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Statistics study of routing and wavelength assignment algorithms in WDM all optical network
15 November 2000
Optics Communications 185 (2000) 315±320
www.elsevier.com/locate/optcom
Statistics study of routing and wavelength assignment algorithms in WDM all optical network Yabin Ye *, Hanyi Zhang, Tao Qin, Wuju Dai, Feifei Feng, Xiaoli Huo, Yili Guo Department of Electronics Engineering, Tsinghua University, Beijing 100084, People's Republic of China Received 24 May 2000; received in revised form 21 September 2000; accepted 25 September 2000
Abstract In this paper, routing and wavelength assignment (RWA) algorithms whose optimum object is the minimum number of wavelength required in wavelength division multiplexing all optical network were studied. Since the number of wavelength required (NWR) obtained by the existing two RWA algorithms is associated with the numbering order of the nodes in the network, a statistics method was proposed to modify the RWA algorithms. With this statistics method, a smaller NWR can be acquired. And, it is the ®rst time that the statistics method was proposed to compare the performance of dierent RWA algorithms. Two RWA algorithms can be compared by contrasting the distribution of the NWRs obtained by each RWA algorithms. Ó 2000 Elsevier Science B.V. All rights reserved. Keywords: WDM all optical network; Routing and wavelength assignment algorithm; Statistics method; Minimum number of wavelength required
1. Introduction In wavelength division multiplexing (WDM) all optical network, the connections between node pairs are established by wavelength channels. For this reason wavelength is the most important resource in networks. Because of the limitation of the gain band of Er doped ®ber ampli®er and the nonlinear eects in the ®ber, the number of wavelength in the ®ber is limited. And, increasing the number of wavelength will also increase costs of many components and the complexity of network management. Therefore it is an important * Corresponding author. Tel.: +86-10-62781358; fax: +86-1062770317. E-mail addresses: [email protected], [email protected], [email protected] (Y. Ye).
issue to minimize the number of wavelength required (NWR) by using appropriate routing and wavelength assignment (RWA) algorithms [1±3]. However, it is proved that the problem of RWA in WDM all optical network is a NP-complete problem [2], it can only be solved by heuristic algorithms. RWA problem can be divided into two subproblems, i.e. routing problem and wavelength assignment problem [1,2]. In Ref. [1], the source± destination node pairs are assigned the ®rst found minimum number of hops (MNHs) paths ®rstly. Because the initial paths may not be the optimal ones, some paths need to be rerouted. The reroute rule is: for each node pair, an alternative MNHs path substitutes the one previously assigned when and only when the number of channels (congestion) of the most loaded link in the alternative path
0030-4018/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 0 ) 0 1 0 1 7 - 8
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Y. Ye et al. / Optics Communications 185 (2000) 315±320
is lower than the congestion of the most loaded link in the previously assigned path. The process is repeated until no further improvement can be made in the performance of the network. The longest paths are assigned wavelength ®rst (it is called the longest priority strategy here), and the strategy to assign wavelength follows the ®rst-®t rule. In Ref. [2], the link weight is considered. If a link is assigned a path, then the link weight will increase. The source±destination node pairs are assigned the minimum link weight path. Then the initial paths are rerouted. Its wavelength assignment strategy is equivalent to randomly selecting the path to assign wavelength (it is called the random strategy here). After RWA, a NWR is gotten. However, for a network, if the node numbering order is dierent, the NWR may be dierent in using these two RWA algorithms. So the NWR obtained by the RWA algorithms under only once numbering may not be the true minimum number of wavelength required (MNWR). A method to solve this problem is to calculate the NWR for the entire possible node numbering, and the minimum value of the NWRs is the true MNWR. However, in a network with N nodes, there exist N! dierent ways to rank the nodes, so it is impossible to compute all kinds of statue in acceptable time. Aiming at solving this problem, we propose a statistics method: for a network, all the nodes are randomly ranked ®rst and a NWR is acquired by the RWA algorithm. Then the nodes of the network are randomly ranked again, and another NWR is acquired. The process is repeated for hundreds of times, then a distribution of NWRs is obtained, and the minimum value of NWRs is the MNWR. From the later numerical simulation, compared with the NWR of once numbering, smaller MNWR can be got by using the statistics method. It is the ®rst time that the statistics method is adopted to compare the performance of dierent RWA algorithms. Usually, the lower bound of wavelength is used as the benchmark, against which the performance of various heuristic RWA algorithms can be compared [1,2]. However, besides this, we should also consider the probability of MNWR that appears and the distribution of
the NWRs by using dierent RWA algorithms. For example, if the MNWRs obtained by two RWA algorithms are the same, but the probability of the MNWR acquired by an algorithm is higher or the distribution of the NWR is more concentrated compared with another algorithm, then the former is better. This paper is arranged as follows: in Section 2, the network topology model is represented. Then we use the statistics method to numerically simulate four real networks in Section 3. The conclusion is given in Section 4. 2. Network model The network topology model we studied is based on the following assumptions: 1. The network physical topology consists of N nodes and L bidirectional ®bers. Each node consists of two parts: the end node and the routing node. The end nodes emit and eliminate the lightpaths, while the routing node routes lightpaths from source to destinations. 2. There is a bidirectional ®ber or two unidirectional ®bers between two directly connected nodes, and this is called a physical link. 3. The communication between two nodes occupies one and only one wavelength channel. 4. Static wavelength routing. 5. Uniform trac. There is a bidirectional call request between arbitrary node pairs, so the total trac of the network is N N ÿ 1 lightpaths. 6. Wavelength conversion is not included, so a connection requires a common idle wavelength channel on all the links that it routes through.
3. Numerical simulation Four real networks are numerically simulated as follows: ARPANet, UKNet, EON, and NSFNet. The network topologies of the four networks are shown in Fig. 1. Their lower bounds of wavelength kL are 33, 19, 18, and 13 respectively. The above four networks were numerically simulated using statistics method, and the routing
Y. Ye et al. / Optics Communications 185 (2000) 315±320
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Fig. 1. Four real networks used for numerical simulation, (a) ARPANet, N 20, kL 33; (b) UKNet, N 21, L 39, kL 19; (c) EON, N 20, L 39, kL 18; (d) NSFNet, N 14, L 21, kL 13. Table 1 The MNWR in Ref. [1] and in this paper ARPANet UKNet EON NSFNet
The lower bound
Ref. [1]
This paper
33 19 18 13
33 22 18 13
33 20 18 13
(it is called routing method 1 here) and wavelength assignment algorithms were adopted from Ref. [1]. The calculation was repeated 400 times. The MNWRs obtained by statistics method were compared with those given in Ref. [1] in Table 1. From Table 1 we can see that the MNWRs of ARPNet, UKNet, NSFNet obtained by statistics method are equal to those in Ref. [1], but the MNWR of UKNet is 20, which is smaller than 22 in Ref. [1]. So it is proved that the statistics method is eective, and the MNWR closer to lower bound of wavelength can be acquired by the statistics method. The distribution of the NWRs obtained by RWA algorithm from Ref. [1] is shown in Fig. 2(a). It can be seen from this ®gure that if the nodes are in a dierent order, the NWRs will accordingly be dierent. In ARPANet, EON and NSFNet, the probability of the MNWRs, which equal the lower bounds of wavelength, are very high, about 95.4%, 10% and 65.2% respectively. So in Ref. [1] the MNWRs of ARPANet, EON and NSFNet, which equal lower bounds, can be acquired only by one calculation. But in UKNet, the
probability of the MNWR 20 is very low, only about 1%, while the probability of NWR 22 is high, about 40.9%. That is why in Ref. [1] the lower NWR of UKNet cannot be obtained. This also indicates that the lower MNWR can be acquired by statistics method. After the paths are determined, only the longest priority wavelength assignment strategy is considered in Ref. [1]. In this paper, the random and shortest priority (where the shortest paths are assigned wavelength ®rst) wavelength assignment strategies were also studied. The distribution of NWRs obtained by the routing method 1 combined with random and shortest priority wavelength assignment strategies are shown in Fig. 2(b) and (c) respectively. The MNWRs of the four networks obtained by statistics method are compared in Table 2. It can be seen from this table that the smallest MNWR can be acquired for the longest priority wavelength assignment strategy, or even the MNWR is equal, the probability of acquiring the MNWR is the highest by this strategy, which can be seen from Fig. 2. So the longest priority strategy has the best performance and the performance of the shortest priority is the worst. This is because it is more dicult to ®nd a common idle wavelength on all the links for longer paths, if the shorter paths were assigned wavelength ®rst. For the purpose of comparison, a routing method from Ref. [2] (it is called routing method 2 here) combined with the three kinds of wavelength assignment strategies was also studied. The distributions of NWRs of the four networks are
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Y. Ye et al. / Optics Communications 185 (2000) 315±320
Fig. 2. The distribution of the NWRs of the four networks obtained by the routing method 1 for three kinds of wavelength assignment strategies. (a) The distribution of the NWRs obtained for the longest priority wavelength assignment strategy by routing method 1. (b) The distribution of the NWRs obtained for the random wavelength assignment strategy by routing method 1. (c) The distribution of the NWRs obtained for the shortest priority wavelength assignment strategy by routing method 1.
Table 2 The MNWRs obtained by the routing method 1 for the three kinds of wavelength assignment strategies ARPANet UKNet EON NSFNet
Longest priority
Random
Shortest priority
33 20 18 13
33 22 20 13
34 24 22 13
shown in Fig. 3, which are acquired by 400 times of calculation. The MNWRs of the four networks obtained by the two routing methods for three kinds of wavelength assignment strategies are compared in Table 3. From this table, we can see that under dierent wavelength assignment strategies, the
MNWRs obtained by the two routing methods are almost the same, therefore it is dicult to judge which routing method is better. But if the distribution of the NWRs obtained by the two routing methods after 400 times of calculation is considered, the dierence of the two routing methods appears. As we can see from Table 4, it is obvious that the distribution of the NWRs obtained by the routing method 2 is more concentrated, i.e., the NWRs are closer to MNWR. So the routing method 2 is better than the routing method 1. This is because the link weight is considered in the routing method 2, so the probability that the next path routes through the links with large weight will decrease, and the network trac will be more evenly distributed in the network.
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Fig. 3. The distribution of the NWRs of the four networks obtained by the routing method 2 for three kinds of wavelength assignment strategies. (a) The distribution of the NWRs obtained for the longest priority wavelength assignment strategy by routing method 2. (b) The distribution of the NWRs obtained for the random wavelength assignment strategy for routing method 2. (c) The distribution of the NWRs obtained for the shortest priority wavelength assignment strategy by routing method 2. Table 3 The MNWRs of the four networks obtained by the two routing methods for the three kinds of wavelength assignment strategies Longest priority ARPANet UKNet EON NSFNet
Random
Shortest priority
Method 1
Method 2
Method 1
Method 2
Method 1
Method 2
33 20 18 13
33 21 18 13
33 22 20 13
33 22 19 13
34 24 22 13
35 24 21 14
The computation times by dierent schemes are given in Table 5, which are obtained through 400 times of calculation in ARPANet. As we can see from this table, the computation time is mainly determined by routing scheme and the wavelength assignment strategy aects the computation time weakly. Using routing method 2 can save much time compared with routing method 1. This also
bene®ts from the link weight considered in routing method 2. 4. Conclusion In this paper, RWA algorithms whose optimal objective is the MNWR were studied. The
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Table 4 The distribution of the NWRs of the four networks obtained by the two routing methods for the three kinds of wavelength assignment strategies Longest priority ARPANet UKNet EON NSFNet
Random Method 2
Method 1
Method 2
Method 1
Method 2
33±34 20±24 18±23 13±15
33±34 21±23 18±21 13±15
33±39 22±29 20±26 13±17
33±38 22±27 19±24 13±17
34±40 24±30 22±26 13±18
35±39 24±27 21±24 14±17
Table 5 The comparison of computation time by dierent schemes obtained through 400 times of calculation in ARPANet
Routing method 1 Routing method 2
Shortest priority
Method 1
Longest priority
Random
Shortest priority
131 min
132 min
135 min
52 min
53.5 min
53 min
equal, performance of the RWA algorithms can still be compared by observing the distribution of the NWRs. If the probability of the MNWR acquired by an algorithm is higher or the distribution of the NWR is more concentrated compared with another algorithm, then the former is better.
Acknowledgements MNWRs of networks are obtained by the RWA algorithms associated with the numbering order of the nodes in networks. When the numbering order of the nodes is dierent, the network NWR obtained by the RWA algorithms is dierent. So a statistics method is proposed to solve the problem. It is proved that lower MNWR can be acquired by this statistics method through numerical calculation. This method can also be applied to other RWA algorithms whose simulation results are affected by the numbering order of the nodes. For the ®rst time RWA algorithms are compared by statistics method. When the MNWRs acquired by two dierent RWA algorithms are
This work is supported by the National Natural Science Foundation of China (69990540) and the 863 High Technology Research Development Program.
References [1] S. Baroni, P. Bayvel, J. Lightwave Technol. 15 (1997) 242± 251. [2] N. Nagatsu, Y. Hamazumi, K. Sato, IEICE Trans. Commun. E78-B (1995) 597±607. [3] W. Chen, H. Zhang, Z. Pan, et al., J. Tsinghua Univ. (Sci. & Technol.) 38 (1998) 43±46 (in Chinese).
Optics Communications 185 (2000) 315±320
www.elsevier.com/locate/optcom
Statistics study of routing and wavelength assignment algorithms in WDM all optical network Yabin Ye *, Hanyi Zhang, Tao Qin, Wuju Dai, Feifei Feng, Xiaoli Huo, Yili Guo Department of Electronics Engineering, Tsinghua University, Beijing 100084, People's Republic of China Received 24 May 2000; received in revised form 21 September 2000; accepted 25 September 2000
Abstract In this paper, routing and wavelength assignment (RWA) algorithms whose optimum object is the minimum number of wavelength required in wavelength division multiplexing all optical network were studied. Since the number of wavelength required (NWR) obtained by the existing two RWA algorithms is associated with the numbering order of the nodes in the network, a statistics method was proposed to modify the RWA algorithms. With this statistics method, a smaller NWR can be acquired. And, it is the ®rst time that the statistics method was proposed to compare the performance of dierent RWA algorithms. Two RWA algorithms can be compared by contrasting the distribution of the NWRs obtained by each RWA algorithms. Ó 2000 Elsevier Science B.V. All rights reserved. Keywords: WDM all optical network; Routing and wavelength assignment algorithm; Statistics method; Minimum number of wavelength required
1. Introduction In wavelength division multiplexing (WDM) all optical network, the connections between node pairs are established by wavelength channels. For this reason wavelength is the most important resource in networks. Because of the limitation of the gain band of Er doped ®ber ampli®er and the nonlinear eects in the ®ber, the number of wavelength in the ®ber is limited. And, increasing the number of wavelength will also increase costs of many components and the complexity of network management. Therefore it is an important * Corresponding author. Tel.: +86-10-62781358; fax: +86-1062770317. E-mail addresses: [email protected], [email protected], [email protected] (Y. Ye).
issue to minimize the number of wavelength required (NWR) by using appropriate routing and wavelength assignment (RWA) algorithms [1±3]. However, it is proved that the problem of RWA in WDM all optical network is a NP-complete problem [2], it can only be solved by heuristic algorithms. RWA problem can be divided into two subproblems, i.e. routing problem and wavelength assignment problem [1,2]. In Ref. [1], the source± destination node pairs are assigned the ®rst found minimum number of hops (MNHs) paths ®rstly. Because the initial paths may not be the optimal ones, some paths need to be rerouted. The reroute rule is: for each node pair, an alternative MNHs path substitutes the one previously assigned when and only when the number of channels (congestion) of the most loaded link in the alternative path
0030-4018/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 0 ) 0 1 0 1 7 - 8
316
Y. Ye et al. / Optics Communications 185 (2000) 315±320
is lower than the congestion of the most loaded link in the previously assigned path. The process is repeated until no further improvement can be made in the performance of the network. The longest paths are assigned wavelength ®rst (it is called the longest priority strategy here), and the strategy to assign wavelength follows the ®rst-®t rule. In Ref. [2], the link weight is considered. If a link is assigned a path, then the link weight will increase. The source±destination node pairs are assigned the minimum link weight path. Then the initial paths are rerouted. Its wavelength assignment strategy is equivalent to randomly selecting the path to assign wavelength (it is called the random strategy here). After RWA, a NWR is gotten. However, for a network, if the node numbering order is dierent, the NWR may be dierent in using these two RWA algorithms. So the NWR obtained by the RWA algorithms under only once numbering may not be the true minimum number of wavelength required (MNWR). A method to solve this problem is to calculate the NWR for the entire possible node numbering, and the minimum value of the NWRs is the true MNWR. However, in a network with N nodes, there exist N! dierent ways to rank the nodes, so it is impossible to compute all kinds of statue in acceptable time. Aiming at solving this problem, we propose a statistics method: for a network, all the nodes are randomly ranked ®rst and a NWR is acquired by the RWA algorithm. Then the nodes of the network are randomly ranked again, and another NWR is acquired. The process is repeated for hundreds of times, then a distribution of NWRs is obtained, and the minimum value of NWRs is the MNWR. From the later numerical simulation, compared with the NWR of once numbering, smaller MNWR can be got by using the statistics method. It is the ®rst time that the statistics method is adopted to compare the performance of dierent RWA algorithms. Usually, the lower bound of wavelength is used as the benchmark, against which the performance of various heuristic RWA algorithms can be compared [1,2]. However, besides this, we should also consider the probability of MNWR that appears and the distribution of
the NWRs by using dierent RWA algorithms. For example, if the MNWRs obtained by two RWA algorithms are the same, but the probability of the MNWR acquired by an algorithm is higher or the distribution of the NWR is more concentrated compared with another algorithm, then the former is better. This paper is arranged as follows: in Section 2, the network topology model is represented. Then we use the statistics method to numerically simulate four real networks in Section 3. The conclusion is given in Section 4. 2. Network model The network topology model we studied is based on the following assumptions: 1. The network physical topology consists of N nodes and L bidirectional ®bers. Each node consists of two parts: the end node and the routing node. The end nodes emit and eliminate the lightpaths, while the routing node routes lightpaths from source to destinations. 2. There is a bidirectional ®ber or two unidirectional ®bers between two directly connected nodes, and this is called a physical link. 3. The communication between two nodes occupies one and only one wavelength channel. 4. Static wavelength routing. 5. Uniform trac. There is a bidirectional call request between arbitrary node pairs, so the total trac of the network is N N ÿ 1 lightpaths. 6. Wavelength conversion is not included, so a connection requires a common idle wavelength channel on all the links that it routes through.
3. Numerical simulation Four real networks are numerically simulated as follows: ARPANet, UKNet, EON, and NSFNet. The network topologies of the four networks are shown in Fig. 1. Their lower bounds of wavelength kL are 33, 19, 18, and 13 respectively. The above four networks were numerically simulated using statistics method, and the routing
Y. Ye et al. / Optics Communications 185 (2000) 315±320
317
Fig. 1. Four real networks used for numerical simulation, (a) ARPANet, N 20, kL 33; (b) UKNet, N 21, L 39, kL 19; (c) EON, N 20, L 39, kL 18; (d) NSFNet, N 14, L 21, kL 13. Table 1 The MNWR in Ref. [1] and in this paper ARPANet UKNet EON NSFNet
The lower bound
Ref. [1]
This paper
33 19 18 13
33 22 18 13
33 20 18 13
(it is called routing method 1 here) and wavelength assignment algorithms were adopted from Ref. [1]. The calculation was repeated 400 times. The MNWRs obtained by statistics method were compared with those given in Ref. [1] in Table 1. From Table 1 we can see that the MNWRs of ARPNet, UKNet, NSFNet obtained by statistics method are equal to those in Ref. [1], but the MNWR of UKNet is 20, which is smaller than 22 in Ref. [1]. So it is proved that the statistics method is eective, and the MNWR closer to lower bound of wavelength can be acquired by the statistics method. The distribution of the NWRs obtained by RWA algorithm from Ref. [1] is shown in Fig. 2(a). It can be seen from this ®gure that if the nodes are in a dierent order, the NWRs will accordingly be dierent. In ARPANet, EON and NSFNet, the probability of the MNWRs, which equal the lower bounds of wavelength, are very high, about 95.4%, 10% and 65.2% respectively. So in Ref. [1] the MNWRs of ARPANet, EON and NSFNet, which equal lower bounds, can be acquired only by one calculation. But in UKNet, the
probability of the MNWR 20 is very low, only about 1%, while the probability of NWR 22 is high, about 40.9%. That is why in Ref. [1] the lower NWR of UKNet cannot be obtained. This also indicates that the lower MNWR can be acquired by statistics method. After the paths are determined, only the longest priority wavelength assignment strategy is considered in Ref. [1]. In this paper, the random and shortest priority (where the shortest paths are assigned wavelength ®rst) wavelength assignment strategies were also studied. The distribution of NWRs obtained by the routing method 1 combined with random and shortest priority wavelength assignment strategies are shown in Fig. 2(b) and (c) respectively. The MNWRs of the four networks obtained by statistics method are compared in Table 2. It can be seen from this table that the smallest MNWR can be acquired for the longest priority wavelength assignment strategy, or even the MNWR is equal, the probability of acquiring the MNWR is the highest by this strategy, which can be seen from Fig. 2. So the longest priority strategy has the best performance and the performance of the shortest priority is the worst. This is because it is more dicult to ®nd a common idle wavelength on all the links for longer paths, if the shorter paths were assigned wavelength ®rst. For the purpose of comparison, a routing method from Ref. [2] (it is called routing method 2 here) combined with the three kinds of wavelength assignment strategies was also studied. The distributions of NWRs of the four networks are
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Y. Ye et al. / Optics Communications 185 (2000) 315±320
Fig. 2. The distribution of the NWRs of the four networks obtained by the routing method 1 for three kinds of wavelength assignment strategies. (a) The distribution of the NWRs obtained for the longest priority wavelength assignment strategy by routing method 1. (b) The distribution of the NWRs obtained for the random wavelength assignment strategy by routing method 1. (c) The distribution of the NWRs obtained for the shortest priority wavelength assignment strategy by routing method 1.
Table 2 The MNWRs obtained by the routing method 1 for the three kinds of wavelength assignment strategies ARPANet UKNet EON NSFNet
Longest priority
Random
Shortest priority
33 20 18 13
33 22 20 13
34 24 22 13
shown in Fig. 3, which are acquired by 400 times of calculation. The MNWRs of the four networks obtained by the two routing methods for three kinds of wavelength assignment strategies are compared in Table 3. From this table, we can see that under dierent wavelength assignment strategies, the
MNWRs obtained by the two routing methods are almost the same, therefore it is dicult to judge which routing method is better. But if the distribution of the NWRs obtained by the two routing methods after 400 times of calculation is considered, the dierence of the two routing methods appears. As we can see from Table 4, it is obvious that the distribution of the NWRs obtained by the routing method 2 is more concentrated, i.e., the NWRs are closer to MNWR. So the routing method 2 is better than the routing method 1. This is because the link weight is considered in the routing method 2, so the probability that the next path routes through the links with large weight will decrease, and the network trac will be more evenly distributed in the network.
Y. Ye et al. / Optics Communications 185 (2000) 315±320
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Fig. 3. The distribution of the NWRs of the four networks obtained by the routing method 2 for three kinds of wavelength assignment strategies. (a) The distribution of the NWRs obtained for the longest priority wavelength assignment strategy by routing method 2. (b) The distribution of the NWRs obtained for the random wavelength assignment strategy for routing method 2. (c) The distribution of the NWRs obtained for the shortest priority wavelength assignment strategy by routing method 2. Table 3 The MNWRs of the four networks obtained by the two routing methods for the three kinds of wavelength assignment strategies Longest priority ARPANet UKNet EON NSFNet
Random
Shortest priority
Method 1
Method 2
Method 1
Method 2
Method 1
Method 2
33 20 18 13
33 21 18 13
33 22 20 13
33 22 19 13
34 24 22 13
35 24 21 14
The computation times by dierent schemes are given in Table 5, which are obtained through 400 times of calculation in ARPANet. As we can see from this table, the computation time is mainly determined by routing scheme and the wavelength assignment strategy aects the computation time weakly. Using routing method 2 can save much time compared with routing method 1. This also
bene®ts from the link weight considered in routing method 2. 4. Conclusion In this paper, RWA algorithms whose optimal objective is the MNWR were studied. The
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Table 4 The distribution of the NWRs of the four networks obtained by the two routing methods for the three kinds of wavelength assignment strategies Longest priority ARPANet UKNet EON NSFNet
Random Method 2
Method 1
Method 2
Method 1
Method 2
33±34 20±24 18±23 13±15
33±34 21±23 18±21 13±15
33±39 22±29 20±26 13±17
33±38 22±27 19±24 13±17
34±40 24±30 22±26 13±18
35±39 24±27 21±24 14±17
Table 5 The comparison of computation time by dierent schemes obtained through 400 times of calculation in ARPANet
Routing method 1 Routing method 2
Shortest priority
Method 1
Longest priority
Random
Shortest priority
131 min
132 min
135 min
52 min
53.5 min
53 min
equal, performance of the RWA algorithms can still be compared by observing the distribution of the NWRs. If the probability of the MNWR acquired by an algorithm is higher or the distribution of the NWR is more concentrated compared with another algorithm, then the former is better.
Acknowledgements MNWRs of networks are obtained by the RWA algorithms associated with the numbering order of the nodes in networks. When the numbering order of the nodes is dierent, the network NWR obtained by the RWA algorithms is dierent. So a statistics method is proposed to solve the problem. It is proved that lower MNWR can be acquired by this statistics method through numerical calculation. This method can also be applied to other RWA algorithms whose simulation results are affected by the numbering order of the nodes. For the ®rst time RWA algorithms are compared by statistics method. When the MNWRs acquired by two dierent RWA algorithms are
This work is supported by the National Natural Science Foundation of China (69990540) and the 863 High Technology Research Development Program.
References [1] S. Baroni, P. Bayvel, J. Lightwave Technol. 15 (1997) 242± 251. [2] N. Nagatsu, Y. Hamazumi, K. Sato, IEICE Trans. Commun. E78-B (1995) 597±607. [3] W. Chen, H. Zhang, Z. Pan, et al., J. Tsinghua Univ. (Sci. & Technol.) 38 (1998) 43±46 (in Chinese).