470 WDM NETWORK DESIGN Figure 8.18 Reuse factor plotted against the number of nodes for random graphs with average degree 4, with full wavelength conversion and no wavelength conversion (from [RS95]). converters, the wavelength assignment algorithm assigns an arbitrary free wave- length on every link in the route to the lightpath; thus we assume full wavelength conversion. In both cases, if the wavelength assignment algorithm is unable to find a suitable wavelength, the lightpath request is blocked. In order to compute the blocking probability for lightpath requests, we make the simplifying assumption that the probability that a wavelength is used on a link is 7r and that this event is independent of the use of other wavelengths on the same link and the use of (the same and other) wavelengths on other links. If the network has W wavelengths on every link and a lightpath request chooses a route with H links, the probability that it is blocked is given by Pb,nc (1 (1 jr)H) w (8.12) when the network does not use wavelength converters. To see this, note that the probability that a given wavelength is free on any given link is (1 - Jr), and thus the probability that it is free on all the H links in the route is (1 - Jr) s by the assumed independence of the use of a wavelength on each link. Therefore, (1 - (1 - Jr) H) is the probability that a given wavelength is not free on some link of the route and, since the use of each wavelength is assumed to be independent of the use of other 8.4 Statistical Dimensioning Models 471 wavelengths, (1 - (1 - zr) H) w is the probability that all W wavelengths are not free on some link of the route, that is, Pb,nc. When the network uses full wavelength conversion, the probability that a light- path request is blocked is given by Pb,fc 1 (1 Jrw) H - - - . (8.13) The derivation of this equation using reasoning similar to that used in the derivation of (8.12) is left as an exercise (Problem 8.23). Given the blocking probability, we denote the solution of (8.12) and (8.13) for Jr by 7rnc and rrfc, respectively. Thus 7rnc (respectively, 7rfc) represents the achievable link utilization for a given blocking probability when wavelength converters are not used (respectively, used). It is easily seen that 1/W (8.14) rCnc l (1 Pb,nc ) 1/ H and 7rfc (1 (1 Pb,fc)l/H) 1/w = - - . (8.15) For small values of Pb,. (which is the case of practical interest) and sufficiently small values of W such that pl(W is not too close to 1, rrnc and rrfc can be approximated by pl/W/H (8 16) 7rnc " b,nc and 7rfc (Pb,fc/H) 1/W . (8.17) Thus for the same blocking probability, the ratio rrfc/Jrnc can be approximated by Hl-1/w. Therefore, this simplified analysis predicts that even for moderately large values of W the achievable link utilization is lower by approximately a factor of H when wavelength converters are not used in the network. Although the preceding analysis is highly simplified, ignores several important effects, and overestimates the efficacy of wavelength converters in improving the link utilization, it does predict correctly that the achievable link utilization is more sensitive to the path length (H) when wavelength converters are not used than otherwise. We now remove the assumption that the probability of a wavelength being used on a link is independent of the use of the same wavelength on other links. However, 472 WDM NETWORK DESIGN we will continue to assume that the events on one wavelength are independent of the events on all other wavelengths. We first consider networks with no wavelength conversion and calculate the probability that a lightpath request that chooses a route with/4 links is blocked. Any lightpath that has already been established and uses one of these H links is termed an interfering lightpath. We assume that an interfering lightpath that uses one of these H links, say, link i, will not use the next link i + 1 with probability Jrl. (So with probability Jrt a lightpath that interferes on link i of the route chosen by the lightpath request leaves after that link.) For any wavelength ~, we also assume that a new lightpath request (one that does not interfere on link i - 1) would interfere on link i of the route chosen by the lightpath request with probability ~r~. This gives us the following conditional probabilities for the use of wavelength )~ on link i: Prob()~ used on link i l)~ not used on link i - 1) = rrn, and Prob()~ used on link il)~ used on link i - 1) = (1 - 7fl) -t 7[lYrn. Note that under the assumption of independent use of the same wavelength on the links, both these conditional probabilities must equal Jr; thus this assumption corresponds to setting Jrl = 1 and Jr = Jrn. Using the same reasoning as that used to derive (8.12), we can show that now Pb,nc (1 (1 7fn)H) W = - - . (8.~8) For networks with full wavelength conversion, the following expression for blocking probability can be derived under a set of assumptions that are similar to that used to derive (8.18): H ( ~y_ (l_~l_~_7.flY.fn)W~iW_l) Pb,fc 1 I- I 1 /=1 1 yrW1 (8.19) where YVi = ~n ~n q-" Y[I ~n YFl 1 - (1 - ( zr z -+- Yr n 7r l Tr n ) ) i ) . For a given blocking probability, we can solve (8.18) and (8.19) for ~nc and Jrfc, respectively. Then we can approximate the conversion gain Jrfc/Jrnc for small blocking probabilities and H >> 1/Jrt by 7Cfc Hl-1/W(~n -ff ~l - Yrl~n). (8.20) 7[nc 8.4 Statistical Dimensioning Models 473 Define the interference length Li 1/rcl. Li is an approximation to the expected number of links that an interfering lightpath uses on the route chosen by a lightpath request. The assumption H >> 1/rrl = Li is thus equivalent to assuming that the number of hops in the path chosen by a lightpath request is much larger than the average number of hops that it shares with an interfering lightpath. This assumption is a good one when the network is well connected, but it is a poorer approximation to the behavior in, say, rings. The conversion gain under the assumption of independent use of a wavelength on each link (rrl = 1) is approximately H 1-1/W. Thus the conversion gain given by (8.20) is lower than this by the factor (rrn + rrl - rrlrrn). This factor is the mixing probability: the probability that at a node along the route chosen by a lightpath request, an interfering lightpath leaves or a new interfering lightpath joins. Thus the conversion gain is more in networks where there is more mixing, for example, in dense mesh networks where the node degrees (switch sizes) are large, as opposed to ring networks where the mixing is small and the interference length is large. In summary, path length is only one of the factors governing the amount of reuse we get by using wavelength conversion; interference length and switch sizes are other important factors. An analysis of WDM ring networks, based on the techniques described above, can be found in [SM00]. Wavelength Assignment and Alternate Routes So far, while studying the RWA problem using a statistical model for the traffic, we have assumed a fixed route between each source-destination pair. We will now present some simulation results to show the effect of using alternate routes. We will also consider two different ways of assigning wavelengths once the route has been selected. Thus we consider the following four RWA algorithms. Random-1. For a lightpath request between two nodes, choose at random one of the available wavelengths on a fixed shortest path between the two nodes. Random-2. Fix two shortest paths between every pair of nodes. For a lightpath re- quest between two nodes, choose at random one of the available wavelengths on the first shortest path between the two nodes. If no such wavelength is available, choose a random one of the available wavelengths on the second shortest path. Max-used-1. For a lightpath request between two nodes, among the available wave- lengths on a fixed shortest path between the two nodes, choose the one that is used the most number of times in the network at that point of time. Max-used-2. Fix two shortest paths between every pair of nodes. For a lightpath request between two nodes, among the available wavelengths on the first shortest 474 WDM NETWORK DESIGN Table 8.3 Reuse factor for 1% block- ing for different RWA algorithms for the 20-node network considered in [RS95]. RWA Algorithm Reuse Factor Random-1 6.9 Random-2 7.8 Max-used-1 7.5 Max-used-2 8.3 path between the two nodes, choose the one that is used the most number of times in the network at that point of time. If no such wavelength is available, among the available wavelengths on the second shortest path between the two nodes, choose the one that is used the most number of times in the network at that point of time. The topology we consider is the 20-node, 39-1ink network from [RS95]. We assume 32 wavelengths are available on each link and that the traffic is uniform (same for every pair of nodes). The reuse factor obtained by using each of the above four RWA algorithms for a blocking probability of 1% is shown in Table 8.3. Observe that the reuse factor improves substantially when an alternate path is considered. Ideally we would like to have more alternate routes for longer routes and less for shorter routes. This will help reduce the blocking probability on longer routes and ensure better fairness overall. Otherwise, short routes tend to have much less blocking than long routes. Having more routes to consider usually increases the control traffic in the network and leads to an additional compuational burden on the network nodes, but this is not significant in networks with a moderate number of nodes where lightpaths are set up and taken down slowly. In addition to the choice of routes, the wavelength assignment algorithm also plays an important role in determining the reuse factor. Note that for the same num- ber of available paths, the max-used algorithms have a distinct advantage over the random algorithms. The intuitive reason for this phenomenon is that the max-used strategy provides a higher likelihood of finding the same free wavelength on all the links along a particular route. A drawback of the max-used algorithm is that it requires a knowledge of the wavelengths in use by all other connections in the network. When the routing and wavelength assignment is performed in a distributed manner, such information typically has to be obtained by means of periodic updates broadcast by each node. This again increases the control traffic load on the network. 8.5 Maximum Load Dimensioning Models 475 8.5 Maximum Load Dimensioning Models As discussed above, from a dimensioning perspective, the fundamental prop- erty that distinguishes wavelength-routing networks from traditional electronic circuit-switched networks is the absence of full wavelength conversion. A number of studies have been undertaken to determine how networks using no, or limited, wave- length conversion should be dimensioned in order to support the same set, or sets, of lightpaths as an optical layer with full conversion. In this section, we will present some of the results obtained in this direction. We assume that both the lightpaths and the network edges are undirected (see Table 8.2). The results can be broadly classified into two categories: offline requests and on- line requests. The offline problem corresponds to a "static" network design problem, where only a single set of lightpaths is to be supported. This set is constrained to be such that it can be supported in a network with nodes capable of full wavelength conversion, with at most L wavelengths per link, since there is a routing that places no more than L routes on any link. Thus, the maximum load of this set of lightpaths is said to be L. In a network with nodes incapable of wavelength conversion, more than L wavelengths per link would be needed, in general, to support the same set of lightpaths. We are interested in determining the additional number of wavelengths that would be required to support every set of such lightpaths, with nodes that do not have any wavelength conversion capability. Online RWA corresponds to the "dynamic" network design case where lightpaths arise one at a time and have to be assigned routes and wavelengths when the request arrives, without waiting for future requests to be known. However, the requests and routing are such that no more than L lightpaths use any link at any given time. Thus a network with fully wavelength-converting crossconnects that provides L wavelengths on each link would be able to support all the requests. In this case, the task is to compare the number of additional wavelengths that would be required to support the same sets of lightpaths with non-wavelength-converting crossconnects. One shortcoming of this maximum load model is that the number of wavelengths required may be excessively large in order to support all sets of lightpaths with max- imum load L. If we are permitted not to support a small fraction of these sets of lightpaths, it may be possible to considerably reduce the number of wavelengths re- quired. In this sense, the maximum load model is a worst-case dimensioning method. 8.5.1 Offline Lightpath Requests In this section, we will survey the results for offline lightpath requests. 476 WDM NETWORK DESIGN L/ ~3 A A ~ A (a) (b) Figure 8.19 (a) A line network with a set of lightpaths, also called an interval graph. (b) Wavelength assignment done by Algorithm 8.3. Theorem 8.1 [ABC+94] Given a routing of a set of lightpaths with load L in a network G with M edges, with the maximum number of hops in a lightpath being D, the number of wavelengths sufficient to satisfy this request is W _ min[(L - 1)D + 1, (2L - 1)~/-M- L + 2]. Proof. Observe that each lightpath can intersect with at most (L - 1)D other lightpaths. Thus the maximum degree of the path graph P(G) is (L - 1)D. Any graph with maximum degree A can be colored using A + 1 colors by a simple greedy coloring algorithm, and hence the path graph can be colored using (L- 1)D + 1 colors. So W _< (L- 1)D + 1. To prove the remainder of the theorem, suppose there are K lightpaths of length >_ ~ hops. The average load due to these lightpaths on an edge is <L M - so that K <_ L~/-M. Assign L~/-M separate wavelengths to these lightpaths. Next consider the lightpaths of length _< ~- 1 hops. Each of these inter- sects with at most (L - 1)(v/-M- 1) other such lightpaths, and so will need at most (L - 1)(x/M- 1) + 1 additional wavelengths. So we have W<_LVr-M+(L-1)(x/M-1)+I=(2L-1)~-L+2, which proves the theorem, m A line network, shown in Figure 8.19, is simply a network of nodes intercon- nected in a line. A sample set of lightpath requests is also shown in the figure. In this case, there is no routing aspect; only the wavelength assignment problem remains. We study this topology because the results will be useful in analyzing ring networks, which are practically important. Our WA-NC problem (see Section 8.2.2) is equivalent to the problem of coloring intervals on a line. The following greedy algorithm accomplishes the coloring using 8.5 Maximum Load Dimensioning Models 477 L wavelengths. The algorithm is greedy in the sense that it never backtracks and changes a color that it has already assigned when assigning a color to a new interval. Algorithm 8.3 [Ber76, Section 16.5] 1. Number the wavelengths from 1 to L. Start with the first lightpath from the left and assign to it wavelength 1. 2. Go to the next lightpath starting from the left and assign to it the least numbered wavelength possible, until all lightpaths are colored. Rings are perhaps the most important specific topology to consider. A ring is the simplest 2-connected topology and has been adopted by numerous standards (FDDI, SONET) as the topology of choice. We expect WDM networks to be first deployed as rings. In a ring, we have two possible routes for each lightpath. Given a set of lightpath requests, there is an algorithm [FNS+92] that does the routing with the minimum possible load Lmin. This algorithm may involve some lightpaths taking the longest route around the ring. A simpler alternative is to use shortest-path routing for light- paths, which, however, yields a higher load, as shown next. Lemma 8.2 [RS97] Suppose we are given a request of source-destination pairs and the minimum possible load for satisfying this request is Lmin. Then shortest-path routing yields a load of at most 2Lmin. Proof. Suppose shortest-path routing yields a load Lsp. Consider a link i with load Lsp. Rerouting k connections using link i on their longer routes on the ring reduces the load on link i by Lsp - k. Note that since all these connections are routed on paths on length _< [N/2J initially, their longer routes on the ring will all use the link [N/2J + i, increasing its load by k. Therefore, the load Lmin of the optimal routing algorithm must satisfy Lmi n > mink max(Lsp - k, k), or Lmin > [Lsp/2]. II It turns out that the joint RWA-NC problem is hard, even in rings. However, we can get good bounds on how many wavelengths are needed. Theorem 8.3 [Tuc75] Given a set of lightpath requests and a routing on a ring with load L, WA-NC can be done with 2L - 1 wavelengths. Proof. Determine the node in the ring with a minimum number 1 of light- paths passing through it (do not count lightpaths starting or terminating at the node). Cut the ring at this point (see Figure 8.20). Now we have an interval 478 WDM NETWORK DESIGN (b) ~2 kl (a) (c) A Figure 8.20 Wavelength assignment in a ring network. (a) A ring network and a set of lightpaths. (b) The ring is cut at a node that has a minimum number of lightpaths passing through it to yield a line network. (c) The lightpaths in the line network are assigned wavelengths according to Algorithm 8.3. The lightpaths going across the cut node are assigned separate additional wavelengths. graph with a maximum load of L, which we can color with L wavelengths, using Algorithm 8.3. However, we still have to deal with the l lightpaths that may wrap around the edge of the line. In the worst case, we can always assign wavelengths to these lightpaths using 1 additional wavelengths, requiring a total of L +l wavelengths. Now with any routing, there is a node in the ring where l < L - 1. To see this, suppose all nodes have at least L paths flowing through them. There exists a node, say, node x, where a path terminates. Let y be the node adjacent to x on this path. Then link xy must have a load of at least L + 1, a contradiction, m It is possible to construct an example of a traffic pattern consisting of 2L - 1 lightpaths, with each pair of lightpaths sharing at least one common link. This implies that all of them have to be assigned different wavelengths regardless of the algorithm used, showing that there are examples for which 2L- 1 wavelengths will be required. However, this is not a scenario that occurs very often. In fact, it has been shown in [Tuc75] that if no three lightpaths in a given traffic pattern cover the entire ring, then 3 L wavelengths are sufficient to perform the wavelength assignment. This is an example where the worst-case nonblocking model results in overdesigning the network. In order to support a few pathological patterns, we end up using approximately ~- additional wavelengths. Let us see what can be gained by having wavelength conversion capabilities in a ring network. Clearly, if we have full conversion capabilities at all the nodes, then we 8.5 Maximum Load Dimensioning Models 479 Figure 8.21 A ring network with fixed-wavelength conversion at one node and no conversion at the others that is able to support lightpath requests with load L < W - 1. One of the nodes is configured to convert wavelength i to wavelength (i + 1) mod W, and the other nodes provide no wavelength conversion. can support all lightpath requests with load L _< W. However, the same result can be achieved by providing much less conversion capabilities, as shown by the following results. Theorem 8.4 [RS97] Consider a ring network that has full wavelength conver- sion at one node and no wavelength conversion at the other nodes. This network can support all lightpath requests with load L <_ W. The proof of this result is left as an exercise (Problem 8.19). Limited-wavelength conversion can help significantly in improving the load that can be supported in many network configurations. The detailed derivations of the results for this case are beyond the scope of this book. We summarize the key results here. Theorem 8.5 [RS97] Consider the ring network shown in Figure 8.21, which has fixed-wavelength conversion at one node where wavelength i is converted to wavelength (i + 1) mod W, and no wavelength conversion at the other nodes. This network can support all lightpath requests with load L <_ W - 1. By having d 2 limited conversion at two nodes and no conversion at the others, it is possible to improve this result to L _< W [RS97], making such a network as good as a network with full wavelength conversion at each node. . such wavelength is available, choose a random one of the available wavelengths on the second shortest path. Max-used-1. For a lightpath request between two nodes, among the available wave- lengths. Wavelength Assignment and Alternate Routes So far, while studying the RWA problem using a statistical model for the traffic, we have assumed a fixed route between each source-destination pair of finding the same free wavelength on all the links along a particular route. A drawback of the max-used algorithm is that it requires a knowledge of the wavelengths in use by all other connections