An approach to wide area WDM optical network design using genetic algorithm

Transcription

1 Computer Communications 22 (1999) An approach to wide area WDM optical network design using genetic algorithm D. Saha a, *, M.D. Purkayastha a, A. Mukherjee b,1 a Department of Computer Science and Engineering, Jadavpur University, Calcutta , India b Pricewaterhouse Coopers Limited, Plot Y 14, Block EP, Sector 5, Salt Lake City, Calcutta , India Received 5 January 1998; received in revised form 19 October 1998; accepted 19 October 1998 Abstract This article presents a systemic method to generate and map an optimal virtual topology onto a certain physical wide area optical fiber network so as to maximize the scaleup which provides an estimate of the network throughput. The problem is to find the best possible virtual topology over a given wavelength-routed all-optical physical topology for wide area coverage. The physical topology consists of wavelength-routing nodes interconnected by fiber links in the network. The virtual topology consists of a set of lightpaths. For a given physical topology and traffic matrix, our objective is to maximize the throughput as well as to minimize the delay. We have also studied the scalability of various parameters, such as queuing delay, propagation delay and average-hop-distance, with increase in throughput. We present a heuristic algorithm for embedding a virtual topology to the given physical fiber network when the number of wavelength channels supported on each fiber is limited. A number of feasible virtual topologies are generated randomly using Prufer number method, and the optimum one is selected. The problem being computationally intractable, we use genetic algorithm (GA)for optimization. As GAs are expected to converge to globally optimal solutions, the proposed approach yields better solutions than previous ones. As a result of immense practical importance of the issues addressed in this work, the algorithms are expected to be significant for practitioners Elsevier Science B.V. All rights reserved. Keywords: Optical network; Physical topology; Virtual topology; Routing and wavelength assignment; Scaleup; Prufer number; Flowdeviation; Genetic algorithm 1. Introduction This work presents a set of principles for designing next generation, optical, wide area network, employing Wavelength Division Multiplexing (WDM). The complete design of such an optical network is usually viewed to consist of the following four steps [1 7,19 25,26]: (1) determine a feasible virtual topology consisting of lightpaths as links and routing centers as nodes, (2) route the lightpaths over the physical topology, (3) assign wavelengths optimally to various lightpaths, and (4) route packet traffic on the virtual topology. This design problem is often referred to as optimal virtual topology design problem in the literature. The problem has been conjectured to be NP-hard [1], which means that it cannot be solved optimally for large problem sizes, unless some form of heuristic search method is used. In this work, genetic algorithm (GA) combined with a * Corresponding author. Tel.: ; fax: ; d.saha@computer.org 1 Tel.: ; fax: ; amitava_- mukherjee@india.notes.pwa.co.in novel formulation of the problem and a new process generating feasible solutions, has been used to design an optimal wide-area WDM all-optical network. The aim of the algorithms is to maximize the throughput of the network. The performance index has multiple attributes and the GA selects among feasible solutions. The following modules of the algorithm are proposed in this article (1) an approach to generate random virtual topologies used in initialization of the the population of the GA, (2) a heuristic algorithm for embedding a virtual topology into a physical topology, and (3) a slightly modified version of the algorithm in [5] for wavelength assignment. The physical topology of the optical network consists of optical wavelength routers interconnected by pairs of pointto-point fiber links [18]. Each pair of links is represented by an undirected edge between nodes. Each end node has a limited number of optical transmitters and receivers (transceivers) called nodal degree. Each link is capable of carrying a certain number of wavelengths. A routing node takes in a signal at a given wavelength at one of its inputs and routes it to a particular output. A network node i is assumed to be equipped with a D p i Š D P i wavelength routing /99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S (98)

2 D. Saha et al. / Computer Communications 22 (1999) switches (WRS), where D p (i) is the physical degree of node i [1,6]. A virtual topology is a directed connected graph of lightpaths. A lightpath consists of a path through the network between end nodes and a wavelength on that path. Lightpaths are set up by configuring the routing nodes in the network. Two lightpaths that share a common link in the network must use different wavelengths. Ideally, in a network with N nodes, we would like to set up lightpaths between all the N N 1 =2 node pairs. However, this is usually not possible because of two reasons. First, the number of wavelengths available are limited. Second, each node can be the source and sink of a limited number of lightpaths because limited amount of optical hardware is available at each node. When it is not possible to establish lightpaths between all pairs of nodes, node pairs, that are not directly connected via lightpaths, the use of a sequence of lightpaths (i.e., multihop paths) through intermediate nodes is a must to communicate. At each intermediate node, packets coming in on a lightpath must be converted to electronic form, switched electronically and then converted back to optical form and then sent out on a different lightpath enroute their destination Previous work The problem of designing an optimal virtual topology has been studied extensively in the literature [1,4 7,19 26]. Most of them have suggested of using a regular topology, such as hypercube, as a virtual topology [1,19,25]. An algorithm for embedding a regular hypercube virtual topology was proposed in [25]. But it does not take into account the number of wavelengths per fiber as a constraint. Moreover, it does not consider topologies other than regular hypercube for embedding into a given physical fiber network. Regular topologies, no doubt, offer several advantages such as 1) simplified routing and congestion control procedures to reduce the amount of electronic processing required at each node, and 2) simplified network implementation because of standard hardware requirements at each intermediate node. But, on the negative side, in regular topologies, asymmetric traffic patterns create bottlenecks, thereby substantially deteriorating the overall system performance (i.e., increasing average delay and average hop length). To alleviate this problem, topology embedding must take into account traffic patterns in the system. Pankaj and Gallager [20] have studied the mapping of regular structures into a WDM star, considering the traffic pattern imposed on the system, so that mapping will optimize any objective function that is a function of the stochastic traffic pattern, such as system throughput. The particular algorithm used in [20] is known as stochastic ruler algorithm. Ton and Du [19] have studied the embedding of unidirectional incomplete hypercubes into the physical topology. The work in [6] proposes a virtual topology design for packet switched networks. Here, heuristic algorithms are proposed, for minimizing congestion, given the traffic distribution matrix. Thus, work in [6] uses the physical topology as a subset of the virtual topology, employing algorithms for maximizing the throughput, subject to bounded delay characteristics. Routing and Wavelength Assignment (RWA) in circuit switched networks has been studied by Zhang and Acampora [5]. They proposed a heuristic based on sequentially assigning a single wavelength to all possible lightpaths in order of decreasing traffic before proceeding to the next wavelength. The objective was to maximize the amount of traffic carried in one hop from its source to its destination, but degree and delay constraints on the virtual topology were ignored. In [23], some theoretical results on the bounds for minimum congestion routing have been studied, while, in [24], an upper bound is derived on the carried traffic for any RWA algorithm. Banerjee and Mukherjee [7] have addressed the problem of dynamically routing lightpaths, which may be thought of as one of the components of the overall virtual topology design problem Our approach Our approach is an extension to the work done in [1] and [25] in the sense that (i) we have removed the assumption of unlimited number of wavelenghts per fiber, and (ii) the virtual topologies are no more restricted to regular hypercubes. Consequently, our algorithm, being more flexible, allows the nodal degree to be nonuniform. Moreover, the present work differs from [1,25] in another important aspect, that is of optimization algorithm. We have used GA [13 17] in place of simulated annealing employed in [1]. Successful application of genetic approaches towards the solution of network design problems has been reported recently in [16] and [27]. Moreover, GA is preferred because of the following reasons. Simulated annealing probabilistically generates a sequence of states based on a cooling schedule to ultimately converge to the global optimum. But GA generates a sequence of populations by using a selection mechanism [13] and use crossover and mutation as search mechanism [14]. Thus, the optimal solution obtained using GA should ideally give better result as it efficiently explores the search space using the genetic operators. As a result, we have been able to achieve: 1. Better throughput (maximum scaleup is 148 for our optimal virtual topology, in comparison with 106 reported in [1], for the NSFNET backbone network with nodal degree 4). 2. Significantly less queuing delay (in our optimal virtual topology, any packet experiences an average queuing delay of ms as compared to ms reported in [1] for the same NFSNET backbone network). However, as we allow irregular virtual topologies, the routing implementation becomes complex. Also, memory requirement for our approach is more, because, unlike simulated annealing, GA keeps some of the previous solutions.

3 158 D. Saha et al. / Computer Communications 22 (1999) Fig. 1. NSFNET s T1 backbone network (not drawn to scale) used for experimentation 1.3. Outline of the article In Section 2, a brief overview on the background of the proposed work is presented. In Section 3, the feasible virtual topology generation problem is subdivided into four subproblems, and each subproblem is discussed elaborately. In Section 4, we discuss how to use GA for virtual topology optimization. In Section 5, we discuss experimental setup and various results. Finally, in Section 6, we conclude the article with a brief discussion on future issues and possible improvement. 2. Overview of the work This work studies optical networks, exploiting wavelength multiplexing and optical switches in routing nodes, so that arbitrary virtual topologies can be embedded on a physical fiber network. In particular, WANs, such as NSFNET s T1 backbone network (Fig. 1, not drawn to scale), are considered so that the design can be targeted for nation-wide coverage. For the problem under consideration, we assume that the following inputs are given: 1. A physical fiber topology, where each node is equipped with WRS. 2. The number of wavelength channels carried by each fiber. 3. A traffic matrix. 4. The number of transmitters and receivers available at each node. The outputs that are to be produced are: 1. A virtual topology of lightpaths. 2. Wavelength assignment for the links of the virtual topology. 3. The sizes and configurations of WRS at intermediate nodes. Similar to [1] and [25], we have two independent objectives (optimality criteria [8]). These are: (1) Maximize throughput, and (2) Minimize network-wide average packet delay, for a given traffic matrix. We mainly concentrate on the first objective. We express throughput in terms of scale factor (scaleup) i.e., by what factor elements of a given traffic matrix can be multiplied so that the network can still carry the scaled up traffic. We only consider integer scale factor. The mathematical formulation of this problem is given in [1] and [25]. We consider any arbitrary virtual topology, which can be regular or irregular, for embedding into a physical topology. So, we overcome the problem of asymmetric traffic flow present in a regular topology. While optimizing throughput, we trade off simplified routing offered by regular topology, as it is not so important in this case. Unlike in [1,25], we consider a fixed number of wavelength channels permitted per fiber. We generate virtual topologies randomly. We first generate a tree, based on a random Prufer number [16]. Next, we add and/or delete edges from the tree until the randomly generated connected graph satisfies transmitter and receiver constraints at all nodes. For embedding a virtual topology into a physical fiber network, we develop a heuristic algorithm which considers the number of wavelengths per fiber as fixed. For assigning wavelengths optimally to various lightpaths, while not violating physical constraints, we use a

4 D. Saha et al. / Computer Communications 22 (1999) we encode these virtual topologies as strings and choose suitable fitness functions for the afore-mentioned optimality criteria (1) and (2) separately. Next, selection, crossover and mutation operations are repeatedly performed on the initial population, after choosing suitable control parameters such as mutation rate, crossover rate and population size. These operations are repeated until termination criterion is reached. The solution obtained at this point is taken as the optimal solution [26]. The major steps of the complete solution procedure shown in a flowchart of Fig Derivation of a feasible virtual topology Fig. 2. Outline of the approach slightly modified version of the heuristic algorithm presented in [5]. In [5], wavelengths are assigned to lightpaths in the decreasing order of traffic, but, in our approach, wavelengths are assigned to lightpaths in the decreasing order of number of disjoint lightpaths (i.e., lightpaths that do not share the same physical link). This slight modification is because of our assumption that the number of wavelengths available is not a constraint for our problem. Barring this little difference, our wavelength assignment algorithm is exactly similar to that of Zhang and Acampora [5]. So, we skip the discussion on the mathematical background of the algorithm in this article, and refer to either [5] or [26] for its detailed description. Similar to [1] and [25], for routing of packets in virtual topology, we use the well-known flow deviation algorithm [9,12]. This algorithm is based on the notion of shortest-path flows. It first calculates the linear rate of increase in delay with an infinitesimal increase in the flow on any particular channel and then uses these lengths for shortest path routing. We do not present the details of the algorithm in this article. It can be found in [9] and [26]. For optimization, we use GA [13 17,26]. We first generate a number of random feasible virtual topologies to be used as the initial population in the genetic method. Then, The optimal topology design problem has usually been formulated as an optimization problem [1] using principle from multicommodity flow for physical routing of lightpath and traffic flow on the virtual topology. This optimization problem has already been proved to be NP-hard in the literature [25]. As discussed earlier, in our approach, we first find a set of feasible solutions and then produce the best solution out of this initial population by GA. For the purpose of deriving a feasible solution, the problem is divided into four subproblems: 1. Determine a virtual connected topology so that every node of it satisfies transmitter and receiver constraints. 2. Route the lightpaths over the physical topology. 3. Assign wavelenghts optimally to various lightpaths. 4. Route packet traffic on the virtual topology. Each of them is solved by a suitable algorithm given in the following section Generation of arbitrary virtual topology The virtual topologies are generated randomly, where each topology must satisfy three constraints: (i) It must be connected i.e., there must exist at least one path between every pair of nodes, (ii) The outdegree of every node must be less than or equal to number of transmitters available at each node, and (iii) The indegree of every node must be less than or equal to number of receivers available at each node. An effective strategy is required to generate random topologies that satisfy all the above constraints. There will be a significant wastage of time if we follow a simple greedy heuristic such as: (a) generate an arbitrary topology, (b) check whether it satisfies all three constraints; (c) if yes, select it; otherwise reject it. In this case, the rejection rate may be very high. So we devise a technique, which generates a topology that automatically satisfies the first two constraints. Therefore, we need to check for the third constraint only. This is obviously faster. The technique is described next. As we require a connected topology, we can start with a tree that is a

5 160 D. Saha et al. / Computer Communications 22 (1999) connected graph. For our problem, we assume that outdegree (indegree) of each node in virtual topology is equal to the maximum number of transmitters (receivers) available at that node. This additional assumption not only simplifies the problem, but also helps find a good solution (because throughput is maximum when we consider maximum degree of nodes). But our genetic approach can also support solutions without this additional assumption [26]. Next, we discuss the Prufer number technique to generate an arbitrary virtual topology. We can associate Prufer number [16] with a tree in the following manner. Let T be a tree of N nodes. The Prufer number, P(T), is an (N-2) digit number where digits are numbers between 1 and N and are defined by the following algorithm: (1) Let i be the lowest numbered leaf (node of degree (1) in T. Let j be the node which is predecessor of i. Then, j becomes the rightmost digit of P(T). We build P(T) by appending digits to the right; thus P(T) is built and lead from left to right. (2) Remove i and the edge (i,j) from further consideration. Thus, i is no longer considered at all, and if i was the only successor of j, then j has become a leaf. (3)If only two nodes remain to be considered, Stop because P(T) has been formed. If more than two nodes remain, return to step (1). Conversely, we can generate a tree from a Prufer number via the following algorithm: (1) Let P(T) be the original Prufer number, and let all nodes, not part of P(T), be designated as eligible for consideration. (2) If no digits remain in P(T), then there are exactly two nodes, i and j, still eligible for consideration [as we remove a digit fromp(t)in step (3) below, we remove exactly one node from consideration, and there aren-2 digits in the originalp(t)]. Add (i,j) tot and Stop. (3) Let i be the lowest numbered eligible node. Let j be the leftmost digit of P(T). Add the edge (i,j)tot. Remove the leftmost digit from P(T). Designate i as no longer eligible. If j does not occur anywhere in what remains of P(T), designate j as eligible. (4) Return to step (2). So, we first generate a Prufer number randomly, Any digit in the Prufer Number is repeated at most one less than the number of transmitters available at the node corresponding to the digit. Then, an undirected tree is constructed from the Prufer number by the algorithm given earlier. We are representing the tree in adjacency matrix form. The 1s in adjacency matrix are made permanent. New edges are then added to the already constructed graph so that number of 1s in any row of adjacency matrix equals the number of transmitters available at that node corresponding to the row. The resultant topology that we get is clearly a connected topology that satisfies transmitter constraint at every node. Next, in order to verify the receiver constraint for each node i, we calculate EXCESS(i), where EXCESS(i) is defined as follows: EXCESS(i) ˆ Indegree of node(i) Number of receivers available at node(i) Indegree is calculated by counting the number of 1s available at ith column of adjacency matrix. Clearly, EXCESS(i) 0 means that node i has an indegree greater than the number of receivers available at that node EXCESS(i) ˆ 0 means that indegree of node i equals the number of receivers available at node i, while EXCESS(i) 0 means that indegree of node i is less than the number of receivers available at node i. So, for each node i having EXCESS(i) ˆ 0, we mark the 1s and 0s at ith column permanent. Next, in each pass, the nonpermanent node i having maximum positive EXCESS value and the node j having maximum negative EXCESS value are identified. Then, in the adjacency matrix, 1s that are in column i are exchanged arbitrarily with as many 0s that are not permanent in column j, keeping row fixed. This can be done at most min (absolute (EXCESS(i)), absolute (EXCESS(j))) times. After the operations are performed, 0s and 1s that are exchanged are made permanent. If no exchange operation can be performed, node j is marked temporarily deleted, and, from the rest nodes, another node having maximum negative value is considered. If all nodes having negative EXCESS values are temporarily deleted, then the current topology cannot be used further. Otherwise (i.e., if, at least, one exchange operation can be performed), all temporarily deleted nodes are undeleted. If, after exchange operation, EXCESS(i) becomes zero, then node is permanently deleted. Also, if EXCESS(j) becomes zero, then node j is permanently deleted. After EXCESS value of node i becomes zero, another node having positive EXCESS value that is not permanently deleted is chosen as i. If all nodes (including permanently deleted, temporarily deleted and undeleted) have EXCESS value either zero or negative, then the topology thus generated satisfies the receiver constraint. If the topology does not satisfy receiver constraint, then another random topology is generated and the operations discussed earlier are repeated. The complete details of the random feasible virtual topology generationcanbefoundin[26] Embedding of virtual topology to physical topology Embedding means mapping of virtual topology to physical topology without violating the constraints. Node exchange operations for embedding arbitrary virtual topology to physical topology is studied in [1,25]. This approach does not take into account number of wavelength channels permitted on a physical fiber as a constraint. But we

6 D. Saha et al. / Computer Communications 22 (1999) consider number of wavelength channels per physical fiber as fixed and develop one heuristic algorithm for embedding. This algorithm uses one heuristic evaluation function. For each successive pass of the algorithm, one link from virtual topology is taken and it tries to find the best path having minimum cost based on the heuristic evaluation function. The complete algorithm is as follows: (1) For each physical link i, calculate COST (i), where COST (i) ˆ Normalized_length (i). The normalized length of a link is calculated by dividing the length of the link by the length of the longest link. set NO_OF_VLINK_ASSIGNED(i) ˆ 0, where NO_OF_VLINK_ASSIGNED(i) indicates, at any point of the algorithm, how many virtual links have used the link i for embedding, thus far. (2) Amongst the virtual links (x,y) that are not yet embedded, find the one which, if embedded along shortest path between node x and node y in physical topology based on cost of all links that are not yet deleted, will take minimum number of physical links. In case of a tie, take that virtual link which will take less cost in embedding in the shortest path. Let this virtual link be (u,w). If embedding is not possible for all virtual links that are not yet embedded, then return FAILURE. Otherwise, virtual link (u,w) is embedded to the shortest path found in physical network. Also, NO_OF_VLINK_ASSIGNED is incremented by 1 for each link in physical topology that lies in the shortest path. (3) If, for any physical link i, NO_OF_VLINK_AS- SIGNED(i) equals number of wavelength channels permitted per physical fiber, then the link i, is considered deleted (i.e. not considered for next passes). (4) Revise COST of each physical link i that is not deleted as follows: COST(i) ˆ W1*NO_OF_VLINK_ASSIGNED(i) W2*Normalized_Length(i), where W1 and W2 are two weights (5) If no more virtual links remain for embedding, then return SUCCESS. Otherwise, go to step (2). The performance of this heuristic algorithm depends on the suitable choice of weights W1 and W2. The rationale behind cost revision operation is that as a virtual link uses a particular physical link for embedding, it becomes costlier so that unassigned virtual links try to avoid that particular physical link so long as a better path is found. The normalized length is used in evaluation function so as to take into account for propagation delay which increases with increase in length of link. The variable NO_OF_VLINK_ASSIGNED indicates feasible part of embedding solution. The shortest path is calculated by Dikjstra s shortest path algorithm [10]. An example of the implementation of this algorithm can be found in [26] Wavelength assignment Given an embedding, we can determine all the virtual paths that pass through a physical link. Next, we need to assign wavelengths to lightpaths in such a way that any two lightpaths passing through the same physical links are assigned different wavelengths. Assigning wavelength colors to lightpaths, so as to minimize number of wavelengths (colors) under the wavelength continuity constraint, reduces to the well-known graph coloring problem [7]. This problem is Np-complete, and the minimum number of colors needed to color a graph G (called chromatic number) is difficult to determine. A mathematical formulation of the problem can be found in [5]. In this article, we assume that number of wavelengths available is not a constraint. But our approach is, as far as possible, to minimize number of wavelengths used. We use a heuristic algorithm for wavelength assignment. This algorithm is, to a large extent, similar to the wavelength assignment heuristic given in [5], where one optical hop traffic is maximized. It is a greedy algorithm which attempts to assign each wavelength to as many connections (i.e. lightpaths) as possible without violating the physical constraints. Here, we use the same wavelength channel for two or more different conversations at the same time, without violating any physical constraint, known as wavelength reusing. The algorithm first generates connection link indication matrix [5]. In this matrix, each connection corresponds to one column (or row). In this algorithm, we use connection and column interchangeably. The algorithm is given below: (1) Generate the connection link indication matrix M ˆ m ij ; lm, where m ij ; lm ˆ 1, if virtual links ij and lm uses a common link, and 0 otherwise. Then, order the rows and columns in the nondecreasing number of 1s in it. (2) Initialize W ˆ 1, where W indicates the wavelength number. (3) Repeat steps (4) (8) until all links in the virtual network are assigned unique wavelength numbers. At last, go to step (9). (4) Assign the wavelength W to connection (row or column) i having smallest number oif 1s (5) Remove temporarily connection j (i.e. row and column j) with m k;j ˆ 1 for j ˆ k 1; k 2; ; N. (6) Assign the wavelength number W to the connection not removed from the matrix which is having the least number of 1s, say connection s, where s k. If such a s exists, set k ˆ s and go to step (5). (7) Otherwise (i.e. no such a exists), the rows and columns (i.e. connection) that were temporarily deleted from M, undelete them and remove the rows and columns corresponding to connection which have been assigned wavelength number W. (8) Set W ˆ W 1

7 162 D. Saha et al. / Computer Communications 22 (1999) (9) Set P ˆ W 1, return P where P is the number of wavelengths used. For details on the implementation of this algorithm, we again refer to [26] Traffic assignment Flow deviation method [9] has already been used for the traffic assignment problem for minimizing network-wide average packet delay in [1,25,26,28]. We also use the same flow deviation algorithm for traffic assignment in a virtual topology. The flow deviation algorithm requires one feasible starting flow, which can be iteratively improved. In [25], starting solution is found by just finding the shortest-path routing. We propose another algorithm for initial assignment where we first sort the node-pairs in the traffic matrix in descending order of traffic. Then, we take the node-pairs (i.e. source node and destination node) in the sorted order one by one and assign the traffic in the shortest path based on minimum number of links. If full amount of traffic cannot be assigned in the shortest path, then the maximum possible amount is assigned and those links that are fully saturated are marked deleted. The rest amount is repetitively assigned in shortest path using the remaining links. Clearly, this approach will tend to optimize throughput as it assigns greater amount of traffic with lesser number of links. For maximum scaleup [1] and [25], we first calculate the theoretical limit of maximum scaleup by dividing total capacity of the virtual network by total amount of external traffic. Next, any convenient search technique (such as binary search technique [11]) can be used to find maximum scaleup in the range 1 to theoretical limit of maximum scaleup [26]. Once we have solved these subproblems, we generate a number of feasible solutions to start with. The feasible soultions are then converted to strings. These are then optimized using the GA. 4. Optimal virtual topology generation using genetic algorithm GA [13 17] has been applied to various network optimization problems such as optimal communication spanning tree (OCST) problem [16], LAN design for delay minimization [27] and topological expansion of networks [29]. We employ GA to search through the space of virtual topologies where each topology satisfy the following constraints: 1. The topology must be connected. 2. Every node must satisfy transmitter constraint. 3. Every node must satisfy receiver constraint. 4. It should be possible to embed the topology to the given physical fiber topology for a given number of wavelength channels permitted per fiber. 5. The virtual topology must be able to support the given traffic matrix. For each virtual topology, maximum scaleup of traffic matrix that can be accommodated by the virtual topology is found out. As our goal is to design a virtual topology that can support the maximum traffic scaleup, we take the maximum scaleup of traffic matrix as the fitness value of a solution in the GA. We have developed our own encoding scheme [16] for mapping a virtual topology into a chromosome [13] which is then subjected to genetic operations [13]. The scheme has been derived out of the similar kind of works previously done by Palmer and Kershenbaum [16] and Elbaum and Sidi [27] Virtual topology encoding Each virtual topology is encoded using a string of N elements, where N is the total number of nodes. Again, each element can assume integer values from 1 to N. A similar encoding scheme has been used in [16] for encoding trees in Optimal Communication Spanning Tree (OCST) Problem. Thus, each virtual topology in the encoded form is represented as concatenation of N segments, where segment i (1 i N) contains directed adjacent nodes of node i. In other words, the complete string is composed of N segements numbered 1 to N, where each number (i.e., segment) corresponds to one unique node. There are k (k 0) entries in segment i (1 i N), if the node i has directed edges to k other nodes (i.e., node i has k adjacent nodes or neighbors). In other words, node r is present in pth segment, if there exists a directed edge (i.e. virtual link) starting from node p and ending at node r. Thus, r can be any number between 1 to N except p. The elements in a segment are placed in ascending order and no duplication of element is allowed within a segment. This representation assumes that outdegree of a node in a virtual topology is equal to the number of transmitters available at the node. So, for a particular problem, each string has the same format and length. For example, if there are 8 nodes in a network with 4 transmitters per node, a solution will be encoded as the following string (chromosome): [ ] As there are 8 nodes, the chromosome consists of 8 segments in order, with 4 entries per segment. So, in total, there are 32 entries. The first four entries in the string indicate that node 1 has directed edges to nodes 2, 4, 6 and 8. Similarly, entries between five and eight indicates that node 2 has directed edges to nodes 1, 3, 5, and 8, and so on. Thus, if total number of nodes in a network be N and maximum number of transmitters available at each node be T, then this encoding scheme is capable of representing N 1 C T N unique virtual topologies. So any feasible topology can be mapped into one and only one representation in this encoding scheme. This encoding scheme is unbiased as all feasible virtual topologies are represented by the same number of encoding. Using this encoding scheme, it is very easy to go back and forth between the encoded representation of a

8 D. Saha et al. / Computer Communications 22 (1999) Table 1 Initial population of virtual topologies Sequence number Virtual topology encoded as chromosomes (strings) Fitness value (maximum scaleup) , solution and its actual representation. So, this encoding scheme is relevant for our problem Other issues The initial population is chosen from a random global space that satisfies all the five constraints stated above. The fitness of a string is set to the maximum scaleup of the virtual topology corresponding to the string. The selection mechanism [13] that we have used is the roulette wheel selection scheme. Sigma truncation method of scaling is used to produce a good competition with other solutions [26]. We have used single point crossover [13]. The crossover rate is chosen as 0.6 because this rate has been found to give good solutions for our problems. After each crossover operation, the algorithm used earlier for satisfying receiver constraint in generating random virtual topology, is applied to satisfy the receiver constraint. Also, elements within a segment are placed in ascending order, if not already placed in order. These operations together serves as a polynomial time repair algorithm, in case an infusible virtual topology is generated. Mutation [13] is used as a secondary operator. In case of binary alphabet, mutation involves changing a 1 to 0 or vice-versa. But, in our case, the alphabet size is N. So, if the position that is to be changed has an initial value j, then the changed value can be any number from 1 to N except j and i where the considered position is the ith segment. Very low mutation rate (0.01) has been chosen. After each mutation operation, the same polynomial time repair algorithm (as used for crossover operation) is used to satisfy receiver constraint and also to order the elements within a segment. In spite of using repair algorithm, if a topology still remains infusible, then we set the fitness value of the corresponding string to 0. So, next selection operation will automatically, reject it. We have chosen a fixed number of generations as the termination criterion. In each generation, we always keep the best two solutions of the previous generation so that we do not loose the best solutions An example using genetic algorithm Below we are presenting one sample experiment on NSFNET s T1 backbone network (Fig. 1) with GA. We take population size, crossover rate, mutation rate, number of generations as 10, 0.6, 0.01 and 1 respectively. We encode the nodes in as shown in Table 10. Assuming 10 wavelength channels per fiber and both indegree and outdegree of each node as 4, we obtain ten following feasible virtual topologies (shown in Table 1) encoded as strings in the initial population. After scaling and selection operations, strings with sequence numbers 4, 3, 1, 5, 8, 5, 9, 9 are placed in a new sequence (i.e., 1, 2,,8) for the next generation, while the best two topologies i.e., strings with sequence number 8 Table 2 Cross over operation details String s sequence numbr from initial population 4&3 45 1&5 32 8&5 15 Crossover point

9 164 D. Saha et al. / Computer Communications 22 (1999) Table 3 Population after crossover operation Sequence number Virtual topology encoded as chromosomes (strings) Fitness value (maximum scaleup) (maximum scaleup 135) and 5 (maximum scaleup 131) are placed in positions 9 and 10 respectively. Thus, we arrive at a new selected sequence which is ready for crossover. For crossover operation, consecutive strings are paired up except the last two in the sequence. With 0.6 probability, crossover operation is performed as given in Table 2. After performing the crossover operation and running the repair algorithm, a new population is generated, which is tabulated in Table 3. The best two chromosomes have sequence numbers 4 (with maximum scaleup 142) and 9 (with maximum scaleup 135). They survive in new sequence as numbers 9 and 10 respectively. The sequence numbering of rest of the population is kept as it is, and the current population is subjected to mutation operation with 0.01 as mutation rate. Mutation operation is performed according to Table 4. After performing mutation operation and executing the repair algorithm, the new population generated is given in Table 5. In this way, a desired number of generations can be calculated before terminating the algorithm. For instance, if we stop using the genetic operation on the above sample population, then the best topology is found to give a maximum scaleup of Time complexity analysis We consider the following notations for the complexity analysis. Total number of nodes ˆ N Total number of physical links ˆ P Highest possible degree (indegree or outdegree) of a node in virtual topology ˆ K Maximum number of virtual links possible ˆ V ˆ K*N Theoretical limit of maximum scaleup of traffic matrix that can be supported by virtual topology ˆ T Maximum string length ˆ K * N Population size ˆ R Maximum number of generation ˆ G With the above notations, we now proceed to find the complexity. Time required to generate initial population ˆ O(RN 3 ) Time required to calculate fitness value for whole population ˆ O(R.KN 5 log 2 T) Time needed to calculate scaled fitness value of each string ˆ O(R) Selection operation take time ˆ O(R) Crossover operation on a pair (repair algorithm embedding fitness calculation) will take time ˆ O N 3 KNP KN 3 KN 5 log 2 T O KNP KN 5 log 2 T. Hence, crossover operation on the whole population for a generation will involve time O(R(KNP KN 5 log 2 T). Similarly, mutation operation on whole population will also involve time O(R(KNP KN 5 log 2 T). Table 4 Mutation operation details Chromosome sequence number Mutation site Initial value Changed value

10 D. Saha et al. / Computer Communications 22 (1999) Table 5 Population after mutation operation Sequence number Virtual topology encoded as chromosomes (strings) Fitness value (maximum scaleup) Hence, total time complexity required for GA allowing G generations is ˆ O(GR(KNP KN 5 log 2 T). Thus, we have reduced the optimization problem to polynomial time complexity. On the contrary, in case of exhaustive search, if we consider only virtual topologies of degree K, then total computation time required for just generating all topologies (ignoring embedding and scaleup calculation) would have been 0 N 1 C K N i.e. O(N KN ). So, it is infeasible to perform exhaustive search in real time for N greater than even 15 nodes. However, it is to be noted that, in almost every stage of the algorithm, the string needs to be adjusted so that it satisfies all the 3 constraints mentioned in Section 3.1. The adjustment given in Section 3.1. may not always yield a string that satisfies the 3 constraints. So, the given time complexities are valid only when the adjusting algorithm yields a valid solution. variations with actual ones is not ruled out. Barring this little discrepancy, the input data set is exactly identical with the dataset for the same problem in [1]. The capacity of each channel is taken as 45 Mbp/s. the queuing delay was calculated using a standard M/M/1 queuing system with a mean packet length bytes/packet. We assume infinite buffers at all nodes. Also, we assume number of transmitter and receivers available at each node is 4 i.e. degree of each node is assumed to be 4. We assume number of wavelength channels permitted per fiber as 10. Using an initial population of 100 and crossover rate 0.6, mutation rate 0.01 and number of generation as 100, the best virtual topology is found out. Table 6 Best virtual topology 5. Experiments and results 5.1. Experimental setup We have tested our algorithm on various design problems of varying input sizes and have found satisfactory results in all cases [26]. However, we present only one result here for the sake of brevity. The physical topology that we choose for this purpose is the NSFNET s T1 backbone [1,25]. The traffic matrix employed is taken from [1]. This will help us to compare our approach with that followed in [1]. The nodal distance used are geographical distances. As we have calculated distances from Oxford Atlas, so distance Source WA CA1 CA2 UT CO TX NE IL PA GA MI NY NJ MD Neighbors CA1, UT, NJ, MD TX, NE, GA, MD WA, UT, NE, GA NE, IL, PA, MI WA, GA, MI, MD CAI, PA, MI, MD CA2, CO, TX, IL CA1, NE, NY, NJ WA, UT, CO, NJ CA2, IL, NY, NJ CO, TX, PA, NY WA, TX, IL, PA CA2, GA, MI, NY CA1, CA2, UT, CO

11 166 Table 7 Total number of packets/virtual link D. Saha et al. / Computer Communications 22 (1999) Table 8 Results of our algorithm Virtual link Number of packets carried Parameter Physical topology Virtual topology WA-CA WA-UT WA-NJ WA-MD CA1-TX CA1-NE CA1-GA CA1-MD CA2-WA CA2-UT CA2-NE CA2-GA UT-NE UT-IL UT-PA UT-MI C0-WA CO-GA CO-MI CO-MD TX-CA TX-PA TX-MI TX-MD NE-CA NE-CO NE-TX NE-IL IL-CA IL-NE IL-NY IL-NJ PA-WA PA-UT PA-CO PA-NJ GA-CA GA-IL GA-NY GA-NJ MI-CO MI-TX MI-PA MI-NY NY-WA NY-TX NY-IL NY-PA NJ-CA NJ-GA NJ-MI NJ-NY MD-CA MD-CA MD-UT MD-CO Results Although the theoretical limit of maximum scaleup is 370 (which is calculated by dividing the maximum possible Maximum scaleup Queuing delay ms ms Propagation delay a ms ms Total delay ms ms Average hop distance Maximum link load % (WA-IL) % (UT-MI) Minimum link load 0% (WA-CA1) % (MI-PA) a As propagation delay computation requires length of links, this calculation is based on our measurement of node distance form Oxford atlas which may not be accurate. value of the total capacity of channels i.e Mbps by the total external traffic), we have reached upto 148 (against 106 of [1]) for the optimum virtual topology obtained by GA. Details of the optimum topology are given next. (i) Best found virtual topology: The best virtual topology is represented in Table 6. (ii) Embedding: The embedding of best virtual to physical network is given in Appendix A. (iii) Wavelength Assignment: The wavelength assignment in various virtual links is given in Appendix B. (iv) Feasible Traffic Assignment: One feasible assignment of traffic to the best found virtual topology, for scaleup equal to 148, is tabulated in Appendix C. Correspondingly, the total number of packets carried by each virtual link is listed in Table 7. It is to be noted from the above table that the maximum number of packets carried by any virtual link (42121 packets/s) is less than maximum capacity of packets/s which is calculated by dividing 45 Mbps by bytes/ packet Comparison of physical topology versus virtual topology For the above design, various experimental results, that we have found, are summarized in Table 8. In order to compare the values obtained by us with those reported in [1], the various parameters that are found in [1] are also tabulated in Table 9. Comparing Table 8 with Table 9, it Table 9 Results of algorithm (Mukherjee et al). Parameter Physical topology Virtual topology Maximum scaleup Queuing delay ms ms Average. Propagation delay ms ms Total delay ms ms Average hop distance Maximum link load 98% 99% Minimum link load 32% 71%

12 D. Saha et al. / Computer Communications 22 (1999) Fig. 3. Comparison of delay Fig. 4. Scaleup Fig. 5. Ave hop distance Fig. 6. Link load

13 168 D. Saha et al. / Computer Communications 22 (1999) Table 10 Encoding the nodes Node WA CA1 CA2 UT CO TX NE IL PA GA MI NY NJ MD Encoded as is clear that the maximum scaleup and average queuing delay that we have found are significantly better than that of [1] for the respective optimum virtual topology (Figs. 3 and 4). But average-hop-distance for both physical and virtual topologies as found in [1,25] is slightly (3% 10%) better than our results (Fig. 5). The maximum link load, however, is almost equal for both algorithms (Fig. 6). This clearly indicates that our GA outperforms the simulated annealing algorithm of [1] in terms of both delay and throughput (i.e., scaleup). Table Effect of scaleup on various parameters We have extensively studied the changes in network parameters, such as queuing delay, propagation delay and average hop distance, for both physical topology and virtual topology against different scaleup of traffic matrix. We note that the average queuing delay increases slightly with increasing traffic until the scaleup nearly reaches its maximum value, after which there is a sharp increase in the queuing delay. The propagation delay increases with increasing scaleup as more traffic is deviated away from the shortest path routes by the flow deviation algorithm. One interesting feature is that the average hop distance decreases as traffic load is increased. This is again because of the flow deviation algorithm which will deviate into longer links, thereby increasing the propagation delay encountered by a packet and decreasing average hop distance. objective is to maximize the scaleup that provides an estimate of the network throughput. We have used GA for optimization, and carried out various experiments on several test as well as real-life networks including NSFNET backbone network. The most original contributions of this work is the development of an approach to generate random initial Virtual Topologies (VT) satisfying three design constraints. The other significant contributions can be identified as: (a) A heuristic algorithm for embedding of VT in a Physical Topology (b) A slightly modified version of the algorithm in [5] for wavelength assignment (c) Used standard GA with previously proposed encoding of chromosomes to find a VT that has maximum throughput. Used (a) to initialize the population in GA and (b) and (c) in calculating the fitness values of chromosomes. This work can be further extended to find an optimal topology with respect to average packet delay, for a given scaleup of traffic matrix, by choosing fitness value as reciprocal of average packet delay. As the problem solved in this article is NP-hard [1] and [25], the optimal solution that are computationally intractable solution is the only way and the GA seems to be the best choice for its inherent capability of attaining globally optimal solutions. However, the smaller network one can use the mixed integer linear programming technique instead of GA because it may be a lengthy process for obtaining optimal solution. 6. Conclusion In this work, we have explored a set of design principles for next-generation optical wide-area networks, employing wavelength-division multiplexing (WDM). Our main Acknowledgements This work is supported in part by AICTE Career Award Grant No. F. 1-52/CD/CA(10)/96-97.