DESIGNING AN OPTIMAL VIRTUAL TOPOLOGY IN WDM BASED TOROIDAL OPTICAL NETWORKS

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1 DESIGNING AN OPTIMAL VIRTUAL TOPOLOGY IN WDM BASED TOROIDAL OPTICAL NETWORKS Dhivya Muralishankar Department of Computer Science North Carolina State University Raleigh, North Carolina Abstract This paper introduces a new approach of designing an optimal virtual topology for Wavelength-division-multiplexed (WDM) based Toroid Optical Networks that support unicast traffic, with a static traffic pattern. A virtual topology is created by setting up a collection of light paths that maybe embedded in a given physical topology. Traffic is routed over the light paths in an optimal manner such that the delay of transmitting data from a single source to destination is reduced. Previous work deals with designing optimal virtual topologies over path, ring or bus topologies. This paper, however, deals with designing an optimal virtual topology to reduce the number of hops while routing over a toroidal topology by introducing a Super node concept. This concept helps limit the number of hops, under any situation, to a maximum of three and also causes aggregation of traffic. The design problem is studied taking into consideration the wavelength continuity constraint and sub-wavelength routing. Index Terms- Optical Networks, WDM, Toroidal topology, Heuristics. I. Introduction: The technology of Wavelength division multiplexing plays an essential role in networking as new developments come to improve existing components and introduce new solutions. It offers the capability of building very large wide area networks comprised of several thousands of nodes.in fact, it is believed that WDM will form the backbone for the next generation Internet.As traffic volumes continue to increase drastically, WDM based optical networking applications are gaining a lot of prominence owing to numerous properties such as high bandwidth, low cost and scalable data services. Wavelength Routed optical networks permit the exploitation of the huge fiber capacity and do not require complex processing functionalities in the optical domain. In these networks, each fiber is divided into multiple channels each of which operates at a different wavelength. In an all-optical network, a single wavelength maybe used to establish communication between a source and destination, but such connections aren t realistically easy to provide because they require many dedicated optical paths and this is difficult to achieve. In WDM Optical Networks, high capacity routers are connected to each other through what are called light paths that extend over several physical links. A light path maybe defined as a path comprising channels on a single wavelength through which packets are routed to their destinations. Within a light path the transmission of data is entirely optical (i.e. intermediate nodes will allow packets to pass through in the optical domain itself).it is only where one light path ends and another begins, that conversion of the light path to electronic domain for extracting and processing packets and then back to optical domain is required (OEO Conversion). A single light path alone is not sufficient to connect every source destination pair in the network. This is because (i) the number of light paths beginning and terminating at each node is limited depending on the number of transmitters and receivers and the processing capabilities of electronics (ii) Degradation might occur if the length of the light path is too much. Thus it is necessary to establish a virtual topology instead. A virtual topology maybe defined as a collection of light paths linking the source to the destination. It is independent of the physical topology. In a wavelength division multiplexed Optical network, the virtual topology whose vertices are routers and the edges are light paths are mapped onto the physical topology whose

2 vertices are the optical cross connects and paths are the optical fibers. In order to fully exploit the capacity of WDM based optical networks, it is necessary to design the most optimal virtual topology for any given traffic pattern. Generally optimization of a topology maybe based on several different criteria such as minimizing use of resources like electronic components used (transceivers/receivers add/drop multiplexers wavelength converters), minimizing the delay, minimizing the congestion etc. In [2], the authors have suggested a novel Light-tree approach for multicast traffic that has been formulated using Mixed Integer Linear programming based on the criteria of minimizing the number of transceivers used and the average II. Previous Work Done Several researchers have developed and analyzed optimization models for designing an optimal virtual topology for a given physical topology. This is considered an intractable problem and is NP complete. The approaches to this problem can be broadly classified as Heuristic in nature or as Integer Linear Programming problems. While integer linear programming allows optimal solutions, they require exponential time to compute.on the contrary, heuristics are fast but the results aren t guaranteed. Thus it is necessary to establish theoretic bounds, so that the results of the heuristic methods maybe compared with these bounds to decide whether they are optimal. In [1], the authors provide a comparative study of the various greedy heuristic and metaheuristic methods of designing optimal virtual topologies for multicast traffic with the optimization criteria of finding the maximum congestion levels in the network. Greedy heuristics namely Source Copy Multicast Algorithms and Route and Remove algorithms are used either to generate initial solutions that maybe used by the meta heuristic methods or in situations where time constraints do not permit computationally intensive calculations. Metaheuristics such as Tabu search and simulated annealing work on the initial solutions generated by the Greedy heuristic methods. The experimental results obtained from various traffic scenarios, as shown in Fig.1 proved that the greedy methods produced results similar to that of Best Random which refers to the lowest experienced maximum congestion levels among all random topologies and the meta-heuristic methods performed better than the greedy methods under the same circumstances. Both cases provided more desirable values of traffic congestion levels than the measured average congestion levels. Fig.1. Maximum Congestion level normalized to the best solution for various traffic scenarios packet hops distance between source and destination. Aneja [3] presents a mixed integer linear programming approach to design a virtual topology and route traffic with the objective of minimizing the network congestion while restricting the average propagation delay between source and destination pairs taking into consideration the degree of the logical topology. A bi-criteria linear program is used.the optimal solution is found at either extreme point of a path with a delay that is less than or equal to or a convex combination of two paths with an average delay equal to.the author has proven that in an optimal routing of traffic, every traffic carrying path with a delay that is greater than can be paired with another traffic carrying path whose delay is less than thus achieving the desired objective of requiring that delay on every traffic carrying path should be limited to. Hartline et al. [4] have proposed efficient algorithms (based on polynomial time approximation techniques) to design an optimal virtual topology for one to much communication so as to minimize the maximum or average number of light paths from source to destination and thus reduce the delay that is caused by OEO conversions. This solution is applicable to path and ring topologies. Jeykumar et al [5] have proposed a genetic algorithm to design virtual topologies with delay constraints. This algorithm, based on heuristic approaches, is proven to produce better results than the conventional heuristic approaches because of its generate-and-test approach that provides a large set of potential solutions that can be tested against certain fitness criteria such as cost,

3 reliability and Delay success probability to produce the most optimal solution. Dutta and Rouskas [6] provide a framework to evaluate the performance of heuristic methods based on a general formulation of the virtual topology problem. A sequence of upper and lower bounds are considered. These bounds are based on decomposing the ring topology into sets of nodes arranged in a path and adopting the locally optimal VT within each set. The main objective in this case is to minimize electronic routing. The results show that decomposition is a considerably more tractable problem than solving the complete problem by itself. In [7] Gerla et al. present the bidirectional shuffle net topology with wormhole routing, which is a very popular logical topology for telecommunication networks, and interconnection networks. In the optical domain the shuffle net application is used to provide endto-end light paths in a physical topology given the physical topology and a set of constraints. They provide a framework where bidirectional shuffle nets can provide reductions in the number of distinct wavelengths required in the physical topology. Finally, the problem of designing an optimal virtual topology has been studied in various other networks such as ATM. Although previous work has produced excellent results to design optimal virtual topologies for multicast and one-to many communications based on several different optimization criteria, they are applicable only to the topologies such as paths, rings, stars, bus etc. This paper, however, aims at designing optimal virtual topology based on minimal hop counts while routing on a toroidal topology. In general, the following steps are involved in designing an optimal virtual topology.the first step involves providing the physical topology and traffic pattern as the input parameters. Also required as input are the optimization constraints. The optimal virtual topology thus obtained by implementing the algorithm is then mapped on to the physical topology. Finally the traffic on the optical networks is studied to verify whether the given constraints are adhered to. The rest of the paper is organized as follows. Section 3 provides an overview of the toroid topology.in Section 4, the problem that this paper specifically deals with is stated formally. In Section 5, a meta-heuristic approach to reduce the number of hops and thus design an optimal virtual topology for toroidal networks, based on the constraints described in Section 4, has been presented. III. Overview of Toroid Topology In this paper, the physical topology that is input to the system is an N*M nodes toroidal topology. Mesh topologies usually have ``wraparound'' connections, e.g. the node at the top of the grid has an ``up'' link that connects to the node at the bottom of the grid. If one visualizes only north-south links in a rectangular mesh, it can be seen that these links turn the 2D mesh into a 3D cylinder. Now if the east-west links are added, it connects the ends of the cylinder to form a toroidal solid. Thus a mesh topology with wraparound connections is often referred to as a TtorusT. Fig. 4 A toroid is a product of circles. In mathematical terms, a torus [Fig, 4] is a doughnut-shaped surface of revolution generated by revolving a circle about an axis coplanar with the circle. Fig. 5 Manhattan Street Network

4 When an n-dimensional grid network is connected circularly in more than one dimension, the resulting network topology is a torus, and the network is called "toroidal". All nodes lie at the points of intersection between the vertical and horizontal rings. This topology when laid out will resemble a Manhattan Street Network [Fig.5], a two connected regular network topology with unidirectional links. The figure represents a 6*6 node network. Thus the toroidal topology considered in this approach is also assumed to be two connected with unidirectional links. IV. Problem Formulation The network is modeled as a directed graph G= (V, E) where V refers to vertices and E refers to edges. In the general formulation of a virtual topology design problem, the inputs provided are the existing physical topology over which the optimal virtual topology is to be mapped, the optimization criteria such as delay constraints or permitted traffic congestion levels or number of electronic components that can be used etc. The general approach to solving a design problem in wavelength division multiplexed network is to break up the problem into sub problems and solve each independently. One fundamental sub problem for the design of wavelength routed WDM networks is the Routing and Wavelength Assignment Problem (RWA). Based on [8], we can formulate the RWA sub problem as follows: Consider a request R=[u, v] for a connection from node u to node v. This request is satisfied by: (i) Assigning a path P(R) consisting of a sequence of links from G that (ii) connects u to v. Assigning a wavelength W to the path. Since we define the toroid as a product of many rings, it is necessary to consider both (i) and (ii) while designing a toroidal topology. If R is a set of requests say {R1, R2 Rn}, then the RWA problem should satisfy all requests in R using minimum number of wavelengths. For any two requests in R, say R1 and R2, the paths assigned say P1 and P2 respectively, which share a common edge, are assigned two different wavelengths W1 and W2 respectively. The RWA problem is found to be NP hard. The traffic is assumed to be static in nature. Hence its is assumed that no requests for light paths or electronic traffic components such as transceivers will be received Thus dealing with a dynamic traffic scenario is out of scope in this paper. Generally in an all-optical network, wavelength converters are used to convert from one wavelength to another within a network, but such converters are considered very expensive. In the optical network considered in this problem, no wavelength converters are allowed. Thus a light path traveling on a given wavelength continues to travel on the same assigned wavelength from the starting node to the terminating node on all physical links along its path. No conversion by intermediate nodes is allowed. This paper also considers the problem of traffic grooming, which allows independent traffic streams to share the bandwidth of a light path. The toroidal topology considered in this paper is considered to have M columns and N rows. All nodes are represented as a pair (i, j) where i represents the order of the horizontal ring and j represents the order the vertical ring. Each node lies on the intersection of a horizontal and vertical ring and thus the naming convention. Given: (i) The physical topology of N*M nodes. (ii) The traffic matrix T=[t(s, d)] where s, d are elements of {0,1 (MN-1); t(s, s)=0; t (sd) element of {0,1 }. The traffic matrix consists of MN*MN elements. Let: (a) Number of wavelengths in a link: W (b) Lij: The light path-count (of integer value) from a node i to a node j. (c) Lij (l, m): Number of light paths traversing the physical link from l to m. (c) cⁿij (l, m) is the link light path wavelength indicator which takes up the value 1 if a light path uses a wavelength n over the physical link from l to m or assumes the value 0 otherwise. (d) n ha a domain of {0,1, W-1} (e) C is the bandwidth of a single wavelength channel. (f) t (s, d) is the amount of traffic that needs to flow between nodes s and d, which collectively forms the traffic matrix T. (g) p (i, j) is the link indicator which assumes the value 1 if a physical link exists in the physical topology

5 Bhave between nodes i and j and assumes the value 0 otherwise. For a toriodal topology, this indicator function will hold good for the following cases: (i) p (i, j) is 1 for each pair of successive nodes i and j in each individual row where which fall within the range (xm-m+1) and xm, for each value of x where: x represents the row number. M represents the number of columns. Also: xm has a link to xm-m+1 Eg: If M=4, i=2, x=1 Then Node 2 can have a link to node 3 as both 2 and 3 fall in the range (xm-m+1) and xm, i.e. (between 1 and 4) (ii) p (i, j) also holds good for links between i and j=i+m. Eg: Node 2 is linked to node 5. The light paths from I to j denote the amount of traffic from node I to j and tij(s, d) denotes the amount of this traffic that is due to the demand t(s, d). The domain for each i, j, l, m, s, d is {0, 1, 2 MN-1} where MN is the total number of nodes in the physical topology. Find: The Optimal Virtual Topology such as to: Reduce the number of hops from source to destination by causing aggregation of traffic. Subject to: Constraints: (i) Physical Topology Constraints (ii) Light path Routing constraints (iii) Light path Wavelength assignment SP Constraints (iv) Traffic Routing SP Constraints The Integer Linear Programming (ILP) formulation for these constraints can be obtained from [6]. V. Approach Since we have defined a toroid to be a product of circles, the first approach in designing an optimal VT for a toroidal network is as follows: Algorithm 1: Step 1: Decompose the toroidal topology into several ring topologies, both in horizontal and in vertical directions. In this case, we have divided the topology into M vertical and N horizontal rings with M*N elements, at the points of intersection of the nodes. Step 2: Apply well-known ring heuristics to each individual ring to obtain the optimal light paths and the corresponding costs of the individual ring. The cost consists of the cost required for traffic routing, light path routing, wavelength assignment and mapping to the physical topology. The working of ring heuristics is considered as a black box and it is assumed that the results are obtained. Step 3: Design a meta-heuristic algorithm to route traffic from a source node to a destination node. For the above step, three different traffic scenarios B to be considered, (h) When the source and destination rings lie on the same ring. (i) When the source lies on a horizontal ring and the destination lies on a vertical ring and viceversa. (j) When the source and destination lie on two different horizontal rings or on two different vertical rings. Routing: Fig. 6 Routing of traffic from source to destination on the same horizontal ring For the first case where the source and destination nodes lie on the same ring, the ring heuristics will return the optimal light paths. Thus routing is simple as seen in Fig.6. However, for the second and third case, we introduce the concept of a Super node, for each horizontal ring i, such that all the traffic that

6 flows out of a node on a ring i: (whether horizontal or vertical) to a destination node on another ring flows only via the Super node. The purpose behind this is to limit the degrees of separation between source and destination to a maximum of three. Super node Concept The Super node of each ring is chosen at random. Traffic from all nodes on the same horizontal ring first flows to the corresponding Super node that in turn will direct traffic to along the vertical ring that it lies on, to the node directly below it, which lies on the same horizontal ring as the destination. Let (0,3), (1,2), (2,5), (3,1), (4,0), (5,4) be the randomly chosen Super nodes of horizontal rings 0, 1, 2, 3, 4 and 5 respectively. Thus for ring 0, if all nodes namely (0,0),(0,1), (0,2), (0,3), (0,4) and (0,5), the traffic to all other rings should flow only via (0,2), the designated Super node for ring 0. Similarly, the same pattern of sending traffic via the designated Super nodes applies to the other rings too. Fig.7. Randomly chosen Super nodes on each horizontal ring for routing between nodes that lie on different rings. Let us consider a simple example of routing from a node (1, 0) located on horizontal ring 1 to a node (5, 5) located on ring 5. Since the designated Super node for ring 1 is (1, 2), traffic from (1, 0) flows to (1, 2) and (1, 2) directs the traffic along the vertical ring 2 to the node locate directly below it (5, 2) which lies on horizontal ring 5, the same as the destination ring. Thus the hop count is maintained as three. (Fig. 8) Fig. 8 Routing of traffic via Super nodes Aggregation of traffic Since all the data from the rings directed to nodes on other rings have to pass through the Super node, aggregation of traffic occurs. In this section, the algorithm used to extract traffic from the MN*MN toroid traffic matrix is explained. The algorithm takes the physical topology (the number of nodes and the links) and the MN*MN traffic matrix as input. The output, in case of routing between a source and destination located on the same horizontal ring, is extracted directly from the traffic matrix. However, when the Super node concept is used, the traffic from the source to destination should also be added on to the traffic along the paths (hops) through which it flows. Thus, with respect to the example explained in Fig.8, when traffic is routed from (1,0) to (5, 5) via (1, 2) and (5, 2), the traffic from (1, 0) to (5, 5) should also be added on to the traffic flowing between (1,0) and (1,2), (1, 2) and (5, 2) and (5, 2) to (5, 5). The algorithm is designed to extract the values of source to destination traffic out of the toroid traffic matrix and add the traffic values to the original values of traffic along the paths through which the traffic is now routed. The following is the pseudo code of the algorithm. Algorithm 2: The following notations are used in the algorithm. Let N be number of horizontal rings and M be number of vertical rings. For each horizontal node, select a random Super node DBi. Thus there are N numbers of Super.B nodes selected.

7 Communications 01, Proc. Let each node be represented as (i, j) where I is the number of the horizontal ring on which it is located and j is the number of the vertical ring on which it is located. Inputs: MN*MN traffic matrix of toroid where each element t(s, d) represents the traffic of source to destination. Note that the traffic from node to itself will always be 0, and thus the diagonal of this MN*MN matrix will be all 0 s. Step 1: Consider the source and destination nodes between which the traffic is to be routed. Routing: Check if the i value of source == i of destination. { If true, then both nodes lay on the same horizontal node i. Thus the routing requires only a single hop. The ring heuristics will itself provide optimal routing on the ring. From the given traffic matrix, locate the element that corresponds to the traffic value between source and destination and output. } Else { Check the super node from the super node sets to a match between the i value of the source and the i value of the Super node. Once the Super node is located, direct traffic from source to Super node. Check the values of all nodes in the nodes array to find a match between the j value of Super node and the j value of the node. The set of all nodes lying below the Super node along the same vertical ring are returned. Let it be a set MatchNodes. For each element in MatchNodes set, check the i value of the nodes to find the one that matches with the i value of the destination node. Select the node returned from the matching process. Let it be K. Direct all source to destination traffic from source node to the new selected K. K now lies on the same horizontal ring as the destination node. Traffic is routed from K to the destination node as in any normal ring, based on the optimal paths returned by the ring heuristics. From the traffic matrix, extract the traffic value for source to destination. Let it be z Add z to the traffic values of source to Super node. Add z to traffic value of Super node to K. Add z to traffic value of K to Destination. Output the new traffic matrix value. The above algorithm has been implemented to perform routing and traffic extraction and updation but the numerical values have not yet been collected. However, it has been proven that the degree of separation between source and destination is maintained as one in case where the source and destination lie on the same horizontal ring and a degree of three where the source and destination lie on different rings, irrespective of the number of rings that the toroid topology is divided into. VI. Conclusion Although the facts that under certain situations, traffic from all nodes on a horizontal ring flowing through the Super node of that ring simultaneously could cause a bottle neck at the Super node or if the Super node fails, then none of the nodes on the horizontal ring could route to rings located on other horizontal rings, has been acknowledged, the solutions for these problems has not been considered within the scope of this research. These issues can be addressed in future work. This approach can be adopted while designing a toroid topology not only for sole purpose of maintaining the maximum hop count while routing as three but also for ease of configuring each ring in the toroid network. In this approach, it is enough if each node on a horizontal ring has routing information of nodes on the same horizontal ring as itself. Only the Super node of each ring need to be configured with routing intelligence to other rings in the network. References: [1] Mellia.M, Nucci, A., A. Grosso, E. Leonardi, M. Ajmone Marsan, TOptimal Design of Logical Topologies in Wavelength-Routed Optical Networks with Multicast TrafficT, T Of IEEE TGlobe COM T San Antonio, TX, pp , TT TTNov [2] TSahasrabudde, L.H.; TMukerjee, BT.; TLight treest: Optical Multicasting for improved performance in wavelength routed Tnetworks, IEEE T Magazine, 37(2), p.67-73, Feb 1999T T[3]TT TTAneja, Y.PTTTTU. UTT, TRouting in wavelength routed optical networks,t 2001 IEEE Workshop on High Performance Switching and Routing (IEEE Cat. No.01TH8552), pgs , T [4] Hartline J.R.K, Drucker et al. Optimal Virtual Topologies for One-to-many

8 Communications ion WDM Paths and Rings, IEEE/ACM Transactions on Networking, 12(2), Pg ,April [5] TJeyakumar, A.E T; TBaskran, K. T; TSumathy, TV., Genetic algorithm for optimal design of delay bounded WDM multicast networks, IEEE TENCON Conference on Convergent Technologies for the Asia-Pacific Region (IEEE Cat. No.03CH37503), 2003, pt.3 (3), p , May 2003 [6] Dutta, Rouskas, Traffic Grooming in WDM Networks: Past and Future, IEEE Network, 16(6), Pg 46-56,Nov-Dec [7] Mario Gerla, Emilio Leonardi, Fabio Neri, Prasasth Palnati, TRouting in the bidirectional shufflenet, TT IEEE/ACM Transactions on Networking (TON), 9(1), Feb [8] Dutta, Rouskas G.N, On Optical Traffic grooming in WDM Rings, IEEE Journals on Selected Areas in Communications, 20(1), Pg , Jan 2004.

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