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4 the same one) is free on each of the links required by the call. Consequently, wavelength allocation is not an issue under circuit-switching. In our study, we have used a number of dierent trac load patterns to compare the four wavelength allocation policies against each other and against circuit-switching. These patterns are representative of the wide range of loading situations that one expects to encounter in practice, and can be found in [17]. To ensure that the results are comparable across the dierent patterns, the load values were chosen so that the total load is the same for all patterns. 2.1 Policy Comparison for a Single Path of a Network We have shown in [15] that the evolution of a 2-hop path with random wavelength allocation can be characterized by the Markov process (n ;n ;n ;f ). The rst three random variables in the state description provide the number of active calls between the three source-destination pairs in the path, and the last random variable gives the number of wavelengths that are free on both links of the path. The state transition diagram of this Markov process is shown in Figure 1 for W = 2wavelengths, and it is straightforward to see that the process is not timereversible [15]. By modifying a few of the transition rates of this process, we were able to derive a time-reversible Markov process with the same state space, which has a product-form solution. We have demonstrated in [15] that the blocking probabilities obtained through the productform solution to the time-reversible Markov process are very close to the blocking probabilities obtained through the numerical solution to the original Markov process for a wide range of trac loads. Let us now consider the same 2-hop path with the most-used wavelength allocation policy. This policy can be modeled as a Markov process with the same state description as the random policy case, i.e., (n ;n ;n ;f ). The key dierence is that, under the most-used policy, if n >n, then we know that there is at least one wavelength that is used on hop 1 but not used on hop 2. Thus, an incoming call that uses the second hop only will be assigned a wavelength that is already in use on the rst hop, and will cause a transition to state (n ;n ;n + 1;f ); similarly for n >n and incoming calls using only the rst hop. (Under the random wavelength allocation policy, the transition could be to either state (n ;n ;n +1;f )orto state (n ;n ;n +1;f, 1) if the numberoffreewavelengths on both hops f > 0 and one of these wavelengths is assigned to the call.) The state transition diagram of the Markov process for the most-used allocation policy is shown in Figure 2 for a 2-hop path with W = 2 wavelengths. Again, it is straightforward to verify that this Markov process is not time-reversible. Comparing to Figure 1, we note that despite having the same state space, the two processes dier in two ways. First, some of the transition rates are dierent; for instance the transition rate from state (0,0,1,1) to state (1,0,1,1) is equal to =2 for the random allocation, but for the most-used allocation. Second, some of the transitions are missing in the new Markov process. For example, there is a transition from state (0,0,1,1) to state (1,0,1,0) under random allocation in Figure 1, but there is no such transition in Figure 2. Furthermore, since there is a transition from state (1,0,1,0) to state (0,0,1,1) in Figure 2, but no transition in the reverse direction, it is not possible to obtain an approximate time-reversible process by simply modifying some of the transition rates, as we did for the random policy. Although we donothave an approximate product-form solution for the most-used allocation policy, the state space for a 2-hop path is small enough so that the solution to the Markov process can be obtained numerically. Based on similar arguments, it can be determined that the least-used wavelength allocation policy can also be modeled by amarkov process with the state description (n ;n ;n ;f ). The state transition diagram for this process is shown in Figure 3, and it can be easily veried that the process is not time-reversible. If a converter is placed at node 1 of a 2-hop path (the only interesting possibility inthis case), the system becomes equivalent to a 2-hop circuit-switched path, and it can be described by

5 1,1,0,0 1,0,0,1 /2 2 1,1,1,0 1,0,1,1 0,1,1,0 /2 0,0,1,1 2 2,0,0,0 2,0,1,0 1,0,2,0 0,0,2, ,0,2,0 2 0,2,0,0 0,1,0,1 0,0,0,2 n = 2 n = 1 n = 0 1,1,0,0 1,0,0,1 2 1,1,1,0 1,0,1,1 0,1,1,0 0,0,1,1 2 2,0,0,0 2,0,1,0 1,0,2,0 0,0,2, ,2,0,0 0,1,0,1 0,0,0,2 n = 2 n = 1 n = 0 1,0,1,0 /2 2,0,2,0 1,0,1,0 /2 Figure 1: State space (n ;n ;n ;f ) of a 2-hop path with W = 2 wavelengths (random allocation) Figure 2: State space (n ;n ;n ;f ) of a 2-hop path with W = 2 wavelengths (mostused allocation) the three-dimensional Markov process (n ;n ;n ). Random variable f becomes redundant because calls using both hops can now use any of the (W, n, n ) available wavelengths on the second hop. It is well-known that this process has a closed-form solution. In Figure 4 we show the state space of a 2-hop circuit switched path with two wavelengths. Although this path is described by the above 3-dimensional process, we include in the state description of Figure 4 the variable f to make it easier to compare to Figures 1{3. For instance, the fact that there are no transitions into state (1,0,1,0) in the gure can be explained by recalling that f = 0 (i.e., that no wavelength is free on both links of the path) implies that calls traversing both hops are blocked. However, since exactly one wavelength is free on each hop (even if it is not the same one), calls using both hops cannot be blocked in the circuit-switched path, and the system will never enter state (1,0,1,0), only state (1,0,1,1). The rst-t wavelength allocation policy can also be modeled as a Markov process, but the size of its state space is in the order of W 5, too large to obtain a numerical solution even for relatively small values of W. In view of this, the blocking probabilities for this policy are obtained by simulation only Numerical Comparisons Let us rst consider the blocking probabilities of the random, most-used, least-used, and circuitswitched systems for calls traversing both links of the 2-hop path. In Figures 1 to 4, the blocking states for these calls are those with f = 0, i.e., those states in which neither of the two wavelengths is free on both links. We also observe that, except for state (1,0,1,0) at the bottom of each of the four gures, the transitions (and transition rates) in and out of all other blocking states are exactly the same for all four cases. Consequently, we expect that the dierence in the blocking probability experienced by calls traversing both links of the path under the dierent policies will be mainly due to the steady-state probability of blocking state (1,0,1,0). Referring to Figure 4, we note that the corresponding Markov process never enters state (1,0,1,0). Thus, we expect that calls traversing both hops will experience the least blocking probability ina circuit-switched path. In Figure 2 (most-used policy) we note that there are two transitions into state (1,0,1,0), and four transitions out of it. The blocking probability will be higher under this policy compared to the circuit-switched case. The Markov process in Figure 1 (random policy) has two additional transitions into state (1,0,1,0) from states (0,0,1,1) and (1,0,0,1) with rates =2 and =2, respectively. Therefore, the blocking probability of

6 1,1,0,0 1,0,0,1 2 1,1,1,0 1,0,1,1 0,1,1,0 0,0,1,1 2 2,0,0,0 2,0,1,0 1,0,2,0 0,0,2, ,0,2,0 2 0,2,0,0 0,1,0,1 0,0,0,2 n = 2 n = 1 n = 0 1,1,0,0 1,0,0,1 2 1,1,1,0 1,0,1,1 0,1,1,0 0,0,1,1 2 2,0,0,0 2,0,1,0 1,0,2,0 0,0,2, ,2,0,0 0,1,0,1 0,0,0,2 n = 2 n = 1 n = 0 2,0,2,0 1,0,1,0 1,0,1,0 Figure 3: State space (n ;n ;n ;f ) of a 2-hop path with W = 2 wavelengths (leastused allocation) Figure 4: State space (n ;n ;n ) of a 2- hop path with W = 2 wavelengths (circuitswitched) these calls under the random policy will be higher than under the most-used policy. Finally, the Markov process in Figure 3 (least-used policy) has the same transitions as the one in Figure 1, but the transition rates into state (1,0,1,0) from states (0,0,1,1) and (1,0,0,1) are and, respectively. Therefore, we expect that these calls will experience the highest blocking probability under the least-used policy. We now note that the lower the blocking probability for calls traversing both hops, the larger the number of such calls accepted, and the larger the number of wavelengths they occupy, thus leaving fewer wavelengths available for calls using a single link (either the rst or the second) of the path. Hence, we expect that the behavior of the four policies in terms of the blocking probability of calls using a single link of the path will be exactly the opposite of what was discussed above. Specically, we expect the least-used policy to provide the lowest blocking probability for these calls, followed by the random, the most-used, and the circuit-switched policies, in that order. The above conclusions, derived by direct comparison of the states of the Markov processes, are in agreement with intuition. We have conrmed these conclusions by numerically comparing the blocking probabilities of the various policies for 8 dierent load values. Figures 5 and 6 show results for two cases corresponding to a uniform and descending load pattern, respectively, and for W = 10 wavelengths. More specically, the arrival rates used to obtain the results in Figure 5 were = 0:2; = 0:1; = 0:2, while for the results in Figure 6 we used =3:0; =2:0; =2:0. In both gures we plot the blocking probability forthe three types of calls, namely, calls using the rst hop only (label \hop 1" in the x-axis of the gures), calls using the second hop only (label \hop 2"), and calls using both hops (label \both hops"). We rst note that the results are aected by the trac pattern used. For instance, under uniform loading (Figure 5), calls using the rst hop only experience the same blocking probability as hops using the second hop only, while in the descending pattern (Figure 6), due to the lower load oered to the second hop, the latter calls experience a much lower blocking probability for all four policies. More importantly, the relative values of the blocking probabilities for the four policies are also consistent with our discussion above. Very similar results have been obtained for all 8 dierent load values. In Figure 7 we compare the most-used and rst-t policies for the same arrival rates as those used for Figure 6. We observe that the blocking probabilities of the rst-t policy are almost identical to those of the most-used policy for all three types of calls. This result can be explained by noting that both policies attempt to maximize the number of wavelengths that

7 circuit-switched most-used random least-used circuit-switched most-used random least-used hop 1 hop 2 both hops Hops used by the calls hop 1 hop 2 both hops Hops used by the calls Figure 5: Policy comparison, 2-hop path, uniform trac pattern Figure 6: Policy comparison, 2-hop path, descending trac pattern most-used random circuit-switched first fit 1e-05 hop 1 hop 2 both hops Hops used by the calls 1e Figure 7: Most-used vs. rst-t allocation, 2- hop path, descending trac pattern Figure 8: Policy comparison, 10-hop path, inverted bowl trac pattern are available for calls that use both hops of the 2-hop path by reducing the \fragmentation" of the set of wavelengths. The most-used policy assigns to an incoming call that requires a single hop of the path a wavelength that is already used on the other hop, if such awavelength exists. On the other hand, the rst-t policy attempts to achieve the same goal by searching the set of wavelengths in a xed order, thus increasing the chances that a wavelength used on a single hop will be assigned to an incoming call using the other hop. As can be seen from Figure 7, the most-used policy is slightly better, but overall the blocking probability values of the two policies are very close. Similar results have been obtained for all 8 trac loads. The relativebehavior of the four policies for longer paths is very similar to shown in Figures 5 and 7 for a wide range of trac patterns. Specically, for calls using one or two hops of a path only, the least-used policy provides the lowest blocking probability, followed by the random policy, the most-used policy, and the circuit-switched case. However, for calls traversing three or more hops of the path, the situation is reversed. Due to space limitations, results for paths longer than two hops are omitted, but can be found in [17]. Since, under the least-used policy, the blocking probability of calls using multiple hops increases signicantly, we will not consider the least-used policy any further. Finally, in Figure 8 we compare the rst-t policy to the random (no converters) and circuitswitched cases for a 10-hop path (more results for 10-hop paths can be found in [17]). Two interesting observations can be made. First, the blocking probability values of the rst-t policy are always between the corresponding values of the random and circuit-switched cases. In other words, the blocking probability values under the random and circuit-switched cases

8 1 most-used Figure 9: The NSFNET topology 8 1e Figure 10: Most-used vs. rst-t allocation, NSFNET, pattern based on actual trac provide lower and upper bounds for the blocking performance of the rst-t policy. Second, the rst-t policy is quite eective in reducing the blocking probability of calls traveling over multiple hops (which are the ones that experience the highest blocking probability under the random policy) close to the level of the circuit-switched case. Very similar results have been obtained for the other trac patterns [17]. 3 Mesh Wavelength Routing Networks In this section we consider the NSFNET irregular topology in Figure 9. Results for other topologies can be found in [17]. Since we use the trac data reported in [4], following that study, we have augmented the 14-node NSFNET topology with nodes 1 and 16 in Figure 9. We present detailed results for the blocking probabilities of calls involving nodes along the path (3,5,6,7,9,,15,16). There are 28 source-destination pairs in this path, and in Figures 10 to 14 they have been labeled so that numbers 1 to 7 refer to pairs with one-hop paths, numbers 8 to 15 correspond to pairs with two-hop paths, etc. Wehave used two trac patterns. For the rst pattern, the call arrival rates are sd =0:6,l, where l is the length of the shortest path from s to d. The second trac pattern was designed to reect actual trac statistics collected on the NSFNET backbone network [4]. Clearly, this data, collected over a packet-switched network, cannot be directly applied to a circuit-switched wavelength routing network. However, our intention is simply to capture the relative trac demands among the dierent source-destination pairs. To this end, we rst divided the entries of the matrix in [4] by the link capacity to obtain the \oered load" sd per source-destination pair. Since the resulting values were too small, we multiplied them by a constant to obtain reasonable values for the oered load. Then, assuming that all calls have a mean holding time 1= = 1, the oered load values become the arrival rates sd used in the experiments. Figure 10 compares the rst-t to the most-used policies, and we again see that that the two policies result in almost identical blocking probabilityvalues for all calls. Figures and demonstrate that the random and circuit-switched cases provide upper and lower bounds on the performance of the rst-t policy, similar to the single-path cases studied above. Finally, in Figures 13 and 14 we compare the rst-t policy to the random policy with converters. The converters were placed in the network using the optimization techniques in [16]. As can be seen, using the rst-t policy is roughly equivalent to employing a signicant number of converters in the network. The overall behavior of the graphs in these gures is very similar to the single path case as well as other topologies (see [17]), indicating that our observations and conclusions are valid for a wide range of topologies and trac patterns.

9 1 1 random circuit-switched random circuit-switched e Figure : Policy comparison, NSFNET, traf- c pattern based on locality Figure : Policy comparison, NSFNET, pattern based on actual trac random with 2 converters random with 5 converters random with 4 converters random with 8 converters Figure 13: First-t policy vs. random policy with converters, NSFNET, trac pattern based on locality Figure 14: First-t policy vs. random policy with converters, NSFNET, pattern based on actual trac 4 Concluding Remarks We have shown that the most-used and rst-t policies have very similar call blocking probabilities for all calls in a network, regardless of the number of hops used by the calls. The two policies tend to favor calls using multiple paths at the expense of calls using a single path. This is a desirable feature, since calls traversing multiple paths experience the highest blocking probability. However, the most-used policy requires that the network nodes exchange information about the network-wide usage of wavelengths, while the rst-t policy only relies on a xed ordering of wavelengths, and is signicantly easier to implement. We have also shown that the random policy and the circuit-switched case provide bounds on the call blocking probability under the rst-t (or most-used) policy. Specically, for calls using one or two hops, the random policy provides a lower bound and the circuit-switched case provides an upper bound, while for calls using longer paths the bounds are reversed. We have presented results which indicate that the call blocking probabilities under the rst- t policy are similar to those under the random policy but employing a number of converters in the network. In most cases, introducing the rst-t policy results in a decrease in the blocking probability of calls traveling over multiple hops to a level very close to the blocking probability experienced under the circuit-switched case. Since, in terms of implementation, there is no signicant dierence between the rst-t and random policies, the gains obtained by employing expensive hardware can be realized by making more intelligent choices in software.

10 It also appears that the benets of the rst-t policy diminish at high loads. It is in these situations that employing converters would benet calls traversing a large number of hops. However, the number of converters to be employed in this case must be very large, close to the number of nodes in the network, and even if all nodes contain converters the blocking probability will remain at (reduced but) high levels. Since it is unlikely that future wavelength routing networks will be designed to operate at such high call blocking probabilities, reducing the call blocking probabilities in this case may not be of practical importance. While previous studies of \sparse" wavelength conversion have measured the improvement obtained by employing converters in conjunction only with the random wavelength allocation policy, we have shown that an equivalent improvement can be achieved merely by using appropriate allocation policies such as rst-t or most-used. References [1] R. A. Barry and P. A. Humblet. Models of blocking probability in all-optical networks with and without wavelength changers. IEEE J-SAC, 14(5):858{867, June [2] A. Birman. Computing approximate blocking probabilities for a class of all-optical networks. IEEE J-SAC, 14(5):852{857, June [3] I. Chlamtac, A. Ganz, and G. Karmi. Lightpath communications: An approach to high bandwidth optical WANS. IEEE Trans. Commun., 40(7):71{82, July [4] B. Mukherjee et al. Some principles for designing a wide-area WDM optical network. IEEE/ACM Trans. Networking, 4(5):684{696, October [5] A. Girard. Routing and Dimensioning in Circuit-Switched Networks. Addison Wesley, Reading, MA, [6] H. Harai, M. Murata, and H. Miyahara. Performance of alternate routing methods in all-optical switching networks. Proc. INFOCOM '97, pp. 517{525, April [7] A. Hill, A. Salek, and K. Sato (Eds.). Special issue on high-capacity optical transport networks. IEEE J-SAC, 16(7), Sep [8] E. Karasan and E. Ayanoglu. Eects of wavelength routing and selection algorithms on wavelength conversion gain in wdm optical networks. IEEE/ACM Trans. Networking, 6(2):186{196, April [9] M. Kovacevic and A. Acampora. Benets of wavelength translation in all-optical clearchannel networks. IEEE J-SAC, 14(5):868{880, June [10] A. Mokhtar and M. Azizoglu. Adaptive wavelength routing in all-optical netowrks. IEEE/ACM Trans. Networking, 6(2):197{206, April [] B. Ramamurty and B. Mukherjee. Wavelength conversion in WDM networking. IEEE J-SAC, 16(7):1061{1073, Sep [] S. Subramaniam, M. Azizoglu, and A. Somani. All-optical networks with sparse wavelength conversion. IEEE/ACM Trans. Networking, 4(4):544{557, Aug [13] S. Subramaniam, M. Azizoglu, and A. K. Somani. On the optimal placement ofwavelength converters in wavelength-routed networks. Proc. INFOCOM '98, pp. 902{909, Apr [14] S. Subramanian, A. K. Somani, M. Azizoglu, and R. A. Barry. A performance model for wavelength conversion with non-poisson trac. Proc. INFOCOM '97, pp. 500{507, Apr [15] Y. Zhu, G. N. Rouskas, and H. G. Perros. Blocking in wavelength routing networks, Part I: The single path case. Proc. INFOCOM '99, pp. 321{328, March [16] Y. Zhu, G. N. Rouskas, and H. G. Perros. Blocking in wavelength routing networks, Part II: Mesh topologies. Proc. ITC 16, pp. 1321{1330, June [17] Y. Zhu, G. N. Rouskas, andh. G. Perros. Bounds on the blocking performance of allocation policies in wavelength routing networks and a study of the eects of converters. Tech. Rep. TR-99-01, NCSU, Raleigh, NC, Jan

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