electronic speed of a few Gbps. An access station may transmit signals on dierent wavelengths, which are coupled into the ber using wavelength multiplexers. An optical signal passing through an optical switch may be routed from an input ber to an output ber without undergoing optoelectronic conversion. The architecture of a wavelength-routed optical network, shown in Fig. 1, consists of wavelength-routing switches (WRSs) (labeled 1 through 15) interconnected by ber links. Each link consists of a pair of unidirectional ber links. Technological constraints dictate that the number of WDM channels that can be supported in a ber be limited to W (whose value is a few tens today, but is expected to improve with time and technological breakthroughs). An access station is connected S. Ramamurthy and B. Mukherjee were supported in parts by NSF Grant No. NCR , and grants from Pacic Bell and UC Micro. 1 c0000 (copyright holder)

2 2 S. RAMAMURTHY AND BISWANATH MUKHERJEE to each WRS. For clarity of exposition, we will consider the access-station/wrs combination as an integrated unit which we will refer to as a network node. A connection request is satised by establishing a lightpath from the source node of the connection to the destination node. A lightpath is an all-optical channel which may span multiple ber links, to provide a circuit-switched interconnection between two nodes. In the absence of wavelength converters, a lightpath would occupy the same wavelength on all ber links that it traverses. This is called the wavelength-continuity constraint. Two lightpaths on a ber link must also be on dierent wavelength channels to prevent the interference of the optical signals. Figure 1 shows the following wavelength-continuous lightpaths: (a) between Nodes 10 and 6 on wavelength 1, and (b) between Nodes 15 and 9 on wavelength WC λ2 10 λ1 1 WC 2 6 WC WC 12 Figure 1. Architecture of a wavelength-routed optical network. A logical topology consists of a set of lightpaths set up to exploit the relative strengths of both optics and electronics - viz., packets of information are carried by the logical topology \as far as possible" in the optical domain using optical switching, but packet forwarding from lightpath to lightpath is performed via electronic packet switching, whenever required. The electronic packet-switching may be performed by an ATM switch (or an IP router), in which case the logical topology is operated as an ATM network (or IP network). In this context, we dene the routing and wavelength assignment problem [2] as follows: Given a trac demand (i.e., the number of lightpaths between each node-pair that comprises a logical topology), route and assign wavelengths to the lightpaths such that some network criteria (such as capacity utilization) is optimized. In such a backbone optical network architecture, the failure of a component such as a ber link can lead to the failure of all the lightpaths that traverse the failed link 1. Since each lightpath is expected to operate at a rate of several Gbps, 1 We assume in this paper that a link is unidirectional, and therefore, a link failure is a unidirectional link failure.

3 FAULT-TOLERANT DESIGN OF WAVELENGTH-ROUTED OPTICAL NETWORKS 3 such a failure can lead to a severe disruption in the network. There are several approaches to ensure network survivability [3]. The approach we consider here is that of restoration, where lightpaths aected by a component failure are rerouted around the failure, by reconguring network components such as wavelength-routing switches, transmitters, receivers, etc. Higher protocol layers (such as ATM or IP) operating on the logical topology may have their own protection schemes, and can recover from link failures. An IP network, for example, recovers from link failures by rerouting data packets around the failed link. However, the recovery time for higher layers is still signicantly large (on the order of seconds), whereas we expect that restoration times at the optical layer will be required to be of the order of milliseconds [8]. It is benecial to consider restoration mechanisms in the optical layer for the following reasons [7]: (a) the optical layer can eciently multiplex protection resources (such as spare wavelengths and bers) among several higher layer network applications, and (b) optical layer restoration provides protection to higher layer protocols that may not have built-in protection. This paper considers single link failures in an optical network. There are two well-known approaches [12, 14] to protecting against link failures: (a) link (or span) restoration, and (b) path restoration. In link restoration, all lightpaths that traverse the failed link are rerouted (utilizing the spare capacity) between the endnodes of the failed link. In path restoration, each lightpath that traverses the failed link is rerouted along a dierent route between the source and destination of the lightpath. In an optical network, path restoration may be preferred over link restoration for reasons that include the follwoing: (a) because of the wavelength-continuity constraint, it may be more capacity-ecient to reroute a lightpath from source to destination, and (b) because the optical layer has little intelligence, failure detection may be possible only at the end-nodes of the lightpath. In this paper we consider path-restoration-based approaches to designing a fault-tolerant optical network. Specically, we address the following problem: Given a set of lightpaths, route and assign wavelengths to the lightpaths, and reserve spare capacity in the network, such that upon any link failure, all the lightpaths aected by the failure can be rerouted without any blocking. The optimization criteria we consider here is the total network capacity used, i.e., the sum of the wavelength utilizations over all the links in the network. We examine two approaches to path restoration: (a) one-on-one protection, and (b) multiplexed-spare-capacity protection. These two approaches perform path restoration upon link failures and are based on precomputed routes. One-on-one protection: One-on-one protection [5] reserves a backup lightpath for every lightpath that needs to be protected against link failures. The backup lightpaths are designed such that they do not share any links with their corresponding primary lightpaths. When a primary lightpath fails because of a link failure, its backup lightpath is activated. Upon detecting a failure, all aected node-pairs (i.e., node-pairs whose primary lightpath route includes the failed link) switch to their backup paths. We dene the oneon-one protection problem as follows: Given a trac demand, nd routes and assign wavelengths to the each primary and backup lightpath, such that some network criteria (such as capacity utilization) is optimized.

4 4 S. RAMAMURTHY AND BISWANATH MUKHERJEE Multiplexed-spare-capacity protection: Given a set of lightpaths to be protected against link failures, we reserve wavelength channels in the links of the network intelligently. Wavelength channels for backup routes are shared among primary lightpaths that are link disjoint. This is possible under the assumption of single-link failures, because the link-disjoint primary lightpaths are assumed not to fail simultaneously and therefore may share wavelength channels for their backup routes. This approach ensures that, upon any link failure, all lightpaths that traverse the failed link can be rerouted around the failed link, without blocking any of them, and at the same time also minimizes the number of wavelength channels that need to be reserved in the links of the network. We dene the multiplexed-spare-capacity protection problem as follows: Given a trac demand, nd routes and assign wavelengths to the each primary lightpath, reserve spare capacity, and determine routes and wavelengths of backup lightpaths, such that some network criteria (such as capacity utilization) is optimized. We note that one-on-one protection is a static scheme in the sense that the wavelength switches are congured at setup time, prior to a link failure, for the working and the backup lightpaths. One-on-one protection also has the feature of being failure-independent, i.e., the restoration process does not need to isolate the faulty link. Multiplexed-spare-capacity protection is a dynamic scheme in the sense that, for each backup lightpath, wavelength switches may need to be congured (depending on the link that fails) after a link failure. Multiplexed-spare-capacity protection is also failure-dependent, i.e., the restoration procedures need to isolate the faulty link prior to conguring the backup lightpaths. The multiplexed-spare-capacity approach trades-o the eciency of spare-capacity allocation with added control complexity needed to isolate the fault and congure wavelength-routing switches (and therefore possibly have increased restoration times), to establish backup lightpaths upon a link failure. We illustrate the above approaches in an example. Consider the network illustrated in Fig. 1. Assume that two lightpaths need to be protected against link failures: (a) the rst lightpath from Node 10 to Node 6 and (b) the second lightpath from Node 15 to Node 9. The routes and wavelengths of working and backup lightpaths in one-on-one protection are illustrated in Table 1. Primary Lightpath Backup Lightpath 10! 6 (10,11,1,6) on 1 (10,9,7,6) on 1 15! 9 (15,6,7,9) on 1 (15,14,12,13,1,11,10,9) on 2 Table 1. The routing and wavelengths of primary and backup lightpaths with one-on-one protection. In the following discussion, a wavelength-link is dened to be a wavelength on a link. A total of 16 wavelength-links are reserved on the links of the network, 6 wavelength-links for the primary lightpaths and 10 wavelength-links for the backup lightpaths. We note that the two working lightpaths { (10,11,1,6), and (15,6,7,9) {

5 FAULT-TOLERANT DESIGN OF WAVELENGTH-ROUTED OPTICAL NETWORKS 5 are link disjoint. As a result, upon any link failure, at most one of the two lightpaths can fail, i.e., both lightpaths cannot fail simultaneously upon a link failure. Therefore, the backup lightpaths can share wavelengths since they will not be activated simultaneously. This observation leads to the routes and wavelength assignments for the working and backup lightpaths illustrated in Fig. 2. We note that a total of 15 wavelength-links are reserved in this case, a reduction of 1 wavelength-link from the one-on-one protection case. Primary Lightpath Backup Lightpath 10! 6 (10,11,1,6) on 1 (10,9,7,6) on 1 15! 9 (15,6,7,9) on 1 (15,14,12,13,1,11,10,9) on 1 Table 2. The routing and wavelengths of primary and backup lightpaths with multiplexed-spare-capacity protection Previous work. The design of a survivable optical network has been studied in [5, 6, 7, 8, 9, 10, 11]. In [5], the authors propose physical protection schemes and a path-restoration scheme based on one-on-one protection. The work in [6] considers fault-tolerant design of optical ring networks. In work in [7, 8] addresses the issues in the design of a survivable optical layer. In [9], the authors propose an algorithm that protects optical mesh networks from link and node failures. In [11], the authors propose analytical methods to estimate capacity utilization in optical networks that are resilient against single link failures. Our work considers a path-restoration-based approach utilizing multiplexed-spare-capacity, for protection against link failure in an optical networks, and in this manner is dierent from previous work. Network survivability, restoration schemes, and optimal spare-capacity design have been studied extensively in circuit-switched transport networks [3, 12, 13, 14] Outline of remaining sections. In Section 2, we develop the Integer Linear Programming (ILP) formulations of the one-on-one and multiplexed-sparecapacity protection problems. A solution approach for the ILP formulation, and the complexity of the solution procedure is considered in Section 3. Section 4 presents numerical results for two representative network topologies. Section 5 concludes the paper with a discussion of the main contributions of this work, and related problems for further research. 2. Problem Formulation In this section, we develop ILP formulations of the dierent protection schemes. Specically, we will develop the following three programs. ILP1: This ILP determines the routing and wavelength assignment for a given set of lightpaths without any failure protection. ILP2: This ILP formulates the one-on-one protection problem. ILP3: This ILP formulates the multiplexed-spare-capacity protection problem.

6 6 S. RAMAMURTHY AND BISWANATH MUKHERJEE In the following subsections, we dene the algorithm to nd routes for lightpaths between node-pairs in the network, and then we develop the notation to describe the ILPs Routing. We assume that alternate routing [4] is utilized to perform the routing of lightpaths in the network. Alternate routing requires each node in the network to have a routing table. The routing table at a node contains a set of precomputed routes to each destination node. Any lightpath (which may be a primary lightpath or a backup lightpath) between a node-pair can be satised by any one of the routes (with a free wavelength) among the set of alternate routes at the source node. We assume that the set alternate routes for each node-pair will contain at-least two routes that are link disjoint, and that the set of alternate routes at each source node exploits the connectivity the network topology (i.e., all link-disjoint routes to each destination are included in the set of alternate routes). Table 3 illustrates the routing table at node 1 for the network in Fig. 1, with two (link-disjoint) alternate routes to each destination node. Destination Route1 Route Table 3. The routing table at Node 1 for the network in Fig. 1, with two alternate routes to each destination Notation. We dene the notation employed to develop the ILPs. We are given the following: (a) the network topology represented as a directed graph G, (b) a demand matrix, i.e., the number of lightpath requests between node-pairs, and (c) alternate routing tables at each node. Given: N: Nodes in the network (numbered 1 through N). Node-pairs are numbered 1 through N (N? 1). E: Links in the network (numbered 1 through E). W : Maximum number of wavelengths on a link. R i : Set of alternate routes for node-pair i. M i = jr i j: Number of alternate routes between node-pair i. R i j : Set of eligible alternate routes between node-pair i after link j fails. d i : Demand for node-pair i, in terms of number of lightpath requests. We require the ILPs to solve for the following variables: s j : Number of spare wavelengths used on link j.

7 FAULT-TOLERANT DESIGN OF WAVELENGTH-ROUTED OPTICAL NETWORKS 7 w j : Number are the primary channels used on link j. g : Number of primary lightpath wavelengths used on route r to satisfy the demand between node-pair i, before any link failures. w takes on the value of 1 if the r th route between node-pair i utilizes wavelength w before any link failures, 0 otherwise. fj : Number of spare wavelengths used on the r th route between node-pair i, after link j fails. w;j takes on the value 1 if the rth route for node-pair i uses spare wavelength w after link j fails, 0 otherwise ILP Formulations ILP1 { Routing and Wavelength Assignment with No Protection.. Minimize the total capacity used: M inimize E j=1 Number of lightpaths on each link is bounded: Demand between each node-pair i is satised: d i = r=1 w=1 w j (2.1) w j W 1 j E (2.2) M i W w 1 i N(N? 1) (2.3) Dene the number of primary lightpaths traversing each link: W w j = w 1 j E (2.4) i=1 r2r i ;j2r w=1 Wavelength-continuity constraint, i.e., only one primary lightpath can use a wavelength w on link j: i=1 r2r i :j2r w 1 1 w W; 1 j E (2.5) ILP2 { One-on-One Protection.. Minimize the total capacity used: M inimize E j=1 Number of lightpaths on each link is bounded: w j (2.6) w j W 1 j E (2.7)

8 8 S. RAMAMURTHY AND BISWANATH MUKHERJEE Demand between a node-pair i is satised, and each primary lightpath has a backup lightpath: 2 d i = M i W r=1 w=1 w 1 i N(N? 1) (2.8) Denition of the number of lightpaths (primary and backup) traversing each link: W w j = w 1 j E (2.9) i=1 r2r i ;j2r w=1 Wavelength-continuity constraint, i.e., only one lightpath can use wavelength w on link j: i=1 r2r i :j2r w 1 1 w W; 1 j E (2.10) When a link fails, demands between all node-pairs can still be satised: W r2r i w=1 j w di 1 j E; 1 i N(N? 1) (2.11) ILP3 { Multiplexed-Spare-Capacity Protection.. Minimize the total capacity used: M inimize E j=1 Number of lightpaths on each link is bounded. Before Fault: Demand between each node-pair is satised. (w j + s j ) (2.12) w j + s j W 1 j E (2.13) M i d i = g 1 i N(N? 1) (2.14) r=1 Denition of the number of primary lightpaths traversing a link: W w j = w 1 j E (2.15) i=1 r2r i ;j2r w=1 Wavelength-continuity constraint for primary lightpaths, i.e., only one primary lightpath can use wavelength w on link j: i=1 r2r i :j2r w 1 1 w W; 1 j E (2.16)

9 FAULT-TOLERANT DESIGN OF WAVELENGTH-ROUTED OPTICAL NETWORKS 9 After Fault: Denition of the total number rerouted lightpaths between node-pair i when link j fails: W W w;j = w r2r i w=1 r2r j i :j2r w=1 1 i N(N? 1); 1 r M i ; 1 j E (2.17) Wavelength-continuity constraint for backup lightpaths, i.e., only one backup lightpath can use (spare) wavelength w on link k: w;j 1 1 w W; 1 j; k E (2.18) i=1 r2r i ;k2r j Spare capacity on each link k meets the restoration demands on that link: i=1 W r2r i :k2r w=1 j w;j <= s k + i=1 W r2r i ;j;k2r w=1 3. Solution Approach w 1 j; k E (2.19) The routing and wavelength assignment problem has been shown to be NPcomplete [2]. We expect the one-on-one protection problem, and the multiplexedspare-capacity protection problems to be NP-complete as well. We have used a freeware LP/ILP solver called lpsolve, to solve the instances of the ILPs generated for two representative network topologies. We note that the number of variables and equations for the ILPs grows rapidly with the size of the network, and therefore, the ILP formulations are practical only for small networks (a few tens of nodes). For larger networks (a few hundreds of nodes), we may need to employ heuristic methods. We also note that any solution to the one-on-one protection problem is also a solution to the corresponding multiplexed-spare-capacity protection problem, and therefore the capacity utilization of the one-on-one protection solution is at-least as much as that for the multiplexed-spare-capacity solution. 4. Illustrative Examples and Discussion The performance of any restoration mechanism is measured by the following criteria: (a) capacity utilization, (b) restoration time, and (c) scalability and implementation complexity. In this paper, we provide numerical results for capacity utilization, and make qualitative comments on the restoration time and complexity of the dierent protection approaches. We performed our studies on two representative networks: (a) a network of interconnected rings illustrated in Fig. 1, and (b) the European optical network [15] illustrated in Fig. 2. These topologies were chosen to be representative of typical mesh topologies employed in telecommunications networks. For each network topology and between each node-pair, we chose a set of three alternate routes, ensuring that at least two routes were link disjoint. For each network topology, we ran ILPs 1, 2, and 3 on random logical topologies, where each logical topology had between 5 and 20 lightpaths.

10 10 S. RAMAMURTHY AND BISWANATH MUKHERJEE Figure 2. Topology of the European optical network Results. We tabulate the results for the interconnected-rings network in Table 4. In each row of the table, we illustrate numerical results for two random logical topologies with the same number of lightpaths. The rst column is the number of lightpaths in the random logical topology. The second column indicates the capacity utilization of the optimal routing and wavelength assignment of the lightpaths obtained from ILP1. The third and fourth column contain three-tuples (l; v; c) for the one-on-one protection (ILP2) and multiplexed-spare-capacity protection (ILP3), respectively, where l is the lower bound for the ILP obtained by relaxing the integer constraints, v is the optimal value of the capacity utilization, and c is the \congestion" value of the optimal solution, where congestion is the wavelength utilization of the link that has the maximum number of lightpaths traversing through itself. The fth column illustrates the gain in capacity utilization of the multiplexed-spare-capacity solution relative to the one-on-one solution. We note that multiplexed-spare-capacity protection provides moderate gains in capacity utilization, as well as in congestion, over one-on-one protection. The numerical results are also illustrated graphically in Fig. 3. Figure 5 illustrates the performance of dierent protection schemes, for the interconnected-rings network, enhanced with two added rings, shown in Fig. 4. We note that the enhanced interconnected-rings network, has a better capacity utilization relative to the interconnected-rings network in 1 for both protection schemes, because there are more link-disjoint and shorter routes between node-pairs. Figure 6 illustrates the performance of dierent protection schemes, for the European optical network illustrated in Fig. 2. Again, we observe that multiplexedspare-capacity protection provides moderate gains in capacity utilization, as well as in congestion, over one-on-one protection.

11 FAULT-TOLERANT DESIGN OF WAVELENGTH-ROUTED OPTICAL NETWORKS Capacity utilization Vs Number of lightpaths No spare capacity One-on-one protection Multiplexed lower bound Multiplexed protection 140 Capacity utilization Number of lightpaths Figure 3. Performance of dierent protection schemes for the interconnected-rings network. Connections No Protection One-on-One (l,v,c) Multiplexed (l,v,c) Gain ,30,2 26,26, ,27,4 25,25, ,32,3 27,27, ,43,3 37,37, ,47,3 41,41, ,44,3 37,37, ,49,3 42,43, ,57,4 48,48, ,55,3 47,47, ,63,5 54,55, ,69,4 63,63, ,67,4 55,55, ,80,5 71,71, ,65,4 55,55, ,77,5 66,66, ,77,6 64,64, ,88,5 74,74, ,88,6 74,74, ,102,7 91,92, ,89,7 74,74, ,93,5 79,79, ,91,7 71,71, ,100,8 83,83, ,116,7 97,97, ,119,7 99,99, ,113,6 93,93, ,114,7 92,92, ,106,7 86,86, ,125,7 99,99, ,122,7 101,101, ,117,7 86,87, ,132,7 104,104,5 28 Table 4. Results for the interconnected-rings network 5. Conclusion Optical networks based on WDM technology can potentially transfer several Gbps of data on each ber link in the network. However, the high bandwidths

12 12 S. RAMAMURTHY AND BISWANATH MUKHERJEE Figure 4. The interconnected-rings network in Fig. 1 with two added rings. Capacity utilization Capacity utilization Vs Number of lightpaths No spare capacity One-on-one protection Multiplexed lower bound Multiplexed protection Extended ring - No spare capacity Extended ring - One-on-one protection Extended ring - Multiplexed lower bound Extended ring - Multiplexed protection Number of lightpaths Figure 5. Performance of dierent protection schemes for the extended interconnected-rings network. carried by links have the drawback that a link failure can potentially lead to the loss of a large amount of data.

13 FAULT-TOLERANT DESIGN OF WAVELENGTH-ROUTED OPTICAL NETWORKS Capacity utilization Vs Number of lightpaths No spare capacity One-on-one protection Multiplexed lower bound Multiplexed protection 140 Capacity utilization Number of lightpaths Figure 6. Performance of dierent protection schemes for the European optical network. This paper considered two restoration mechanisms for protecting a logical topology from single-link failures: (a) one-on-one protection, and (b) multiplexed-sparecapacity protection. One-on-one protection is based on precomputed link-disjoint alternate paths and is failure independent. The backup paths in one-on-one protection are specied and congured statically prior to the occurrence of the failure. Multiplexed-spare-capacity protection is also based on precomputed alternate paths and is failure dependent. The backup paths are computed at setup time but they congured dynamically after a link-failure. This paper formulated the one-on-one protection problem, and the multiplexedspare-capacity protection problem as integer linear programs. The numerical results obtained for two representative network topologies and for random logical topologies indicate that multiplexed-spare-capacity protection provides moderate gains in capacity utilization over one-on-one protection. We are currently performing simulation-based studies on the restoration times for dierent protection schemes, and we will report these results in the future. This paper assumed the existence of fault-detection and isolation capabilities in the network. However, fault-detection and isolation is a challenging problem in optical networks, because of the lack of optical processing within the network. In addition to single-link failures, other failure scenarios in optical networks that merit further study include: channel failures, node failures, and multiple failures of nodes and/or links. The design of the control network, and control protocols for optical layer restoration are important areas for further research. References [1] B. Mukherjee, Optical Communication Networks, New York: McGraw-Hill, July [2] R. Ramaswami and K. N. Sivarajan, \Routing and Wavelength Assignment in All-Optical Networks," IEEE/ACM Transactions on Networking, vol. 3, no. 5, pp , October 1995.

14 14 S. RAMAMURTHY AND BISWANATH MUKHERJEE [3] T. Wu, Fiber Network Service Survivability, Artech House, [4] S. Ramamurthy and B. Mukherjee, \Modeling and Simulation of Fixed Alternate Routing and Wavelength Conversion in Wavelength Routed Optical Networks," Technical Report CSE-97-16, Dept. of Computer Science, UC Davis, November [5] J. Armitage, O. Crochat, and J.-Y. Le Boudec, \Design of a Survivable WDM Photonic Network," Proc., IEEE INFOCOM '97, Kobe, Japan, pp , April [6] O. Gerstel, R. Ramaswami, and G. Sasaki, \Fault Tolerant Multiwavelength Optical Rings with Limited Wavelength Conversion," Proc., IEEE INFOCOM '97, Kobe, Japan, pp , April [7] O. Gerstel, \Opportunities for optical protection and restoration," Proc., OFC '98, San Jose, CA, vol. 2, pp , February [8] P. Bonenfant, \Optical layer survivability: a comprehensive approach," Proc., OFC '98, San Jose, CA, vol. 2, pp , February [9] S. G. Finn, M. Medard, and R. A. Barry, \A new algorithm for bi-directional link selfhealing for arbitrary redundant networks," Proc., OFC '98, San Jose, CA, vol. 2, pp , February [10] E. Karasan, and E. Goldstein, \Optical restoration at the wavelength-multiplex section level in WDM mesh networks," Proc. OFC '98, San Jose, CA, vol. 2, pp , February [11] E. Limal, S. L. Danielsen, and K. E. Stubkjaer, \Capacity utilization in resilient wavelengthrouted optical networks using link restoration," Proc. OFC '98, San Jose, CA, vol. 2, pp , February [12] R. R. Iraschko, M. H. MacGregor, and W. D. Grover, \Optimal Capacity Placement for Path Restoration in Mesh Survivable Networks," Proc., ICC '96, Dallas, Texas, pp , June [13] M. Herzberg, S. J. Bye, and A. Utano, \The Hop-Limit Approach for Spare-Capacity Assignment in Survivable Networks," IEEE/ACM Transactions on Networking, vol. 3, no. 6, pp , December [14] J. Anderson, B. T. Doshi, S. Dravida, and P. Harshavardhana, \Fast Restoration of ATM Networks," IEEE Journal on Selected Areas in Communications, vol. 12, no. 1, pp , January [15] M. Garnot, M. Sotom, and F. Masetti, \Routing Strategies for Optical Paths in WDM Networks," Proc., ICC '97, Montreal, Canada, pp , June Department of Computer Science, University of California, Davis, CA 95616, U.S.A. address: ramu@cs.ucdavis.edu Department of Computer Science, University of California, Davis, CA 95616, U.S.A. address: mukherje@cs.ucdavis.edu