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Computer Communications 30 (2007) 3392–3402 www.elsevier.com/locate/comcom

Genetic evolutionary algorithm for static traffic grooming to SONET over WDM optical networks Kuntal Roy *, Mrinal K. Naskar Department of Electronics and Tele-Communication Engineering, Jadavpur University, Kolkata 700032, India Received 10 April 2007; accepted 13 June 2007 Available online 16 June 2007

Abstract In recent years, minimization of SONET-ADMs (Synchronous Optical NETwork-Add-Drop Multiplexers) in WDM (Wavelength Division Multiplexing) optical networks has gained a lot of attention in both the research and commercial arenas. This motivates the research presented in this article. The enhanced searching capability of genetic evolutionary algorithm has been exploited for this purpose. The individuals (chromosomes) have been represented by different sequence of the calls in the traffic matrix. A simple algorithm that minimizes the number of required ADMs based on the shortest path and a possible alternate shortest path has been applied. Some good chromosomes based on some intuitive reasoning have been introduced in the initial population to enhance the convergence of the proposed genetic evolutionary algorithm. The distinguished feature of the proposed algorithm is in introducing the catalyst to direct the convergence of genetic evolutionary algorithm towards its solution. However, the catalyst has been kept small enough to be able to bias the solution. To establish the effectiveness of the proposed algorithm, the simulation results are compared with that of presented in literature with same network configuration and traffic matrix.  2007 Elsevier B.V. All rights reserved. Keywords: WDM optical networks; Static traffic grooming; SONET add-drop multiplexer; Genetic evolutionary algorithm

1. Introduction Wavelength division multiplexing (WDM) coupled with Synchronous Optical Network (SONET) has emerged as a promising technology for use in backbone networks. Multiple signals distinguished by their wavelengths can be carried over through a fiber using WDM technology [1,2]. The reason behind the bandwidth-division of a fiber is that its bandwidth is very high and hence, suitable mechanisms should be employed to efficiently use the same. As the technology progresses, transmission speed of fiber is also increasing from OC-48 (2.5 GBps) to OC-192 (10 GBps). Accordingly, the recent trend is to employ TDM (Time Division Multiplexing) slots in the wavelength channels.

*

Corresponding author. Tel.: +919831592866. E-mail addresses: [email protected] (K. Roy), mrinalnaskar@ yahoo.co.in (M.K. Naskar). 0140-3664/$ - see front matter  2007 Elsevier B.V. All rights reserved. doi:10.1016/j.comcom.2007.06.009

The resulting network configuration is known as WDM– TDM network or WDM grooming network. A route through which a specific wavelength is assigned to each link is called a lightpath. Given a set of connections between source–destination node-pairs, the algorithm of establishing the corresponding lightpaths is called Routing and Wavelength Assignment (RWA) [3]. At each node of the network, there are SONET [4,5] Add-Drop Multiplexers (SADM) for each wavelength to add or drop signal streams. In this section, we will use ‘‘ADM’’ and ‘‘SADM’’ interchangeably. An SADM has the capability to combine lower-rate signals into a higherrate signal. For example, four OC-48s can be multiplexed into an OC-192. But, the cost of SADMs dominates the total cost of designing WDM optical networks. As the number of nodes in the optical network increases, the number of required ADMs also increases by a number equal to the number of wavelength channels in the network per node. So, if a WDM optical network has N nodes and there

K. Roy, M.K. Naskar / Computer Communications 30 (2007) 3392–3402

are W wavelength channels between the node-pairs, it would require a total of N * W SADMs. Fortunately, it is not necessary for every node to be equipped with SADMs for all wavelengths. An SADM corresponding to a wavelength is required only to transmit or receive signals at that wavelength. Hence, it is possible to decrease the number of required SADMs in WDM optical networks by availing of this opportunity. Accordingly, the challenge is to find out the minimum number of required SADMs for some traffic matrix at hand. So, RWA (Routing and Wavelength Assignment) algorithm has to be formed with an eye to minimize the number of SADMs in WDM optical networks. Therefore, tremendous efforts have been exploited to optimize the number of SADMs in SONET-WDM networks. 1.1. Previous approach Different approaches for SADM minimization have been reported in the literature. The approaches proposed in [6–8] are the first-stage works in this field. Some theoretical approaches to compute the lower-bound of ADMs are reported in [9,10]. Traffic-grooming in optical network has been illustrated in [11,12]. In [13–16], heuristic algorithms are proposed for static traffic to minimize the number of ADMs. In [17], a linear programming solution has been proposed. In [18,19], Integer Linear Programming (ILP) and Simulated-Annealing (SA) were used to solve the problem. In [19], there are exhaustive comparisons among greedy heuristic, ILP and simulated-annealing for different types of traffic and grooming capabilities. It is observed in [19] that the ILP solver fails to reach a conclusion for the assumed non-uniform traffic. In [20], a simple heuristic solution based on sequencing of the traffic requests in a certain way has been proposed by the authors. Almost for all the cases, it outperforms the results in [19]. In [21], a genetic algorithm for SADM minimization is proposed. But, it is applicable to uni-directional ring networks only. Accordingly, there exists no efficient solution that can be applied in general. 1.2. Proposed approach In this article, genetic evolutionary algorithm has been applied for optimization purpose, i.e., to efficiently allocate the ADMs at the nodes of WDM optical networks. The proposed algorithm is applicable to any network configuration and traffic requests. The traffic requests between the nodes are represented in the form of a chromosome. Moreover, a good chromosome based on intuitive reasoning has been constructed and injected in the initial population with a hope to have an early convergence. In general, selection of initial population in a genetic algorithm plays an important role in respect to its convergence. However, the quantity of such injection has been kept small (5%) so that it does not bias the solution. This good sequence (chromosome) is formed by placing the calls between the node-pair

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(j, i) followed by the calls between the node-pair (i, j) sequentially. The rationale behind such good chromosome arises considering the case with a uni-directional ring network. In a uni-directional ring network, a link is completely utilized if (i, j)th call follows a (j, i)th call. During mutation, the discontinuity that may arise is considered and taken care of. For each chromosome, the same RWA mechanism has been applied. It has been established that proper sequencing of the calls can result lower number of required ADMs with the same RWA mechanism. In this article, a simple RWA mechanism that eyes to ADM minimization employing the shortest path and a possible alternate shortest path has been introduced. The standard deviation of the individuals in a population is taken as a performance index for the generation. The algorithm converges when the performance index becomes zero. The results according to the proposed approach are compared with the results as in [19] to establish its effectiveness. The rest of the paper is organized as follows. In Section 2, the proposed scheme is outlined in a simple algorithmic form. Time complexity analysis of the proposed algorithm is also presented in this section. Simulation results are provided in Section 3. Also, in Section 3, comparison with the previous approach (as in [19]) is done. Finally, Section 4 concludes the paper. 2. Proposed algorithm In this section, the proposed algorithms are given in subsequent sub-sections. In this section and onwards, we will use ‘‘ADM’’ and ‘‘SADM’’ interchangeably. Also, ‘‘call’’, ‘‘request’’ and ‘‘traffic’’ all will indicate some connection that needs to be established between a source–destination node-pair. Genetic evolutionary algorithm is a simple but effective population-based algorithm for solving optimization problems [22–24]. Genetic Algorithms (GAs) are originated from the studies of cellular automata conducted by John Holland [22] and his colleagues at the University of Michigan. Its applications include diverse areas such as job-shop scheduling, training neural nets, image feature extraction or recognition, pattern recognition, data mining, bio-informatics etc. Genetic algorithms [25–33] are modeled on the concept of natural selection and evolution as in the same way the species adopt their environment. GAs are not random search algorithms rather it searches the solution-space towards global optimization in some directed manner and exploits the advantages of natural genetic system. Genetic evolutionary algorithm [34–39] maintains a population of individuals, P ðtÞ ¼ fxt1 ; xt2 ; . . . ; xtn g for generation t. Each individual in a population is termed as chromosome (or genotype). Next, a new population P(t + 1) is formed in (t + 1)th generation from P(t) with the help of some genetic evolutionary mechanisms (e.g., selection, crossover, mutation). There are no hard rules for the mechanisms selection, crossover, mutation – they can be formed depending on the problem at hand. However, there exist

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and introduced in the initial population. The rationale behind is that the good chromosome may act as a catalyst for earlier convergence of the genetic evolutionary algorithm. However, we should make sure that the good chromosome does not bias the solution. Therefore, a small percent (5%) of such good chromosome has been introduced in the initial population. The positions of the injection of good chromosomes are selected randomly in the initial population. Now, the good property of the good chromosome will be explained. It can be understood intuitively that having some traffic matrix at hand, the sequence of servicing the calls has effect over the required number of ADMs. If a (j, i)th call is followed by an (i, j)th call for a uni-directional ring network, then a link is completely utilized by the two calls and it requires an ADM corresponding to only one wavelength for both ith and jth nodes. So, for uniform traffic (i.e., number of requests between nodepairs are more-or-less of same value) and uni-directional ring network, it is quite obvious that the good chromosome will give more-or-less the optimum sequence of requests corresponding to minimum number of required ADMs. For non-uniform traffic and arbitrary mesh network, the good property of the good chromosome is not as directly feasible as for uniform traffic in uni-directional ring network. The good chromosome corresponding to the traffic matrix, TM given in the previous sub-section is as below.

some standards for the same. If the number of individuals in a population is quite high (e.g., 100), we can assume that the algorithm on its convergence reaches at the globally optimal solution. 2.1. Chromosome representation Problem-specific representation of chromosomes is the very first step of any genetic evolutionary algorithm. In this article, each chromosome is represented as a two-dimensional matrix of order NC · 2, where NC is the total number of calls. To explain the same, let us consider a traffic matrix, TM for a network containing four nodes, TM ¼ ½0;

3; 4; 2;

3;

0; 1; 2;

4; 2;

2; 0; 2; 2; 1; 0

An entry (i, j) in the traffic matrix denotes the number of calls from ith node to jth node. It can be noticed that the diagonal elements of the matrix are all zero which signifies that there exist no connection requests from a node to the node itself. The total number of calls corresponding to the traffic matrix is 28 that is the sum of all the entries in the matrix, TM. Accordingly, a chromosome can be represented for this case as a two-dimensional matrix of order 28 · 2, where an entry in the ith row denotes a source–des

1

1

1

2 2

2

1

1 1

1

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4 4

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1 1

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3 3

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1 1

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1 1

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2 4

4

2

2 4

4

3

‘T’ denotes the transpose of a matrix.

tination node-pair corresponding to a request. The calls are placed row-wise sequentially in the chromosome as in the traffic matrix. Accordingly, row 1 in the chromosome corresponds to source–destination node-pair (1, 2), row 4 corresponds to source–destination node-pair (1, 3) and row 28 corresponds to source–destination node-pair (4, 3). The representation of the chromosome is as below. 

T

2.3. Objective function – ADM allocation In genetic evolutionary algorithm, objective function refers to the objective of the problem at hand. In can be minimization or maximization of a parameter or a function

1

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1 1

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1 1

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1

1 2

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T

2.2. A good chromosome

of parameters specific to the problem. In this article, the objective function is considered as minimization of the total number of allocated ADMs. Determination of the number of allocated ADMs is based on the following step-by-step algorithm.

In general, initial population generation in a suitable manner plays an important role in respect to the convergence of genetic evolutionary algorithm. In this article, an intuitive reasoning-based chromosome has been formed

Step1: Firstly, it is tried so that no extra ADM needs to be allocated at source or destination node to establish the lightpath between the node-pair. Accordingly, a wavelength channel is searched for which an

‘T’ denotes the transpose of a matrix. We can produce different chromosomes by shuffling the calls in different possible ways.

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ADM is already allocated to both the source and destination nodes. Shortest path between the node-pair is considered in this step. Step2: If the lightpath can not be established in this way because of the non-availability of such free wavelength channel, alternate shortest path (the hops of the shortest and alternate shortest path are mutually exclusive), if exists is tried for the same. Step3: Even if the lightpath can not be established, the proposed algorithm tries to utilize one extra ADM to establish the call. The node (source/destination) for which the current number of ADMs is greater than the other is searched first to establish the lightpath. If it fails, the other node is tried for the same. Shortest path between the node-pair is considered to establish the lightpath in this step. Step4: If the endeavor to establish a lightpath is not successful even utilizing one extra ADM for the shortest route, the alternate shortest path (if it exists) is tried for the same. Step5: Even if it fails, the algorithm assigns a new wavelength channel and allocate one ADM corresponding to the new wavelength channel for both the source and destination nodes.

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output: initialPopulation ‹ chromosomes corresponding to the initial population. Step1: Form a chromosome (i.e., sequence of calls) between the node-pairs by traversing the ‘traffic’ matrix row-wise (see Section 2.1). Step2: initialPopulation ‹ Generate NP number of chromosomes by shuffling the sequence as generated in Step1. Step3: Select randomly 5% of the individuals in the population and replace those by ‘goodChromosome’.

2.6. Selection Selection is a genetic evolutionary mechanism to proceed forward towards the next generation. The procedure is to select some individuals from the previous generation for subsequent genetic evolutionary mechanisms, e.g., mutation, crossover to be applied. In this article, two different individuals are selected randomly from a population. These two individuals are mutated and thereby generating a new individual in the next generation depending on the crossover mechanism. 2.7. Mutation

2.4. Fitness function – acceptance criteria Depending on the fitness of an individual over another individual, the firstly said individual can be accepted or rejected in the next generation. In this article, acceptance of an individual over another is done on the basis of the number of required ADMs (i.e., whose ADM requirement is less). If both individuals have same ADM requirements, then the previous individual is retained in the next generation. 2.5. Initial population Producing initial population (i.e., population of first generation) is of immense importance as it determines the speed of convergence of the evolutionary algorithm. As said, a population is formed of individuals or chromosomes. In this article, different chromosomes are produced by shuffling the calls (identified by source–destination node-pairs) randomly inside a chromosome (see Section 2.1). Those different chromosomes are introduced in the initial population and then the ‘good chromosome’ (described in Section 2.2) is injected inside the population by randomly selecting some positions with a small percentage (5%). The algorithm to generate the initial population is described briefly below. inputs: N ‹ number of nodes in the network; traffic ‹ static traffic demand; NP ‹ number of individuals in a generation; goodChromosome ‹ the good chromosome (as described in Section 2.2);

In this step, the chromosomes selected at the selection step undergo mutation. A fixed value called mutation perturbation scale factor (F) is assumed beforehand and depending on its value and a random number generated at runtime, the mutation mechanism is performed. To perform mutation between two individuals, we select some portion (depending on the generation of a random number) of both the selected individuals. Otherwise, the individual which is more fit than the other is selected for crossover operation. However, it can be noticed that the mutation can result discontinuity in the chromosome. The reason is quite obvious. Since, we are selecting a portion of a chromosome, it is not guaranteed that all the calls that should be present in a chromosome would retain in the new chromosome. Some call may be duplicated and some call may not appear. It can be easily observed in this respect that if the two selected individuals are identical, the mutation will generate the same individual. The mutation mechanism that has been employed in this article is as follows. inputs: F ‹ mutation perturbation scale factor; xx1t, xx2t ‹ two selected chromosomes at tth generation; NP ‹ number of population in a generation; NC ‹ number of calls in the given static traffic matrix. output: xxDerivedt ‹ a derived chromosome after mutation at tth generation. Step1: rnd ‹ Generate a random number from ‘0’ to ‘1’. Step2: If rnd P F Then CP ‹ Select randomly a number between 1 and NC.

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xxDerivedt ‹ xx1t (1:CP-1) [ xx2t (CP:NC). Remove any duplicate calls those exist in the derived chromosome xxDerivedt. For all the absent calls in xxDerivedt, select randomly the position of the call and place it accordingly in the derived chromosome xxDerivedt. Else xxDerivedt ‹ xx1t or xx2t whichever is more fit. //According to Section 2.4 In this article, the value of ‘F’ has been chosen as 0.1. 2.8. Crossover In this article, we depend on mutation as primary search mechanism. Thus, the crossover mechanism has been used just to determine if a current individual will be retained or replaced in the next generation depending on the crossover (recombination) constant. The crossover operation that has been employed in this article is as follows. inputs: CR ‹ crossover (or recombination) constant; xxPrevt ‹ a previous individual at tth generation; xxDerivedt ‹ a derived chromosome after selection and mutation operation. output: xxNewt + 1 ‹ new chromosome after crossover in the next generation. Step1: rnd ‹ Generate a random number between ‘0’ to ‘1’. Step2: If rnd P CR Then xxNewt + 1 ‹ xxDerivedt. Else xxNewt + 1 ‹ xxPrevt. In this article, the crossover constant (CR) has been chosen as 0.5. 2.9. Performance index – convergence criteria Performance index of a generation depends on the similarity of individuals in the population. In this article, we have assumed ‘standard deviation’ of the individuals in a population to be the performance index for that generation. When performance index reaches zero, the algorithm converges since, executing more generations does not produce any new result. 2.10. Algorithm So far, we have described different modules of the proposed algorithm. In this sub-section, the complete algorithm is presented with the help of the aforesaid modules as follows. inputs: traffic ‹ static traffic demand; N ‹ number of nodes in the network; W ‹ number of wavelength channels between the nodes in the network; C ‹ number of slots in each wavelength channel of the network; connect ‹ connection matrix between the nodes.

output: ADM ‹ ADM assignment matrix (of order N · W); wv ‹ a matrix (of order N · N · W · C) that denotes the established connections between the nodes through a wavelength channel and a consequent slot; noADM ‹ total number of required ADMs.

Step1: Initialize no_of_generations to 0. Set some value of maximum allowable no_of_generations (i.e., no_of_generations max) Step2: Generate initial population // Section 2.5// Step3: Determine performance index, // Section 2.9 // PI for the initial population Step4: While PI is not 0 or no_of_ generations < no_of_generations max Do selection. // Section 2.6 // Do mutation. // Section 2.7 // Do crossover. // Section 2.8 // Calculate performance index, // Section 2.9 // PI for this generation. Increment no_of_generations by 1.

2.11. Time complexity analysis The following notations have been used for time complexity analysis of the genetic evolutionary algorithm proposed in this article. Number of nodes in the WDM optical network = N. Number of wavelength channels between nodepairs = W. Number of TDM-slots in a wavelength channel = C. Number of individuals in a generation = NP. Number of calls in the traffic matrix = NC. Average number of hops for a source–destination nodepair = H.

Consequently, Time required to generate initial population = O(N2 + NP*NC). Maximum time required to search WADM wavelength channels for a free TDM-slot = O(H*WADM*C), which is quite obvious. Time required for selection operation = O(NP*NC). Average time required for mutation operation = O(N P  N 2C ). Time required for crossover = O(NP*NC). Time required to calculate performance index = O(NC).

// Section 2.5 //

// Section 2.6 // // Section 2.7 // // Section 2.8 // // Section 2.9 //

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The overall time complexity of the proposed algorithm depends on how many generations, the program is executed. Since, the solution of the problem needs not to be real-time and due to mismatch in working environment with the previous approach (in [19]), the corresponding time-complexities are not compared. 3. Results The proposed algorithm has been tested for different kind of traffics and network configurations. Results for both the uni-directional and bi-directional ring networks have been provided. All results correspond to single-hop connections. We have chosen the number of individuals in a generation as 100 for all the experiments performed. Also, the maximum number of allowable generations for execution has been chosen as 100. However, it has been observed that almost all the cases, the proposed algorithm converges before 60th generation. We will denote the number of nodes in the network as ‘N’ and the number of TDM-slots in a wavelength channel as ‘C’ throughout this section.

Fig. 1. Established Connections between the nodes for an all-to-all uniform traffic (N = 8, C = 3).

parison between the best results found in [19] and the results according to the proposed approach in this article. From Table 1, it can be observed that the results provided by the proposed algorithm of this article are better than the previous results. The algorithm reaches the best possible solutions for C = 64 and below node 17 (it needs 24 and 30 ADMs according to the proposed algorithm for N = 17, C = 64 and N = 18, C = 64, respectively) whereas the algorithm in [19] could not reach it for nodes greater than 11. The busy links in the network showing the ADMs required at the nodes for the case N = 8, C = 3 are shown in Fig. 1. From Fig. 1, it can be observed that it needs six wavelength channels to establish all the connections with N = 8 and all-to-all uniform traffic. A wavelength channel has been shown to be composed of three different links as

3.1. All-to-all uniform traffic in uni-directional ring network This type of traffic has been well studied in literature. For this kind of traffic, we have only one request for a source–destination node-pair. Moreover, the source–destination node-pairs are characterized by the directivity of the uni-directional ring network and on other direction of a source–destination node-pair, there is no traffic. If we have a traffic matrix, TM (of order N · N) denoting the traffic demands then, for all-to-all uniform traffic,  1 for i < j TMi;j ¼ 0 otherwise where, we are considering the directivity of the uni-directional ring as 1 fi 2 fi    fi N. The all-to-all traffic has been extensively studied in [19] for different proposed algorithms. Table 1 shows the com-

Fig. 2. Established Connections between the nodes for an all-to-all uniform traffic (N = 4, C = 4).

Table 1 Comparison of results between the proposed algorithm and the algorithm in [19] for all-to-all uniform traffic

C=3 C=4 C = 12 C = 16 C = 48 C = 64

N

4

5

6

7

8

9

10

11

12

13

14

15

16

Algorithm in [19] Proposed Algorithm Algorithm in [19] Proposed Algorithm Algorithm in [19] Proposed Algorithm Algorithm in [19] Proposed Algorithm Algorithm in [19] Proposed Algorithm Algorithm in [19] Proposed Algorithm

7 6 7 4 4 4 4 4 4 4 4 4

12 9 10 8 5 5 5 5 5 5 5 5

17 12 15 12 9 6 6 6 6 6 6 6

21 18 21 15 12 7 11 7 7 7 7 7

31 25 28 20 16 12 14 8 8 8 8 8

36 31 36 27 18 15 18 13 9 9 9 9

48 40 45 35 24 20 20 17 10 10 10 10

67 47 55 42 30 24 26 19 16 11 11 11

69 61 66 50 36 28 33 27 19 12 15 12

78 72 78 61 39 36 37 31 22 13 19 13

95 83 91 73 49 40 42 37 24 16 22 14

105 94* 105 85 57 53* 46 45 31 22 25 15

124 107* 120 96* 64 64 57 49* 34 26* 28 16

The * marked fields correspond to the cases where the proposed evolutionary algorithm does not converge after 10 h of simulation and the value placed is the best value so far found in the simulation.

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C = 3. ADMs required at the nodes are shown by an ellipse. For example, the nodes 2, 3 and 4 do not have ADM corresponding to wavelength 2 which means that the nodes do not transmit or receive signals at that wavelength (wavelength 2). We can calculate the number of busy links for all-to-all uniform traffic in a uni-directional ring networks as there is only one way of reaching one node from another node and also the traffic matrix is already known. The corresponding calculation is given in Appendix A. In Fig. 1, total number of busy links is 84 according to Appendix A. The busy links in the network showing the ADMs required at the nodes for the case N = 4, C = 4 are shown in Fig. 2. The explanation for the configuration is same as described for the Fig. 1. In this case, total number of busy links is 10 according to Appendix A. Some of the graphs showing the convergence of the proposed genetic evolutionary algorithm are shown next and some interesting points therein are explained subsequently. As said, we have chosen both the number of individuals in

a generation and the maximum number of generations that can be allowed for execution as 100. In Fig. 3, a minimum appears when the individuals with ADM = 26 dominates corresponding to the ‘good chromosome’ (see Section 2.2). However, it is only a local minimum which has been cancelled by the evolutionary mechanism as it proceeds. Therefore, at 14th generation, all the individuals have solution corresponds to ADM = 25 and the algorithm converges. We might debate that the solution ADM = 25 can be also a local minimum. But, since we have chosen a population size of 100 which is large enough and all the 100 individuals in 14th generation have the same solution, we can conclude that this is the optimal assignment of ADMs. In Fig. 4, it can be observed that the first minimum in the curve corresponding appears when the individuals with ADM = 63 dominates. But, several chromosomes corresponding to ADM = 62 and ADM = 61 appear simultaneously and the algorithm converges at generation 17 (with ADM = 61) without any local minima in between.

Fig. 3. Convergence curve corresponding to N = 8, C = 3.

Fig. 5. Convergence curve corresponding to N = 14, C = 12.

Fig. 4. Convergence curve corresponding to N = 13, C = 4.

Fig. 6. Convergence curve corresponding to N = 13, C = 16.

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In this case, the number of ADMs corresponding to ‘good chromosome’ was 64. In Fig. 5, we can observe some ups and downs in the curve as the generation moves on. The algorithm starts with the number of ADMs equal to 47 corresponding to the ‘good chromosome’ at first generation. At last, it converges with ADM = 40 for all the individuals at 54th generation. The proposed genetic evolutionary approach may not be able to improve upon the result corresponding to the ‘good chromosome’ at first generation. Fig. 6 highlights this point. However, it should be pointed out that the evolutionary mechanism converges quickly. We can debate that the incorporation of the ‘good chromosome’ in initial population, however, small (5%) has biased the solution. Such possibility can be nullified by experimenting without introducing the ‘good chromosome’ in the initial population and getting the same result, however, possibly higher number of generations has to be executed.

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3.2. Non-uniform traffic in uni-directional ring network As in [19], the proposed algorithm has been tested with a non-uniform traffic matrix to show its effectiveness. The corresponding network configuration and traffic matrix are given below. Number of nodes in the network, N = 4. Number of time-slots in a wavelength channel, C = 3. Traffic matrix is characterized by the two-dimensional matrix, Traffic = {0, 1, 8, 4; 12, 0, 3, 9; 1, 2, 0, 2; 4, 1, 7, 0}. The corresponding established connections are shown in Fig. 7. The four nodes are denoted as Node1, Node2, Node3, and Node4. Since it is a ring network, Node1 again appears after Node4. In the previous sub-section, we did not show up such thing because for all-to-all uniform traffic there is no request from node N to node 1. So, we can consider that the links between the nodes N and 1 are all unused for all-to-all uniform traffic. The explanation for the configuration is same as described for the Fig. 1. In this case also, we can determine the number of busy links provided with the traffic matrix. Accordingly, total number of busy links between the nodes is 123 as given in Appendix B. The number of ADMs (number of ellipses) required is 31. In Fig. 7, it can be observed that the wavelength 7 (i.e., w7) holds a total of eight connections. Table 2 indicates the slots that are busy in wavelength 7 for the same.

Table 2 Established connections thorough different slots of wavelength 7 (w7) corresponding to Fig. 7

Slot-1 Slot-2 Slot-3

Fig. 7. Established Connections between the nodes for a non-uniform traffic in a uni-directional ring network.

Node 1–2

Node 2–3

Node 3–4

Node 4–1

Node 1–3 Node 1–3 Node 1–4

Node 1–3 Node 1–3 Node 1–4

Node 3–4 Node 3–4 Node 1–4

Node 4–1 Node 4–1 Node 4–1

Fig. 8. Convergence curve corresponding to N = 4, C = 3 and a nonuniform traffic in a uni-directional ring network.

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Fig. 10. Convergence curve corresponding to N = 4, C = 3 and a nonuniform traffic in a bi-directional ring network.

tionary mechanism and al last the algorithm converges at 18th generation with the number of ADMs equal to 20. Fig. 9. Established Connections between the nodes for a non-uniform traffic in bi-directional ring networks (N = 4, C = 3).

The corresponding convergence curve is given in Fig. 8. In Fig. 8, the evolutionary approach can not improve upon the result (ADM = 31) corresponding to the ‘good chromosome’ in first generation. It converges at eighth generation. We can explain it in the same way as explained for Fig. 6. 3.3. Non-uniform traffic in bi-directional ring network Today’s most WDM ring networks are bi-directional. So, we have applied the proposed algorithm for the case of bi-directional ring networks as well. For this purpose, we have considered the same network configuration and traffic as in Section 3.2. Unlike uni-directional ring networks, a bi-directional ring network has two possible paths for a lightpath to be established between the node-pairs. We can utilize this alternate path to possibly reduce the number of required ADMs. The proposed algorithm when allocating ADMs is given in Section 2.3. The algorithm first tries via the shortest path and then it does the same for a possible alternate shortest path while eyeing minimization of the required ADMs. The corresponding busy links are shown in Fig. 9. The explanation for the configuration is same as described for Fig. 1. In Fig. 9, it can be observed that all the links have been made busy for the wavelength channels 1–7. The number of ellipses (number of ADMs) is 20 which signifies that we have achieved more than 30% reduction of ADMs compared to the uni-directional ring network. The convergence curve for this case is shown in Fig. 10. In Fig. 10, the number of ADMs corresponding to the ‘good chromosome’ was 22. However, as the generation proceeds, some better chromosomes appear due to evolu-

4. Conclusions In this paper, a genetic evolutionary approach has been proposed to optimize the number of required SADMs for static traffic grooming in WDM optical networks. The simulation results show better performance than the previous approach. Moreover, the proposed scheme is applicable to any network configuration. The results have been presented for both the uni-directional and bi-directional ring networks. For mesh networks, the proposed algorithm will consider at most two shorter paths to establish lightpaths between the source–destination node-pairs while eyeing ADM minimization. Appendix A. Number of busy links for uni-directional ring networks with all-to-all uniform traffic Let us consider a general case of having N nodes in a WDM uni-directional optical network with all-to-all uniform traffic. In uni-directional ring network, there is only one path from one node to another node in the network. For all-to-all uniform traffic, the upper triangular portion of the N · N traffic matrix contains all 1s and other entries in the matrix are 0. It is obvious that the number of busy links for the connections 1 fi 2, 2 fi 3,    , N  1 fi N are 1, for the connections 1 fi 3, 2 fi 4,    , N  2 fi N are 2 and for the connection 1 fi N is N  1. Accordingly, the total number of busy links NBLUDR;AAUT ¼ðN  1Þ  1 þ ðN  2Þ  2 þ    þ 1  ðN  1Þ 8 ðN1Þ=2 P > > > x  ðN  xÞ N is odd <¼ 2 x¼1 NBLUDR;AAUT ¼ ðN2Þ=2 >   P > > :¼ 2 x  ðN  xÞ þ N  N N is even x¼1

2

2

K. Roy, M.K. Naskar / Computer Communications 30 (2007) 3392–3402

We will use the following arithmetic summation rules N X



x¼1 N X x¼1

x2 ¼

N  ðN þ 1Þ 2 N  ðN þ 1Þ  ð2N þ 1Þ 6

For N is odd, NBLUDR;AAUT

N1

  N1 Nþ1  Nþ1  2  ðN Þ 2 2 2 ¼ 2N  2 6 ðN 2  1Þ 1 ðN 2  1Þ  N ¼N 4 3 4 1 ¼  N  ðN 2  1Þ 6 2

For N is even,

N2

  N2 N   N2  2  ðN  1Þ N 2 2 2 þ 2 6 4 2 ðN  2Þ 1 ðN  1Þ  ðN  2Þ N ¼ N2   N þ 4 4 3 4 ðN  1Þ 1 ðN  1Þ  ðN  2Þ 2 ¼N   N 4 3 4 1 2 ¼  N  ðN  1Þ 6

NBLUDR;AAUT ¼ 2N 

2

Accordingly, NBLUDR;AAUT ¼

1  N  ðN 2  1Þ 6

regardless N is odd or even. Appendix B. Number of busy links for uni-directional-ring networks with non-uniform traffic In general, for a uni-directional ring network, the number of busy links, NBLUDR ¼

N1 X

ni  i

i¼1

where, ni is the number of i-hop path. For a traffic matrix, traffic = {0, 1, 8, 4; 12, 0, 3, 9; 1, 2, 0, 2; 4, 1, 7, 0} as in Section 3.2, obviously, N = 4. Also, n1 ¼ traffic1;2 þ traffic2;3 þ traffic3;4 þ traffic4;1 ¼ 1 þ 3 þ 2 þ 4 ¼ 10 n2 ¼ traffic1;3 þ traffic2;4 þ traffic3;1 þ traffic4;2 ¼ 8 þ 9 þ 1 þ 1 ¼ 19 n3 ¼ traffic1;4 þ traffic2;1 þ traffic3;2 þ traffic4;3 ¼ 4 þ 12 þ 2 þ 7 ¼ 25

Accordingly, for the aforesaid traffic, NBLUDR ¼ 10  1 þ 19  2 þ 25  3 ¼ 123 We can write a computer program that takes a traffic matrix as input and outputs the number of such busy links. References [1] B. Mukherjee, Optical Communication Networks, McGraw-Hill, New York, 1997.

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[2] R. Ramaswami, K.N. Sivarajan, Optical Networks: A Practical Perspective, Morgan Kaufman Publishers, 1998. [3] K. Roy, M.K. Naskar, Adaptive Dynamic Wavelength Routing for WDM Optical Networks, 3rd International conference on Wireless and Optical Communications Networks – 2006 (WOCN’06), IEEE Communication Society, Bangalore, India, April 11–13, 2006. [4] U. Black, S. Waters, SONET and T1 Architectures for Digital Transport Networks, Prentice Hall, New Jersey, 1997. [5] S.V. Kartalopoulos, Understanding SONET/SDH and ATM Communications Networks for the Next Millennium, IEEE press, New York, 1999. [6] O. Gerstel, G. Sasaki, Cost effective traffic grooming in WDM rings, Proc. IEEE INFOCOM 0 98 (1998) 69–77. [7] O. Gerstel, P. Lin, G. Sasaki, ‘‘Wavelength Assignment in a WDM Ring to Minimize Cost of Embedded SONET Rings’’, Proc. of IEEE INFOCOM 0 98 1 (1998) 94–101. [8] O. Gerstel, P. Lin, G. Sasaki, Combined WDM and SONET Network Design, Proc. of IEEE INFOCOM ’99, vol. 2, pp. 734–743, NY, March 1999. [9] A. Chiu, E. Modiano, Reducing Electronic Multiplexing Costs in Unidirectional SONET/WDM Ring Networks via Efficient Traffc Grooming, IEEE/IEICE Global Telecommunication Conference, vol. 1, pp. 332–327, Sydney, Australia, November 1998. [10] A. Chiu, E. Modiano, Traffic Grooming Algorithms for Reducing Electronic Multiplexing Costs in WDM Ring Networks, IEEE/OSA Journal of Lightwave Technology 18 (2000) 2–12. [11] E. Modiano, P.J. Lin, Traffic grooming in WDM networks, IEEE Communication Magazine 39 (2001) 124–129. [12] E. Modiano, A. Narula-Tam, Mechanisms for providing optical bypass in WDM-based networks, SPIE Optical networks magazine (2000). [13] X. Zhang, C. Qiao, An effective and comprehensive solution to traffic grooming and wavelength assignment in SONET/WDM rings, Proc. SPIE 3531 (1998) 221–232. [14] L. Liu, X. Li, P. Wan, O. Frieder, Wavelength Assignment in WDM Rings to Minimize SONET ADMs, Proc. of IEEE INFOCOM 0 00 2 (2000) 1020–1025. [15] M. Sridharan, A.K. Somani, Revenue Maximization in Survivable WDM Networks, Optical Networking and Communications, Proc. of SPIE 4233 (2000). [16] P. Wan, L. Liu, O. Frieder, Grooming of Arbitary Traffic in SONET/ WDM Rings, IEEE/IEICE Global Telecommunication Conference 1B (1999) 1012–1016. [17] J. Hu, Traffic grooming in wavelength-division-multiplexing ring networks: a linear programming solution, Journal of Optical Networking 1 (11) (2002) 397–408. [18] W. Cho, J. Wang, B. Mukherjee, Improved approaches for costeffective traffic grooming in WDM ring networks: uniform-traffic case, Photonics Network Communication 3 (2) (2001) 245–254. [19] J. Wang, W. Cho, V. Rao Vemuri, B. Mukherjee, Improved Approaches for Cost-Effective Traffic Grooming in WDM Ring Networks: ILP Formulations and Single-Hop and Multihop Connections, Journal Of Lightwave Technology 19 (11) (2001) 1645–1653. [20] K. Roy, M.K. Naskar, A Heuristic Solution to SONET ADM Minimization for Static Traffic Grooming in WDM Uni-Directional Ring Networks, Photonic Network Communications, Springer, 12, no. 2, 2006, pp. 153–160. [21] Y. Xu, S. Xu, B. Wu, Strictly nonblocking grooming of dynamic traffic in unidirectional SONET/WDM rings using genetic algorithms, Computer Networks: The International Journal of Computer and Telecommunications Networking 41 (2) (2003) 227–245. [22] J.H. Holland, Adaptation in Natural and Artificial Systems, MIT Press, Cambridge, 1975. [23] E. Horowitch, S. Sahani, Fundamentals of Computer Algorithms, Galgotia, New Delhi. [24] R. Horst, P.M. Pardalos (Eds.), Handbook of Global Optimization, Kluwer Academic Publishers, 1995. [25] D.E. Goldberg, Genetic algorithms in search optimizations and machine learning, Addison Wesley, New York, 1989.

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[26] M. Srinivas, L.M. Patnaik, Genetic algorithm: a survey, IEEE Computer Magazine (1994) 17–26. [27] S. Bandyopadhyay, U. Maulik, Non-parametric genetic clustering: comparison of validity indices, IEEE Transactions on Systems, Man and Cybernetics Part-C 31 (1) (2001) 120–125. [28] U. Maulik, S. Bandyopadhyay, Fuzzy partitioning using real coded variable length genetic algorithm for pixel classification, IEEE Transactions on Geosciences and Remote Sensing 41 (5) (2003) 1075–1081. [29] S.K. Sarkar, A. Moi, C. Puttamadappa, A.K. De, M.K. Naskar, Application of genetic algorithm to determine the optimized system parameters of GaAs quantum wells for better high-grequency performance under hot electron condition, Physica B (Elsevier) 325 (2003) 189–194. [30] I. Saha Misra, M. Chakrabarty, M.K. Naskar, B. Banerjee, D. Dutta, Solution of unicast QoS routing using Genetic Algorithm, Int. Conf. CIT 2001, India (2001). [31] A. Mukhopadhyay, U. Biswas, M.K. Naskar, A genetic algorithm for traffic grooming in unidirectional SONET/WDM rings, IEEE India Annual Conference INDICON (2000) 252–255. [32] M.K. Naskar, B. Mukherjee, J.N. Majumder, S.K. Sarkar, A genetic approach for design protection in WDM optical networks, International Conf. PHOTONICS – 2002 Mumbai (2002). [33] A. Mukhopadhyay, U. Biswas, M.K. Naskar, U. Maulik, S. Bandyopadhyay, in: S. Olariu, A.Y. Zomaya (Eds.), Handbook of Bioinspired Algorithms and Applications, first ed., Chapman & Hall/ CRC, 2005. [34] T. Ba¨ck, D.B. Fogel, Z. Michalewicz (Eds.), Handbook of Evolutionary Computation, Institute of Physics Publishing and Oxford University Press, 1997. [35] R. Storn, K. Price, Differential evolution – a simple and efficient adaptive scheme for global optimization over continuous spaces, Journal of Global Optimization 11 (4) (1997) 341–359. [36] J. Liu, Y.Y. Tang, Y.C. Cao, An evolutionary autonomous agents approach to image feature extraction, IEEE Transactions on Evolutionary Computation 1 (2) (1997) 141–158. [37] X. Yao, Y. Liu, G. Lin, Evolutionary programming made faster, IEEE Transactions on Evolutionary Computation 3 (2) (1999) 82–102. [38] K.C. Tsui, J. Liu, An evolutionary multiagent diffusion approach to optimization, International Journal of Artificial Intelligence and Pattern Recognition 16 (6) (2002) 715–733.

[39] K. Roy, M.K. Naskar, Genetic Evolutionary Algorithm for Optimal Allocation of Wavelength Converters in WDM Optical Networks, unpublished.

Kuntal Roy received his B.E.(Hons) from Electronics and Tele-Communication Engineering Department, Jadavpur University, Kolkata, India in the year 2003. His research interests include Microelectronics, Embedded Systems Design, WDM Optical Networks, Robotics, Vision and Image Processing. Currently, he is pursuing Master of Science in Embedded Systems Design at Advanced Learning and Research Institute (ALaRI), University of Lugano, Switzerland. He is a co-author of the article ‘‘Adaptive Dynamic Wavelength Routing for WDM Optical Networks’’, 3rd International Conference on Wireless and Optical Communications Networks – 2006, IEEE Communication Society, Bangalore, India, April 11–13, 2006 and ‘‘A Heuristic Solution to SADM minimization for Static Traffic Grooming in WDM uni-directional Ring Networks’’, Photonic Network Communications, vol. 12, No. 2, September, 2006.

Mrinal Kanti Naskar received his B.Tech. (Hons) and M.Tech from E&ECE Department, IIT Kharagpur, India in 1987 and 1989 respectively. He served as a faculty member in NIT, Jamshedpur and NIT, Durgapur during 1991–1996 and 1996–1999 respectively. Currently, he is a professor in the Department of Electronics and Tele-Communication Engineering, Jadavpur University, Kolkata, India. His research interests include Computer Networks and Embedded Computing. He is a co-author of the article ‘‘Adaptive Dynamic Wavelength Routing for WDM Optical Networks’’, 3rd International Conference on Wireless and Optical Communications Networks – 2006, IEEE Communication Society, Bangalore, India, April 11–13, 2006 and ‘‘A Heuristic Solution to SADM minimization for Static Traffic Grooming in WDM uni-directional Ring Networks’’, Photonic Network Communications, vol. 12, No. 2, September, 2006.

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