Multihop Virtual Topology Design in WDM Optical Networks for Self

Photonic Network Communications, 10:2, 199–214, 2005 © 2005 Springer Science+Business Media, Inc. Manufactured in The Netherlands.

Multihop Virtual Topology Design in WDM Optical Networks for Self-Similar Traffic Sujoy Ghose, Rajeev Kumar∗ , Nilanjan Banerjee, Raja Datta Department of Computer Science and Engineering, Indian Institute of Technology Kharagpur, Kharagpur, WB 721302, India Email: [email protected] Received April 23 2004; Revised March 14 2005; Accepted March 18, 2005

Abstract. In this paper, we consider the problem of designing virtual topologies for multihop optical WDM networks when the traffic is self-similar in nature. Studies over the last few years suggest that the network traffic is bursty and can be much better modeled using self similar process instead of Poisson process. We examine buffer sizes of a network and observe that, even with reasonably low buffer overflow probability, the maximum buffer size requirement for self-similar traffic can be very large. Therefore, a self-similar traffic model has an impact on the queuing delay which is usually much higher than that obtained with the Poisson model. We investigate the problem of constructing the virtual topology with these two types of traffic and solve it with two algorithmic approaches: Greedy (Heuristic) algorithm and Evolutionary algorithm. While the greedy algorithm performs a leastcost search on the total delay along paths for routing traffic in a multihop fashion, the evolutionary algorithm uses genetic methods to optimize the average delay in a network. We analyze and compare our proposed algorithms with an existing algorithm via different performance parameters. Interestingly, with both the proposed algorithms the difference in the queuing delays, caused by self-similar and Poisson traffic, results in different multihop virtual topologies. Keywords: WDM, multihop, virtual topology, lightpaths, self-similar, RWA, evolutionary algorithm (EA)

1 Introduction Due to the limitations in the number of wavelengths per fiber, it may not be possible to connect all the node-pairs of a large optical network with direct lightpaths. As such, a considerable amount of traffic may have to hop a few lightpaths to reach their destinations, giving rise to multihop optical networks. Fig. 1 shows an example of a virtual topology formed with seven lightpaths established with two wavelengths. As there is no direct optical connection for example between node 1 and node 8, data from node 1 can travel to node 8 in two optical hops (from node 1 to node 4 in wavelength Λ1 and from node 4 to node 8 in another wavelength Λ2 . It has been shown that the problem of virtual topology design can be subdivided into four subproblems [1]. They are: (1) determining a good virtual topology, i.e., which node ∗ Corresponding author.

should be optically connected to which node, (2) routing the lightpaths over the physical topology, (3) optimal assignment of wavelengths to the lightpaths, and (4) routing packet/cell traffic on the virtual topology. The subproblems (2) and (3) define the routing and wavelength assignment (RWA) problem in optical networks which has been shown to be NP-hard [2]. This RWA problem is studied by many researchers in the literature and many heuristics/methods exist for obtaining approximate solutions in polynomialtime [2–5]. The subproblems (1) and (4), which determine the node pairs that should be connected by direct lightpaths, and routing of the packet/cell traffic, form an important part of the design. By properly choosing the node-pairs for lightpath connection and routing the packet/cell traffic, we can reduce the average packet/cell delay of the network. Therefore, the above subproblems are not independent in nature and the problem as a whole is NP-hard [1,6]. The virtual

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Fig. 1. An example of a multihop optical network. A eight-node virtual topology formed by establishing optical connections with two wavelengths Λ1 and Λ2 .

topology design problem can thus be formulated as an optimization problem where the objective function can be the minimization of networkwide average packet/cell delay that consists of both propagation and queuing delays. Researchers of optical networks have attempted such problems of designing virtual topologies and have obtained solutions using heuristics/methods in polynomial time [1,7–10]. Mukherjee et al. in [1] formulated the virtual topology design problem as a nonlinear optimization problem where the objective was minimization of average network delay. In [10], Banerjee and Mukherjee formulated the virtual topology design problem as a linear program where the queuing delays were intentionally ignored in the formulation. They were of the opinion that the queuing delays are negligible with respect to the propagation delays when the load per channel is reasonably low and cited the results obtained in [1] as reason to neglect the queuing delay. The results were obtained considering the queuing delay calculated using standard M/M/1 queuing systems. Whereas, it has been shown that for both local-area and wide-area network traffic, the distribution of packet/cell inter-arrivals clearly differs from exponential. In the past few years, studies of high-resolution traffic measurement from different working communication networks have provided ample evidence that actual network traffic is selfsimilar in nature [11–13]. Schwefel et al. [14–16] showed that the buffer sizes and queuing delay in a network calculated using self-similar traffic model is much higher than that using the Poisson model. Our work is motivated by the fact that,

the queuing delay considering self-similar traffic model is substantially higher than that considering Poisson model, and intuitively this difference will influence the design of virtual topology. In this paper, we consider both the queuing delay and the propagation delay of a network while designing a virtual topology and propose two different approaches for solving the problem. We first propose a Greedy algorithm that greedily finds the connection with least delay from an initial topology constructed using first-fit wavelength assignment policy. As evolutionary algorithms are emerging as a good alternative for solving hard optimization problems, we next propose an Evolutionary algorithm that tries to optimize the average delay of a network. The algorithm uses a hybrid routing and wavelength assignment policy for the initial topology and then acts upon it in the evolutionary way to construct the final topology. We construct virtual topologies where the network queuing delays have been calculated using both Poisson and self-similar traffic model. Interestingly, the results obtained are quite different in these cases which validate our thesis that the virtual topology design will be different when we consider self-similar traffic. We analyze and compare the performances of our proposed algorithms with an existing heuristic algorithm proposed in [10] which is found to be closer to our work. The rest of the paper is organized as follows: In Section 2, we review some previous work on virtual topology design, self-similar traffic and the use of evolutionary algorithms in optical network design problem. We then give expressions for estimating the queuing delays and the buffer


sizes in a network for the self-similar traffic in Section 3. Herein, we also examine the impact of self-similar traffic on the buffer sizes of an optical network. In Section 4, we discuss the delay based optimization and the formulation of the problem. We propose a greedy algorithm for the design of virtual topology in Section 5. In Section 6, we describe a suitable model for the representation of virtual topology and its implementation with an evolutionary algorithm. Numerical results are reported in Section 7 and the conclusions are given in Section 8.

2 Previous Work 2.1 Designing Multihop Virtual Topology Many researchers have worked on the construction of optimal structures based on minimizing the maximum link flow and optimizations based on the minimization of the mean networkwide packet delay. Labourdette and Acampora [6] formulated the flow and wavelength assignment problem, when minimizing the maximum flow in the network, as a mixed integer linear program subject to linear constraints. The virtual topology design problem has been formulated as an optimization problem in [1] where the objective functions are (a) minimization of network-wide average packet delay for a given traffic matrix, and (b) maximization of the scale factor by which the traffic matrix can be scaled up (to provide the maximum capacity upgrade for future traffic demands). The problem was formulated as mixed integer linear programming (MILP). Showing the problem as NP-hard, they proposed one heuristic for the virtual topology design based on simulated annealing and another heuristic for traffic flow optimization based on flow deviation algorithm. A major drawback of simulated annealing is that it will be computationally expensive for designing a large network. Bannister et al. [7] proposed the mean network-wide packet delay as an objective function for designing optimal virtual topology. The authors argued that in a high-speed environment, where the channel capacity C is large and the link utilization are expected to be in the light to moderate range, the queuing delay component can be ignorable compared to the propagation

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delay. A general linear formulation which considers routing traffic demands and routing and wavelength assignments as a combined optimization problem has been presented by Krishnaswamy and Sivarajan in [8]. In addition to usual constrains (such as number of available wavelengths per physical link, number of transmitters and receivers per node, underlying physical topology etc.), the paper considered the “maximum number of hops a lightpath is allowed to take” as an important parameter in optimizing virtual topology. In [17], Kumar and Kumar proposed new Integer Linear Program (ILP) formulations for maximizing network traffic for uniform and non-uniform traffic patterns. The paper also proposed two heuristic algorithms for large networks with uniform and non-uniform traffic.

2.2 Self-Similar Traffic Studies of high-resolution traffic measurement from different working communication networks have provided ample evidence that actual network traffic in Internet is self-similar or fractal in nature [11–13]. Network arrivals are often modeled as Poisson processes for analytic simplicity. However, a number of studies with the Internet traffic has shown that both local-area and wide-area network inter-arrival packet distribution clearly differs from exponential [18–20]. Leland et al. [21], in their paper, showed that the LAN traffic is better modeled using statistically Self-Similar processes. Paxson and Floyd in their now famous paper [11] showed how the Poisson model grievously underestimates the burstiness of the wide-area TCP traffic. They analyzed every type of network connections (TELNET, SMTP(e-mail), NNTP(network news), FTP(File transfer) etc.) and offered results to show how it relates to the self-similarity of wide-area traffic. Crovella and Bestavros [22] showed that the self-similarity in world wide web traffic is due to the underlying distributions of its document sizes, the effects of caching and user preference in file transfer, the effect of user “think time” and the superimposition of many such transfers. Since a self-similar process has observable bursts on all time scales, it exhibits long-range dependence (LRD), i.e., values at any instant are

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typically correlated with all future values. The authors observed that WWW file transmission times appear to show heavy-tailed characteristics. A distribution is heavy-tailed if regardless of the behavior of the distribution for small values of the random variable, the asymptotic shape of the distribution is hyperbolic. After lot of analysis with different formats of traffic they came to the conclusion that this heavy tail is primarily due to the distribution of the web file sizes. The “think time” and caching are the reasons for increase in the tail weight of the data traffic. The bursty nature of the Internet traffic has been shown to affect the queuing delay by Schwefel et al. in [14,15]. Schwefel in his Ph.D thesis [16] gave an excellent performance analysis of intermediate systems serving aggregated On/Off traffic with LRD properties. The thesis provided a thorough discussion of the N-Burst On/Off traffic model. A special family of MatrixExponential (ME) distributions—called Truncated Power-Tail distributions—has been used for the On time distribution in order to mimic LRD properties, while still remaining tractable for queuing analysis via Matrix-Analytic methods. Schwefel developed adequate procedures for parameter estimation to make the model applicable to realistic sources. He then applied the estimated values to a set of actual data from measurements of intercell times in an IP-over-ATM network. The results of the steady-state analysis of the queuing models with LRD properties showed marked difference in behavior than the models without LRD properties (we discuss more about it in Section 3.)

2.3 Use of Evolutionary Algorithm in Optical Networks Evolutionary algorithms (EA)/genetic algorithms (GA) have been extensively used in optimization problems for many communication network related optimization problems. For example Abuali et al. [23] assigned terminal nodes to concentrator sites to minimize costs while considering maximum capacity. In [24], Kumar and Banerjee proposed a multiobjective evolutionary algorithm to solve multi-criteria multi-constrained network design problems. The multi-criteria included overall cost, average per packet delay, reliability and provision for expansion of the designed network.

A solution to mesh network design using genetic algorithm approach was presented by Ko et al. in [25]. The method has shown effectively optimized routing and capacity assignments while optimizing network topologies. There has been a number of attempts to use genetic algorithms for designing optical networks as well. In [26], Gazen and Ersoy investigated the optimization of logically rearrangeable multihop lightwave networks using genetic algorithms. They considered regular rearrangeable topologies and the objective was to minimize the largest traffic flowing on any link. The problem was divided into two independent problems: the connectivity problem and assignment problem. The genetic algorithm worked only on topologies without considering the optimization of flows. The individuals of the genetic algorithm’s population were graph topologies. Saha et al. in [27] used a genetic algorithm to generate and map an optimal virtual topology onto a wavelength-routed all-optical physical network. Basically the work was an extension to [1] where the authors used genetic algorithm in place of simulated annealing to achieve better throughput and less delay. They also allowed irregular virtual topologies and included the wavelength constraint in their work but mainly concentrated on the objective of maximizing throughput. The queuing delay was calculated assuming the Poisson model. The authors generated virtual topologies randomly by first generating a tree, and then adding/deleting edges from the tree until the randomly generated connected graph satisfies transmitter and receiver constraints at all nodes. They then used a heuristic to embed a virtual topology into a physical fiber network. The wavelengths were assigned using a heuristic similar to [3] and feasible traffic-flow were assigned in already established lightpaths using a flow-deviation algorithm before finding optimum solution using a genetic algorithm. Yang et al. [28] used a multi-objective genetic algorithm based a methodology for optimizing multi-service convergence in a metro WDM network. They considered arrayed-waveguide grating (AWG) based single-hop WDM networks and developed a genetic algorithm to solve the multi-objective optimization problem of maximizing throughput and minimizing delay. Few

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reconfiguration algorithms for multihop lightwave topology based on meta-heuristics were presented in [29] by Kato and Oie. They considered regular topologies with uniform node degrees for reconfiguration. Xin et al. [30] proposed a set of heuristic algorithms to address the combined problem of physical as well as logical topology design. They used a genetic algorithm to design large scale networks of around 1000 nodes. Bisbal et al. [31] in a recent paper proposed a genetic algorithm for solving the dynamic routing and wavelength assignment problem in wavelength-routed optical networks. It is aimed at achieving low call blocking probability while keeping the computation time short. The authors also proposed an extension of the algorithm to provide fault tolerance capability at the optical layer.

3 Queuing Delay and Buffer Size Estimation for Self-Similar Traffic For self-similar traffic, traffic bursts appear on a wide range of time scales. The paper [11] shows that WAN arrival processes appear better modeled using self-similar processes. Among the many models multiplexed On/Off traffic with LRD properties is one of the models for the actual real data traffic. Hans-peter Schwefel in his Ph.D thesis [16] developed techniques for the analysis of queuing models with such a model as input. In this section we summarize the queuing delay and buffer size estimates with bursty traffic that we have used in our work in designing the virtual topology. For additional details readers may refer [15] and [16]. In the On/Off models the traffic is generated only during On periods and each On period is followed by an Off period during which no traffic is transmitted. The distribution of the duration of the On periods can be any ME distribution. In the N -burst Independent Source Model, cells from N On/Off sources are multiplexed together. If κ is the average cell-rate of the On and Off times together and λ0 is the rate of the background Poisson traffic, then the average cell rate λ for N sources is given by λ = Nκ + λ0 . Let λ p be the peak rate at which the cells are generated during the On period (bursts) of a single source.

Then the burstiness parameter b can be defined as b = 1 − λκp . Long-Range Dependent (LRD) properties have been found in a large number of measurements of network traffic. Schwefel’s work is distinguished with the inclusion of LRD - properties in the N burst On/Off model. LRD effects can be achieved when Power- Tail (PT) distributions are introduced either for the duration of the On or Off periods or both. Such PT distributions are characterized by a complimentary distribution function that drops off very slowly by a Power Law with exponent α > 0. But as PT distributions do not have a finite-dimensional ME representation, it cannot be integrated in the N -burst model. But as there are physical limits on the file sizes that are transferred over the networks, the Power-Tail behavior of the On time distributions can be expected to be cut off at some point. Also, the busy hours of a network, for which the performance modeling is of particular interest, is limited to a maximum of 8 hours a day. Therefore, one can do away with the Powertail distributed On periods with durations longer than those busy periods. Thus, truncated PowerTails (TPT) distributed On times is found to be more reasonable. For the N-burst model with TPT distributed On times, the auto-correlation can be computed numerically in the Markov Modulated Poisson Process (MMPP) representation [16]. 3.1 Queuing Delay Estimation with Self-Similar Traffic If µ is the service-rate and ρ = µλ the utilization, then for M/M/1 queue, the mean delay of ρ . But this observation does cells increases as 1−ρ not hold any more if LRD properties are introduced into the traffic by the use of TPT distribution. Instead, peculiar jumps called blow-ups can be observed. For λ0 = 0 the blow-up region i 0 is given by 1−ρ 1−b · i0 = N . ρ b Schwefel introduced another parameter i as follows i = i0 − N .

1−ρ 1−b . ρ b

which has a range 0 ≤ i < 1. For larger i the performance is worse as the queuing model

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operates closer to the blow-up region. PowerTail distributed burst-lengths can cause very long over-saturation periods in the N -Burst/M/1 queue. Those over-saturation periods are caused by simultaneously long-term active sources and they have a major impact on the performance behavior. It has been shown that the tail exponent β (where β = i 0 (α − 1) + 1) of the duration of the over-saturation period determines the queuing behavior. In addition to this, the tail-constants of Power-Tail is also necessary for quantitative results. As exact formulae for computation of the tail-constant of the queue length distribution of the N -Burst/M/1 model are unknown, bounds for related scenarios can be used as approximations for the true values. An approximation for an upper bound of the tail constant CmC D can be obtained as follows: cm P D ≈

1 ρ i · · λ 1 − ρ i 2−β 0 2 i 0 −1 × n β−1 · p b (1 − b)

i0 [c(1) P T α] (2 − β)(α − 1)i0

where n p is the mean number of cells per burst (1) and c P T (α) is the tail constant of normalized PT distribution. The Maximum Burst Size (MBS) is also an important quantity for deriving the queuing delay of N -Burst traffic model. Schwefel defines the Maximum Burst Size (MBS) as the approximate Power Tail (PT) Range of the distribution of the number of cells. The bursts with more than MBS cells only happen with very small probability. Therefore, if the burst-length distribution is truncated at MBS cells, the asymptotic behavior of mean cell delay can be given by mC D(M B S) ∼ cmC D M B S 2−β Hence, by choosing realistic values of MBS and (1) c P T (α), one can estimate the mean cell queuing delay for bursty Internet traffic. This estimate of delay will be much closer to the real value than that calculated with the Poisson model. In this paper, we have taken n p = 11.1 which are results from standard T X 3 sources [16] and assumed M B S = 2.1 × 105 cells.

3.2 Buffer Size Estimation with Self-Similar Traffic The optimal buffer size in a node of a network can be determined according to some QoS requirements such as Buffer Overflow Probability (BOP). Therefore, while estimating the buffer size, full power-tails for the On time distribution is assumed i.e., power-tails without any truncation. With a target BOP, the buffer size (B) for the bursty traffic can be estimated from the equation [16]: B≈

c

BO P

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BO P

where c B O P is the tail constant and can be approximated from the equation as given below: i0 bβ (1 − b)i0 −1 (i n p )α−1 (1) c P T (α) cB O P ≈ 1−ρ α−1 Choosing realistic values of the various parameters mentioned in the equations above we can estimate the approximate buffer size that will be required in the network nodes for a target buffer overflow probability. We have estimated the maximum buffer sizes required in a node for the optical virtual topology formed over the 14-node NSFNET backbone network (Fig. 6) with a heuristic presented by Banerjee and Mukherjee in [10]. The capacity of a lightpath in the virtual topology was assumed to be 6000 cells/ms and traffic between the nodes were generated by a Gaussian distribution function with mean = 800 cells/ms and std.dev. = 50. Fig. 2 shows the maximum buffer sizes required in the network with respect to different target BOP. It is interesting to note that the maximum buffer sizes that may be required in a node with self-similar traffic, when we consider full power-tails with a buffer overflow probability of 10−7 , is in the order of 108 cells. This is much larger compared to the maximum buffer sizes calculated with the Poisson model, which can be in the order of tens or hundreds of cells at the most. The requirement of the maximum buffer size decreases from the order of 108 cells to an order of 103 cells when the buffer overflow probability is increased from 10−7 to 10−5 . Fig. 2 also shows that the maximum buffer sizes decrease with the increase in number of wavelengths, as more number of optical paths are

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Fig. 2. Maximum buffer sizes required with different buffer overflow probabilities (BOP) in the virtual topologies constructed over NSFNET with mean = 800 and std. dev. = 50 traffic.

Fig. 3. Maximum buffer sizes required in the virtual topologies constructed over NSFNET with high, moderate and low traffic for buffer overflow probability (BOP) = 10−7 .

created, thereby handling more traffic in the optical domain. We have also estimated the requirement of maximum size of buffer in a node varying the volume of self-similar traffic in the network (Fig. 3). Three different traffic patterns were used keeping the buffer overflow probability as 10−7 . The traffic between the nodes were generated with the Gaussian distribution function taking mean = 1000 cells/ms with std. dev. = 20 for high traffic, mean = 800 cells/ms with std. dev. = 50 for moderate and distributed traffic, and mean = 200 cells/ms with std. dev. = 10 for low traffic. It is observed that the maximum buffer size decreases from the order of 108 cells to the order of 106 cells as the traffic is varied from high to low. Therefore, for self-similar traffic the buffer size is more a factor of the burstiness rather than the volume of the track. From the above discussion we find that the buffer size can be very large in case of self-similar model even for reasonably low traffic. This has a significant impact on the queuing delay of the network which cannot be neglected. We now re-investigate delay based optimization design of virtual topology considering self-similar traffic and total delay (propagation and queuing delays).

physical topology (fiber interconnection pattern) G = (V, E) which is represented by an undirected graph, where V is the set of network nodes, and E is the set of links connecting the nodes. The number of wavelengths carried by each fiber is W . The two components that form the network packet delay are (a) propagation delay and (b) queuing delay. The propagation delay is due to the packet/cell traveling from source to destination through the intermediate nodes in a network. The queuing delay is due to queuing of packets/cell in the intermediate nodes. We briefly discuss here the formulation of the optimization problem. The readers may refer to the paper [1] for details from which it is extracted with two modifications. In case of traffic routing we have assumed that traffic from a source to a destination node cannot be bifurcated, i.e., all the packets/cells follow the same route. This is due to the fact that, in GMPLS, which is an emerging technology for wide area optical networks, traffic is supposed to take the same route from the source to the destination. The second modification is in the queuing delay, which is calculated using both self-similar bursty traffic and standard M/M/1 queuing model (for the purpose of comparison).

4 Delay Based Optimization In this section we formulate the problem of virtual topology design by optimizing the delay in the network. The network structure is a given

4.1 Formulation for Delay Based Optimization The variables for the formulation are b(i, j), isdj , ij

Pmn and ck (i, j). b(i, j) = 1 denotes the exis-

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tence of a lightpath from node i to node j in the virtual topology; b(i, j) = 0 otherwise. Traffic flow from node s to node d employing b(i, j) as an intermediate virtual link is denoted by ij isdj . The variable Pmn = 1 denotes the existence of the fiber link Pmn (physical link from node (i, j) m to node n) in the lightpath b(i, j); Pmn = 0 otherwise. ck (i, j) = 1 denotes that the wavelength k is assigned to the lightpath from node i to node j; ck (i, j) = 0 otherwise. Here k = 1, 2, 3, . . . , W , where W is the maximum number of available wavelengths per physical link. Therefore, the optimization can be stated as follows: Given the physical topology, the traffic matrix, the distance matrix (dmn ) and the capacity (C) of each lightpath, minimize the network packet/cell delay given by ij

isdj

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mn

sd

+Q

where Q is the queuing delay component and υ is the velocity of light in a fiber. The constraints being: (a) s,d isdj ≤ b(i, j) ∗ C, i.e., the total traffic flowing through lightpath (i, j) cannot increase its capacity. (b) k ck (i, j) = b(i, j), i.e., the same wavelength is used throughout the lightpath (wavelength continuity constraint). ij (c) i j Pmn · ck (i, j) ≤ 1 ∀ m, n, k., i.e., no two lightpaths in a physical link can be assigned the same wavelength. The constraints that are implicit to an optical network (e.g., constraints regarding receivers, transmitters, physical links and traffic flows) are same as in [1] and not shown here. The queuing delay using standard M/M/1 queuing model (Poisson model) [1,32,33] is given by ij

sd

isdj

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5 The Greedy Algorithm In this section we present a greedy algorithm that constructs a virtual topology considering the constraint of the number of wavelengths per fiber and also the average delay of the network. If we consider any node in the path of a source-destination pair, the idea is to find the path with the minimum delay from that node to the destination node, considering all the traffic that can flow through the concerned links between any nodepair in the path. We now describe the algorithm, which has two phases for establishing lightpaths and routing of traffic. The First Phase: In this phase the algorithm establishes the lightpaths according to the firstfit wavelength assignment policy. Initially, the number of wavelengths considered is one less than the available number of wavelengths per fiber. The remaining wavelength is utilized later to ensure connectivity. The wavelengths are numbered and the node-pairs are sorted according to the decreasing order of traffic between them. Now, each sorted node-pair is taken and a direct lightpath (i.e., single hop optical connection) is tried between them. The path chosen is the shortest path and the first available wavelength is assigned to the connection. There may be quite a few node-pairs that cannot be given a direct lightpath due to wavelength constraint. The process is continued till all the node-pairs are tried. The lightpaths established form the initial virtual topology that is not guaranteed to be a connected topology. Therefore, to ensure that the constructed virtual topology is connected, we establish a lightpath for each edge E in G with the remaining wavelength. We now run the second phase of the algorithm to find single or multihop connections between every pair of source and destination. The Second Phase: The second phase performs a least-cost search on the path delay for each of the source-destination pairs whose traffic must be routed in a multihop fashion. We use the following variables to describe the algorithm:

ij

C − sd sd

For the network queuing delay considering selfsimilar traffic, we use the equations given in Section 3.1.

(i) newsource and parent indicating network nodes. (ii) A set S of nodes whose elements are the variables indicating network nodes.


(iii) delayx,y and newdelayx,y where x and y are variables indicating network nodes. [delayx,y represents the delay (propagation + queuing) of the traffic as it flows from node x to node y]. The Algorithm: FOR all source-destination pairs (Si , Di ) in G for which there is no lightpath { newsour ce = φ; S = {Si }; parent of (Si ) = φ; WHILE newsource = Di { sort the set S in ascending order of delay; remove first element of S and assign to newsource; IF N = {N1 , N2 , . . . , Nk } are neighbors of newsource, then S = S ∪ N ; FOR all neighbor {N1 , N2 , . . . , Nk } of newsource { IF Ni ∈ S { newdelay Si ,Ni = min delay Si ,Ni , delay Si ,newsource + delaynewsource,Ni ; IF delay Si ,Ni > delay Si ,newsource +delaynewsource,Ni { par ent (Ni ) = newsour ce; delay Si ,Ni = newdelay Si ,Ni ; } } ELSE { delay Si ,Ni = delay Si ,newsource +delaynewsource , Ni ; par ent (Ni ) = newsour ce; } } } construct path by tracing parent from Di to Si ; } The above algorithm produces the least delay multihop path for a given source-destination pair. However, since the optimal order for considering the source-destination pairs remain unknown the overall algorithm may give sub-optimal result.

6 The Evolutionary Algorithm We now propose a Evolutionary Algofithm which is a hybrid genetic algorithm where the initial population is not entirely random. The approach

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generates an initial population based on the kshortest paths for a given source-destination pair. This approach is different from the initial encoding in Saha et al. [27], where an initial virtual topology is generated using Prufer sequence technique. 6.1 Initial Population and Encoding A hybrid approach is used for the initialization of the population. For every source-destination pair the k-shortest paths connecting them are evaluated using Yen’s algorithm as in [34–36]. Each gene in a chromosome represents one of the k-shortest paths selected randomly. Every gene in the chromosome has a pointer to an entry in a look up table which contains the actual path. Thus, a single chromosome contains a set of plausible paths for all the source-destination pairs. The structure of the chromosome for k = 6 is depicted in Fig. 4. Hence, if there are n source destination pairs the total length of the chromosome is log2 k × n bits. 6.2 Description of Evolutionary Algorithm Used We use a two phase simple single objective fitness based evolutionary algorithm to find the single hop and multihop connections between all the given source-destination pairs in the network. In the first phase we aim to assign single hops to as many source destination pairs as possible. Hence, following an initial encoding as described above we try to find the set of plausible paths between the given source-destination pairs. The fitness of an individual chromosome is evaluated based on the number of wavelengths required to assign those paths. Therefore, the chromosome which finds paths for the maximum number of source-destination pairs under the given constraint on the number of wavelengths will be the fittest. At the end of the first phase a partial virtual topology is constructed by assigning single hop links to the source-destination pairs for which paths has been found by the GA under the constraint on the wavelength. However, to make the topology connected, the links from the physical topology (which are not in the partial virtual topology) are inserted into the partial virtual topology. This ensures that the virtual topology is connected as the physical topology is always connected.

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Chromosome Encoding : Each Gene points to a path in corresponding SourceDestination Table Fig. 4. Example of the chromosome structure for k = 6.

In the next phase another population is generated on the partial virtual topology created above using the same initial encoding technique described above. Now, we aim to find single or multihop connections between various sourcedestination pairs such that the average delay of the network is optimized. Thus, the chromosome which depicts the connection with the minimum delay is selected as the fittest. Therefore, at the end of this phase we have all the single and the multihop connections between all the source-destination pairs in the given network. We search the objective space using the conventional evolutionary operators of mutation and crossover.

Fig. 5. Example of a 4-point crossover for 5 source destination pairs.

6.3 Evolutionary Operators Crossover: A crossover mechanism is adopted in this work so that if a path is assigned to a particular SD pair, during crossover the identity of the path is maintained in entirety. Thus, a (n − 1) -point crossover is used for a chromosome where n depicts the number of connections to be established. The crossover process is illustrated in Figure 5. If a random uniform crossover is used instead of the (n − 1)-point crossover, the evolutionary material in the chromosome from the previous iteration will be lost during the search procedure. Thus, such a crossover helps in speeding up the convergence process. The individuals for crossover are chosen on the conventional roulette wheel selection scheme where the fitness is assigned by interpolating between the best

individual (whose fitness is least) to the worst individual (whose fitness is largest) according to a simple monotonic function which maps it to the roulette wheel. After crossover an additional individual is created in the population. If the population size is µ, then the first µ individuals are selected according to their best fitness values. Hence, the individual with worst fitness is discarded. The probability of crossover in a particular iteration is set to 1.0. Mutation: A random uniform mutation with a probability of pm = 1s is used, where s is the length of a chromosome. In mutation the created chromosome replaces itself regardless of the fitness function. The random uniform mutation helps in crossing hamming cliffs and hamming valleys which might be inherent in the

ORIGINAL CHROMOSOMES : 00010010111100111000 01100010000001111010 4 Point Crossover Chromosome 1 ;

0001 0011 1111 0011 1000

Chromosome 2 :

0110 0010 0000 0111 1011

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problem. Moreover, it also reduces the chances of the algorithm getting stuck in a local optimum. The individual with the worst fitness function is selected for mutation. This is done with the idea to produce a fitter individual from a bad individual in the population.

6.4 Wavelength Assignment The evolutionary algorithm finds the individuals with the best fitness. However appropriate wavelengths need to be assigned to each connection. The wavelength assignment to the fittest individuals is done using the Brelaz heuristic [37] which is based on a well-known graph algorithm. The algorithm initially generates an undirected graph for the connections, where the nodes denote the connections to be made. There is an edge between two vertices only if there is an edge common between the paths of the two connections. In the graph so formed the vertices are colored using the minimum number of colors using the following scheme: 1. Arrange the vertices of the graph in decreasing order of degrees. 2. Color a vertex of maximal degree with a color (say color 1), which corresponds to the wavelength number 1. 3. Choose a vertex with a maximal saturation degree. The saturation degree of a node is the number of colors to which it is adjacent. If there are two vertices with the same saturation degree, choose any vertex of maximal degree in the uncolored subgraph. 4. Color the chosen vertex with the least possible (lowest numbered) color. 5. If all the vertices are colored, stop. Else go to step 3.

7 Simulation Results We consider a traffic model where the long term average (relative) traffic demand is given by an N × N traffic matrix T . Here N is the number of network nodes, and the (i, j)th element, ti j the long term average traffic demand from Node i to Node j. The diagonal elements of the matrix are zero. Though the average traffic demand between

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any two nodes may be asymmetric, i,e., ti j = t ji , for simulation we assume symmetric traffic matrices by considering the larger traffic from either side. This is due to the fact that a lightpath established between any two nodes is expected to carry traffic for both ways. Three types of traffic matrices are used in the simulation. These are generated with the help of a Gaussian distribution function taking mean = 1000 with std. dev. = 20 for high traffic, mean = 800 with std. dev. = 50 for moderate and distributed traffic, and mean = 200 with std. dev. = 1.0 for low traffic. The traffic between the nodes thus generated are considered to be in cells/ms. We also assume that the capacity of a lightpath is 6000 cells/milliseconds (approximately 2.5 Gbps) and that the establishment of lightpaths will not be blocked by the lack of receivers and transmitters at the nodes. We have simulated results by constructing virtual topologies over the 14-node NSFNET physical network using Poisson and self-similar traffic models. Fig. 6 shows the 14-node NSFNET backbone network with approximate distances between the nodes in kilometers. The virtual topologies are constructed using the greedy algorithm presented in Section 5 and the evolutionary algorithm presented in Section 6. While the standard M/M/1 queuing system has been used for estimating the queuing delays and buffer sizes of the network for the Poisson traffic model, we have used the equations of Section 3 to estimate these quantities for the self-similar traffic model. To compare the performances of both the algorithms proposed in this paper, we have selected a heuristic presented in [10] by Banerjee and Mukherjee. Though the heuristic does not optimize the delay in a network, yet it is nearest to our work presented in this paper. The heuristic initially connects all the physical links using a wavelength. It then calculates the product of the multihop traffic from each source to destination and the number of hops in the path (preferably the shortest path). This is done for all possible source-destination pairs; from all such values it chooses the source-destination pairs with the largest value and tries establishing a lightpath along the minimum hop distance between the nodes. For the nodepairs where direct lightpaths cannot be established, the shortest path (in number of lightpaths) is used to transfer data.

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Fig. 6. The fourteen-node NSFNET backbone network showing approximate distances between the nodes in kilometers.

The plots in the figures of this section are named as follows: G ss and G p : Results with the Greedy algorithm presented in Section 5 using the self-similar model and the Poisson model, respectively. E ss and E p : Results with the Evolutionary algorithm presented in Section 6 using the self-similar model and the Poisson model, respectively. Hss and H p : Results with the Heuristic algorithm of [10] using the self-similar model and the Poisson model, respectively. We analyze the results obtained with the above three algorithms using three parameters, namely average queuing delay, maximum buffer size and the percentage of single-hop traffic in the network.

7.1 Average Queuing Delay The average queuing delays in the network with the Greedy, Evolutionary, and the Heuristic [10] algorithms for Poisson and Self-similar traffic models are shown in Figs. 7 and 8, respectively. We observe that, while the average queuing delay of the network considering Poisson model is less than 2 milliseconds in most cases (Fig. 7), it can be almost five times higher when the traffic is self-similar in nature (Fig. 8). The heuristic algorithm of [10] shows much higher queuing delays using the self-similar traffic model compared to the Poisson model as it constructs the virtual topology independent of the network delay. From the figures it can be seen that the results improve in the cases of Greedy and

Evolutionary algorithms, because these algorithms optimize the topology considering both propagation and queuing delays. The virtual topology constructed with the Greedy algorithm shows high average queuing delays for both Poisson and self-similar models when the number of wavelengths is 4, but it improves as the number of wavelengths increases. This is because, the virtual topology initially constructed with the firstfit wavelength assignment policy which does not necessarily optimize delay in the network. As the number of wavelengths increases, more singlehop connections are established thereby reducing the delay. We also observe that when estimated with the standard M/M/1 queuing model all the three algorithms give low (almost negligible) and similar queuing delays (Fig. 7), whereas, it is not the case with self-similar traffic model (Fig. 8). This shows that though algorithms independent of queuing delays (or neglecting queuing delays) can be used effectively for designing a good virtual topology with Poisson traffic model, for bursty or self-similar traffic it is necessary that the algorithms consider queuing delays of the network.

7.2 Maximum Buffer size In our discussion in Section 3.2, we showed the estimated maximum buffer sizes required in virtual topologies constructed over NSFNET with the multihop design heuristic proposed in [10]. In Fig. 9, we show the maximum sizes of buffer required in an intermediate node of NSFNET

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Fig. 8. Average queuing delay for self-similar traffic with different algorithms.

when the virtual topologies are constructed with our two proposed algorithms Greedy and Evolutionary and compared it with that of the earlier heuristic. The traffic matrix has been generated by a Gaussian distribution with mean = 800 and std. dev. = 50. BOP considered here is 10−6 . We observe that there is a considerable reduction in the maximum buffer sizes with our proposed algorithms as compared to the heuristic of [10] even as the number of wavelengths is increased from 4 to 16. This is due to the reduced average delay achieved in the virtual topologies constructed with our proposed algorithms. For the Greedy algorithm we find that there is no requirement for buffer in the intermediate nodes with 16 wavelengths because with these many wavelengths the algorithm achieves 100% direct traffic from source to destination.

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Fig. 9. Maximum buffer size required in NSFNET with self-similar traffic for virtual topologies constructed with different algorithms.

7.3 Single-Hop Traffic By single-hop traffic we mean traffic traveling from the source node to the destination node through a single lightpath i.e., without any optical/electronic/optical (O/E/O) or wavelength conversion. Figs. 10 and 11 show the percentage of total traffic that can be carried in single-hop achieved with the Greedy and Evolutionary algorithms, respectively, compared with that of the earlier heuristic. We have shown here the plots for a mean = 1000 and std. dev. = 20 traffic matrix. The traffic in single-hop increases with the increase in the number of wavelengths as more direct lightpaths can be connected. It is interesting to observe that the single-hop traffic is different for self-similar and Poisson traffic models when the virtual topology is designed with our proposed algorithms. It is to be noted that for the heuristic of [10] the single-hop traffic would be same for both the cases of traffic as the virtual topology is constructed without considering the delays in the network. From both the Figs. 10 and 11, it can be observed that the virtual topology designed using self-similar traffic achieves better single-hop traffic than the Poisson traffic model. One reason for this could be the higher queuing delay encountered in the nodes while using the self-similar traffic model which helps the delay based algorithms to optimize the virtual topology in a better way. This is more so with the Greedy algorithm, as the decisions regarding routing of the lightpaths are made in the intermediate nodes encountered in the path

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model, will be more effective in handling the bursty Internet traffic.

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Fig. 11. Percentage of the total traffic in single-hop with the evolutionary algorithm compared to the heuristic algorithm of [10].

of the algorithm. It is further observed that in most of the cases about 100% single-hop traffic is achievable with 16 wavelengths per fiber. From the above discussion it follows that, when we construct virtual topologies with the help of delay based optimization algorithms, apart from the queuing delays and the buffer sizes, the resultant traffic that travel in single-hop is also quite different for self-similar and Poisson models. This difference in the single-hop traffic indicates that the virtual topology is different for these two models which is a result of the differences in their queuing delays. Therefore, we conclude that the virtual topologies constructed with the help of the self-similar model, which are different from that constructed with the Poisson

In this work, we have considered self-similar traffic for designing a multihop virtual topology. We show that the bursty nature of self-similar traffic influences the design of such topologies for WDM networks. We have also proposed two different algorithms: Greedy and Evolutionary for delay-based optimization design of optical virtual topologies. With the help of the algorithms we designed a number of virtual topologies, varying the traffic patterns and the number of wavelengths per fiber. The performance of the Greedy algorithm, though better than the traffic independent algorithms, is limited by the fact that the initial virtual topology is constructed with the first-fit wavelength assignment policy. On the other hand, the evolutionary algorithm, due to its evolutionary characteristics, constructs virtual topologies that gives steady and reduced average queuing delays as well as a high percentage of single-hop traffic in a network. The simulation results show that virtual topologies designed considering self-similar traffic are quite different from the virtual topologies designed using the standard Poisson model and therefore more effective in handling the present-day bursty Internet traffic.

Acknowledgment We thankfully acknowledge help of Ashok Turuk in preparation of the final manuscript.

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Nilanjan Banerjee received the B.Tech. degree in Computer Science and Engineering from the Indian Institute of Technology (IIT), Kharagpur, India in 2004. He is presently a graduate student in the department of Computer Science, University of Massachusetts, Amherst. His research interests include multiobjective genetic optimization, high speed networks and mobile systems. Raja Datta received the B.E. degree in Electronics and Telecommunications Engineering from Regional Engineering

College, Silchar, India, in 1988 and the M.Tech. degree in Computer Engineering from the Indian Institute of Technology, Kharagpur. Currently, he is working towards the Ph.D. degree at the Indian Institute of Technology. Since 1990, he has been a faculty member of North Eastern Regional Institute of Science and Technology (NERIST), Itanagar, India. His research interests include optical WDM networks. Sujoy Ghosh received the B.Tech. degree in Electronics and Electrical Communication Engineering from the Indian Institute of Technology (IIT), Kharagpur, India in 1976, the M.S. degree from Rutgers University, Piscataway, NJ, and the Ph.D. degree from the Indian Institute of Technology, Kharagpur. He is currently a Professor in the Department of Computer Science and Engineering, Indian Institute of Technology, Kharagpur. His research interests include design of algorithms, artificial intelligence, and computer networks. Rajeev Kumar is an Associate Professor of Computer Science and Engineering at Indian Institute of Technology (IIT), Kharagpur, India. Prior to joining IIT, he worked for Birla Institute of Technology and Science (BITS), Pilani and Defence Research and Development Organization (DRDO), India. He received his Ph.D. from University of Sheffield, and M.Tech. from University of Roorkee (now, IIT - Roorkee) both in Computer Science and Engineering. His research interests include QoS and multimedia systems, multiobjective optimization and evolutionary algorithms, programming languages and type system, and software tools for embedded system design. He is a member of ACM, senior member of IEEE and a fellow of IETE.