480 WDM NETWORK DESIGN Table 8.4 Number of wavelengths required to perform offline wavelength assign- ment as a function of the load L with and without wavelength converters. The fixed conversion result for arbitrary topologies applies only to one- and two-hop lightpaths. Network Conversion Type None Fixed Full Limited Arbitrary min[(L- 1)D + 1, L L (2L - 1)xfM- L + 2] Ring 2L- 1 L + 1 L L Star ~ L L L Tree 3 EL L L Other topologies such as star networks and tree networks have also been consid- ered in the literature. In star and tree networks, ~-L wavelengths are sufficient to do WA-NC [RU94]. In star networks, L wavelengths are sufficient for WA-FC [RS97]. The same result can be extended to arbitrary networks where lightpaths are at most two hops long. Table 8.4 summarizes the results to date on this problem. It is still a topic of intense research. Multifiber Rings The wavelength assignment problem in multifiber rings is considered in [LS00]. In a multifiber ring, each pair of adjacent nodes is connected by k > 1 fiber pairs: k > 1 fibers are used for each direction of transmission instead of i fiber. Recall that we are considering undirected edges and lightpaths, and each edge represents a pair of fibers, one for each direction of transmission. Thus, such a multifiber ring is represented by k edges between pairs of adjacent nodes. There is no wavelength conversion, but it is assumed that the same wavelength can be switched from an incoming fiber to any of the k outgoing fibers at each node. The following results on multifiber rings are proved in [LS00]. Theorem 8.6 [LS00] Given a set of lightpath requests and a routing on a k-fiber-pair ring with load L on each multifiber link, the number of wavelengths, summed over all the fibers, required to solve the wavelength assignment problem is no more than [k~ !lL- 11. 8.5 Maximum Load Dimensioning Models 481 Thus, for a dual-fiber-pair ring (k = 2), the number of wavelengths required is no than [3L -1~, which is a significant improvement over the bound of 2L -1 more ! / for a single-fiber-pair ring. As in the case of the single-fiber-pair ring, you can come up with a set of lightpath requests with load L for which this upper bound on the number of wavelengths is tight, for all values of the fiber multiplicity, k. 8.5.2 Online RWA in Rings We next consider the online wavelength assignment problem in rings. Assume that the routing of the lightpaths is already given and that lightpaths are set up as well as taken down, that is, the lightpaths are nonpermanent. Here, it becomes much more difficult to come up with smart algorithms that maximize the load that can be supported for networks without full wavelength conversion. (With full wavelength conversion at all the nodes, an algorithm that assigns an arbitrary free wavelength can support all lightpath requests with load up to W.) We describe an algorithm that provides efficient wavelength assignment for line and ring networks without wavelength conversion. Lemma 8.7 [GSKR99] Let W(N, L) denote the number of wavelengths re- quired to support all online lightpath requests with load L in a network with N nodes without wavelength conversion. In a line network, W(N, L) <_ L + W(N/2, L), when N is a power of 2. Proof. Break the line network in the middle to realize two disjoint subline networks, each with N/2 nodes. Break the set of lightpath requests into two groups: one group consisting of lightpaths that lie entirely within the subline networks, and the other group consisting of lightpaths that go across between the two subline networks. The former group of lightpaths can be supported with at most W(N/2, L) wavelengths (the same set of wavelengths can be used in both subline networks). The latter group of lightpaths can have a load of at most L. Dedicate L additional wavelengths to serving this group. This proves the lemma. I The following theorem follows immediately from Lemma 8.7, with the added condition that W (1, L) = 0 (or W (2, L) = L). Theorem 8.8 [GSKR99] In a line network with N nodes, all online lightpath requests with load L can be supported using at most L [log 2 N] wavelengths without requiring wavelength conversion. 482 WDM NETWORK DESIGN The algorithm implied by this theorem is quite efficient since it is possible to come up with lightpath traffic patterns for which any algorithm will require at least 0.5L log 2 N wavelengths [GSKR99]. Theorem 8.9 [GSKR99] In a ring network with N nodes, all online lightpath requests with load L can be supported using at most L [log 2 N] + L wavelengths, without requiring wavelength conversion. The proof of this theorem is left as an exercise (Problem 8.21). When we have permanent lightpaths being set up, it is possible to obtain some- what better wavelength assignments, as given by the following theorem, the proof of which is beyond the scope of this book. Theorem 8.10 [GSKR99] In a ring network with N nodes, all online perma- nent lightpath requests with load L can be supported using (a) at most 2L wave- lengths without wavelength conversion, and (b) with at most max(0, L - d) + L wavelengths with degree-d (d >_ 2) limited wavelength conversion. Table 8.5 summarizes the results to date on the offline and online RWA problem for ring networks, with the traffic model characterized by the maximum link load. For this model, observe that significant increases in the traffic load can be achieved by having wavelength converters in the network. For the offline case, very limited conversion provides almost as much benefit as full wavelength conversion. For the online cases, the loads that can be supported are much less than the offline case. The caveat is that, as illustrated in Figure 8.13, this model represents worst-case scenarios, and a majority of traffic patterns could perhaps be supported efficiently without requiring as many wavelengths or as many wavelength converters. Summary We studied the design of wavelength-routing networks in this chapter. We saw that there is a clear benefit to building wavelength-routing networks, as opposed to simple point-to-point WDM links. The main benefit is that traffic that is not to be terminated within a node can be passed through by the node, resulting in significant savings in higher-layer terminating equipment. The design of these networks is more complicated than the design of traditional networks. It includes the design of the higher-layer topology (IP or SONET), which is the lightpath topology design problem, and its realization in the optical layer, which is the routing and wavelength assignment problem. These problems may need to be Summary 483 Table 8.5 Bounds on the number of wavelengths required in rings to sup- port all traffic patterns with maximum load L for different models, offline and online, from [GRS97, GSKR99]. d denotes the degree of wavelength conver- sion. The upper bound indicates the number of wavelengths that are sufficient to accommodate all traffic patterns with maximum load L, using some RWA algorithm. The lower bound indicates that there is some traffic pattern with maximum load L that requires this many wavelengths regardless of the RWA algorithm that is employed. For the online traffic model, we consider two cases, one where lightpaths are set up over time but never taken down, and another where lightpaths are both set up and taken down over time. Conversion Degree Lower Bound on W Upper Bound on W Offline traffic model No conversion 2L- 1 2L- 1 Fixed conversion L + 1 L + 1 >2 L L m Online model without lightpath terminations No conversion 3L 3L Fixed conversion L 3L Full conversion L L Online model with lightpath terminations No conversion 0.5L [log 2 N] L [log 2 N] + L Full conversion L L solved in conjunction if the carrier provides IP or SONET VTs over its own optical infrastructure. However, this is difficult to do, and a practical approach may be to iteratively solve these problems. We then discussed the wavelength dimensioning problem. The problem here is to provide sufficient capacity on the links of the wavelength-routing network to handle the expected demand for lightpaths. This problem is solved today by periodically forecasting a traffic matrix and (re)designing the network to support the forecasted matrix. Alternatively, you can employ statistical traffic demand models to estimate the required capacities, and we discussed two such models. The absence of wavelength conversion in the network can be overcome by pro- viding more wavelengths on the links. In the last section, we studied this trade-off under various models. 484 WDM NETWORK DESIGN Further Reading The issue of how much cost savings is afforded by providing networking functions within the optical layer is only beginning to be understood. For some more insights into this issue, see [RLB95, Ba196, GRS98, SGS99, CM00, BM00]. The material in this chapter is based on [GRS98]. See [Wi196, WW98, Ber96] for a discussion of the problem of setting up connections between all pairs of nodes in a WDM ring network. The lightpath topology design problem is discussed in [RS96, KS98, CMLF00, MBRM96, BG95, ZA95, JBM95, GW94, CGK93, LA91]. Our discussion is based on [RS96]. This is an example of a network flow problem; these problems are dealt with in detail in [AMO93]. Several papers [ABC+94, RU94, RS95, CGK92, RS97, MKR95, KS97, KPEJ97, ACKP97] study the offline routing and wavelength assignment problem. There is also a vast body of literature describing routing and wavelength assignment heuristics. See, for example, [CGK92, SBJS93, RS95, Bir96, WD96, SOW95]. The statistical blocking model for dimensioning is analyzed in [SS00, BK95, RS95, KA96, SAS96, YLES96, BH96]. The worst-case analysis of the maximum load model with online traffic is con- sidered in [GK97]. 8.1 8.2 Problems In general there are several valid design options even for a three-node network. Consider the designs shown in Figure 8.1(c), but now assume that the number of dropped lightpaths is six instead of five as discussed in the text. The advantage of this design is that it provides more flexibility in handling surges in A-B and B-C traffic. For example, this design not only can handle the traffic requirement of 50 Gb/s between every pair of nodes, it can also handle a traffic requirement of 60 Gb/s between nodes A-B and B-C, and 40 Gb/s between nodes A-C. This latter traffic pattern cannot be handled if only five lightpaths/wavelengths are dropped. Consider the design of Figure 8.1(c), and assume that x wavelengths are dropped at node B and y wavelengths pass through. Determine the range of traffic matrices that this design is capable of handling as a function of x and y. Consider the network design approach using fixed-wavelength routing in a four-node ring network with consecutive nodes A, B, C, and D. Suppose the traffic requirements are as follows: Problems 485 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 A B C D A B C D - 3 - 3 3 - 2 3 - 2 - 2 3 3 2 - (a) Do a careful routing of traffic onto each wavelength so as to minimize the number of wavelengths needed. (b) How do you know that your solution uses the minimum possible number of wavelengths required to do this routing for any algorithm? (c) How many ADMs are required at each node to support this traffic? (d) How many ADMs are required at each node if instead of fixed-wavelength routing, you decided to use point-to-point WDM links and receive and re- transmit all the wavelengths at each node? How many ADMs does wave- length routing eliminate? Derive (8.1). What is the value when N is odd? Derive (8.5). What is the value when N is odd? Derive (8.8) for the case where there is one full-duplex lightpath between each pair of nodes. Hint: Use induction. Start with two nodes on the ring, and determine the number of wavelengths required. Add two more nodes so that they are diametrically opposite to each other on the ring and continue. Show that when N is odd, (8.8) is modified to W= N-1 8 " Derive (8.9). What is the value when N is odd? Develop other network designs besides the ones shown in Examples 8.2, 8.3, and 8.4, and compare the number of LTs and wavelengths required for these designs against these three examples. Consider the network shown in Figure 8.9(a). Assume that each undirected edge can be represented by a pair of directed edges as in Figure 8.9(c). Represent each undirected lightpath in Figure 8.9(a) by a pair of directed lightpaths with oppo- site directions. Consider the RWA problem in the resulting network and show that two wavelengths are sufficient to support these directed lightpaths. Note that three wavelengths were required to support the corresponding undirected lightpaths. This problem illustrates the complexity of wavelength assignment in networks where the transmission is bidirectional over each fiber. Consider the two networks shown 486 WDM NETWORK DESIGN Figure 8.22 Two different scenarios of wavelength assignment in networks with bidi- rectional links. 8.11 8.12 8.13 8.14 8.15 in Figure 8.22. In Figure 8.22(a), the network uses two fibers on each link, with two wavelengths on each fiber, with unidirectional transmission on each fiber. In Figure 8.22(b), the network uses one fiber on each link, with four wavelengths. Transmission is bidirectional on each fiber, with two wavelengths in one direction and two in the other. No wavelength conversion is allowed in either network. Both networks have the same nominal capacity (four wavelengths/link). Which network utilizes the capacity more efficiently? Show that a network having P fiber pairs between nodes and W wavelengths on each fiber with no wavelength conversion is equivalent to a network with one fiber pair between nodes with P W wavelengths, and degree P wavelength conversion capability at the nodes. Generalize the example of Figure 8.13 to the case when the number of nodes is arbitrary, say, N. Compare the number of wavelengths required in this general case to the upper bound given by Theorem 8.1. In order to prove that W _< (2L - 1)x/M- L + 2 in Theorem 8.1, we supposed that there were K lightpaths of length >__ ff-M hops. Instead suppose there are K(x) lightpaths of length >_ x hops, and derive an upper bound for W that holds for every x. Now, optimize x to get the least upper bound for W. Compare this bound with the bound obtained in Theorem 8.1. Show that Algorithm 8.3 always does the wavelength assignment using L wave- lengths. Hint: Use induction on the number of nodes. Consider the following modified version of Algorithm 8.3. In step 2, the algorithm is permitted to assign any free wavelength from a fixed set of L wavelengths, instead of the least numbered wavelength. Show that this algorithm always succeeds in performing the wavelength assignment. Problems 487 8.16 8.17 8.18 8.19 8.20 8.21 8.22 8.23 8.24 Prove that Theorem 8.3 can be tight in some cases. In other words, give an example of a ring network and a set of lightpath requests and routing with load L that requires 2L - 1 wavelengths. Hint: First, give an example that requires 2L - 2 wavelengths and then modify it by adding an additional lightpath without increasing the load. Note that the example in Figure 8.20 shows such an example for the case L = 2. Obtain an example for the case L > 2. Consider a ring network with a lightpath request set of one lightpath between each source-destination pair. Compute the number of wavelengths sufficient to support this set with full wavelength conversion and without wavelength conversion. What do you conclude from this? Give an example of a star network without wavelength conversion where 3 L wave- lengths are necessary to perform the wavelength assignment. Prove Theorem 8.4. Prove Theorem 8.8. Based on this proof, write pseudo-code for an algorithm to perform wavelength assignment. Prove Theorem 8.9. This problem relates to the wavelength assignment problem in networks without wavelength conversion. Let us assume that the links in the network are duplex, that is, consist of two unidirectional links in opposite directions. A set of duplex lightpath requests and their routing is given. In practice, each request between two nodes A and B is for a lightpath 1 from A to B and another lightpath 1 f from B to A, which we will assume are both routed along the same path in the network. One wavelength assignment scheme (scheme 1) is to assign the same wavelength to both 1 and l I. Give an example to show that it is possible to do a better wavelength assignment (using fewer wavelengths) by assigning different wavelengths to 1 and 1 t (scheme 2). Show using this example that scheme I can need up to ~- W wavelengths, where W is the number of wavelengths required for scheme 2. Hint: Consider a representation of the path graph corresponding to directed lightpaths. Derive the expression (8.13) for the probability that a lightpath request is blocked when the network uses full wavelength conversion. Derive the approximate expressions for ~nc and ~fc given by (8.16) and (8.17). Plot these approximations and the exact values given by (8.14) versus W for Pb = 10 -3, 10 -4, and 10 -5, and H = 5, 10, and 20 hops to study the behavior of ~nc and ~fc, and to verify the range of accuracy of these approximations. 8.25 Derive (8.18). 488 WDM NETWORK DESIGN Figure 8.23 Network topology for Problem 8.26. 8.26 Consider the five-node fiber topology shown in Figure 8.23 on which IP bandwidth is to be routed between IP router node pairs over a WDM network. The bandwidth demands are given for each node pair in the following table. Assume that all demands are bidirectional, and both directions are routed along the same path using the same wavelengths in opposite directions. Gb/s B C D E A 15 25 5 15 B 5 35 15 C 15 25 D 5 (a) Assuming OC-192c (10 Gb/s) trunks are used, complete an equivalent table for the required number of lightpaths (that is, wavelengths) between each pair of nodes. (b) Using the given physical topology, and assuming that there are no wave- length conversion capabilities contained within the optical crossconnects at the nodes, specify a reasonable wavelength-routing design for each light- path. Clearly label each wavelength along its end-to-end path through the network. (c) What is the maximum load on any link in the network, and how does it compare with the number of wavelengths you are using in to- tal? References [ABC+94] A. Aggarwal, A. Bar-Noy, D. Coppersmith, R. Ramaswami, B. Schieber, and M. Sudan. Efficient routing and scheduling algorithms for optical networks. In References 489 Proceedings of 5th Annual A CM-SIAM Symposium on Discrete Algorithms, pages 412-423, Jan. 1994. [ACKP97] V. Auletta, I. Caragiannis, C. 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Reducing electronic multiplexing costs in SONET/WDM rings with dynamically changing traffic. IEEE Journal of Selected Areas in Communications, 18:1961-1971, 2000. [Cah98] R. Cahn. Wide Area Network Design: Concepts and Tools for Optimization. Morgan Kaufmann, San Francisco, 1998. . for optical networks. In References 489 Proceedings of 5th Annual A CM-SIAM Symposium on Discrete Algorithms, pages 41 2-4 23, Jan. 1994. [ACKP97] V. Auletta, I. Caragiannis, C. Kaklamanis, and. number of wavelengths you are using in to- tal? References [ABC+94] A. Aggarwal, A. Bar-Noy, D. Coppersmith, R. Ramaswami, B. Schieber, and M. Sudan. Efficient routing and scheduling algorithms. Bermond et al. Efficient collective communication in optical networks. In 23rd International Colloquium on Automata, Languages and Programming ICALP '96, Paderborn, Germany, pages 57 4-5 85,