Splitter Placement in AllOptical WDM Networks

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1 plitter Placement in AllOptical WDM Networks HwaChun Lin Department of Computer cience National Tsing Hua University Hsinchu 3003, TAIWAN hengwei Wang Institute of Communications Engineering National Tsing Hua University Hsinchu 3003, TAIWAN Abstract In alloptical WDM networks, splitters at branch nodes are used to realize multicast trees. It is expensive to place splitters at all of the nodes in an alloptical WDM network. To reduce the cost, splitters can be placed at a subset of nodes. The problem of selecting a subset of nodes to place the splitters such that certain performance measure is optimized is called the splitter placement problem. plitter placement problems in alloptical WDM networks in which a single light tree is constructed to realize each multicast connection have been studied in previous researches. This paper studies the splitter placement problem in alloptical WDM networks in which a light forest consisting of a collection of light trees is used to realize a multicast connection. The goal is to place a given number of splitters in the network such that the average per link wavelength resource usage of multicast connections is minimized. An upper bound and a lower bound on the per link average wavelength resource usage for a given number of multicast connections are derived. Two splitter placement methods are proposed for this problem. The two proposed splitter methods are shown to yield significant lower average wavelength resource usage than the random placement method. One of the methods is shown to produce near minimum average wavelength resource usage. I. INTRODUCTION A number of multimedia applications such as video on demand, video conference, distance education and etc. require a vast amount of bandwidth. The demand for networks with high bandwidth is increasing. An optical fiber can provide a large amount of bandwidth (nearly 50 Tb/s [3]) by using wavelength division multiplexing (WDM) [] technology to satisfy the increasing bandwidth demand. Wavelength division multiplexing (WDM) technology is able to divide the vast bandwidth of a fiber into a number of highspeed channels each of which is located at different wavelength. Each channel is capable of operating at the peak rate of an electronic interface. Optical WDM networks are promising transport networks for providing vast amount of bandwidths. If some or all of the switches in an optical WDM network are electronic switches, the bandwidth of the optical network is limited by the speed of the opticalelectricaloptical (OEO) conversion. Alloptical WDM networks in which the signals remain in the optical domain throughout the networks are desirable for providing large amount of bandwidths. In all optical WDM networks, each of the connections going through a link is assigned a wavelength. If a connection is assigned the same wavelength on all of the links along its path, the signal is able to travel from the source node to the destination node using the same wavelength. In case that the connection has to be assigned two or more wavelengths on different links, one or more nodes along its path must have the capability of converting the signal from one wavelength to another wavelength. If all nodes along the path of a connection are incapable of converting a wavelength to another wavelength, the connection must be assigned the same wavelength on all links along its path. Otherwise, the connection is blocked. This is known as the wavelength continuity constraint. A connection that is assigned the same wavelength on all links along its path is called a light path [2]. The WDM networks considered in this paper are alloptical WDM networks without wavelength conversion. In alloptical WDM networks without wavelength conversion, multicast communications can be supported by establishing a light path from the source node to each of the destination nodes. However, two or more light paths of the same multicast connection may go through the same link resulting in a waste of bandwidth. It is desirable to establish a treeshaped path that uses the same wavelength on all links on the treeshaped path from the source node to all destination nodes in order to conserve bandwidth. A treeshaped path using only a single wavelength is called a light tree []. In order to realize a light tree in an alloptical WDM network, the node(s) at the branching point(s) of the light tree must be capable of splitting an input light signal into two or more output signals of the same wavelength. A node with such capability is called a multicast capable node or a splitter [3]. A node without such capability is called a multicast incapable node. When a signal goes through a splitter, the power of the output signals is degraded by a factor of the number of output signals. To maintain the power level of the signal, a costly active amplification device is required to be installed in the splitter [3]. Therefore, a multicast capable node is much more expensive than a multicast incapable node. An alloptical WDM network in which all nodes are multicast capable is very expensive. To reduce the cost, splitters can be placed at a subset of selected nodes. Given that only a subset of nodes in the network are equipped with splitters, a multicast routing protocol needs to construct a light tree for each multicast connection according to the locations of the splitters in order to improve the probability of successfully establishing the multicast connection. Another approach is to construct multiple light trees, which is collectively called a light forest, to realize a single multicast connection [5], [6], [7]. Each of the light trees in the light forest is assigned a different wavelength. Multicast routing algorithms using this approach are known as sparse light splitting multicast routing algorithms. Note that the performance of the multicast routing algorithm is highly affected by the locations of the splitter in the network. How to place a given number of splitters at suitable locations IEEE Globecom /05/$ IEEE
2 such that the performance of the multicast routing algorithm can be improved is an important issue. This problem is referred to as the splitter placement problem which is the focus of this paper. plitter placement problems in alloptical WDM networks in which a single light tree is constructed to realize each multicast connection have been studied in [], []. Note that each of the branch nodes on a light tree must be a splitter. If such a light tree cannot be found to fulfill a multicast request, the multicast request is blocked. The objective in [] is to place a given number of splitters in the network such that the blocking probability of a multicast request is minimized. The problem was proved to be NPcomplete. Integer linear programming was used to solve the problem. everal heuristic algorithms were also studied. A different objective function was considered in []. The objective in [] is to place the splitters in the network such that the probability that a destination node of a multicast connection cannot be reached by the light tree for the multicast connection is minimized. A heuristic algorithm was proposed for the problem. This paper considers the splitter placement problem in alloptical WDM networks in which a light forest consisting of a collection of light trees rooted at the source node is used to realize a multicast connection. In such a network, in the worse case a multicast connection can be realized by multiple light paths without requiring any splitter. Thus, a multicast request will never be blocked due to poor placement of the splitters. Instead, it will be blocked because of insufficient wavelength resource in the network. The advantage of placing splitters in such a network is that the required wavelength resource for multicast connections can be reduced. The objective of this paper is to place a given number of splitters in the network such that the wavelength resource utilized by the multicast connections is minimized. To our knowledge, this problem has not been studied in previous researches. In this paper, a lower bound and an upper bound on the per link average wavelength resource usage for a given number of multicast connections are derived. Two splitter placement methods, namely, kmaximum degree and kmaximum WR methods are proposed. The performances of the proposed splitter placement methods are compared with a random placement method and the optimal placement. Our simulation results show that the two proposed splitter placement methods yield significantly lower average wavelength resource usage than the random placement method. In particular, the kmaximum WR method is able to produces near optimal average wavelength resource usage. The rest of this paper is organized as follows. The splitter placement problem is given in the next section. The lower and upper bounds on the per link average wavelength resource usage are derived in ection III. The two proposed splitter placement methods are described in ection IV. The performances of the proposed splitter placement methods are studied in ection V. Finally, some concluding remarks are given in ection VI. II. THE PLITTER PLACEMENT PROBLEM The objective of this paper is to place a given number of splitters in an alloptical WDM network such that the wavelength resource utilized by the multicast connections is minimized. When the number of connections in the network is large, some of the links may become out of wavelengths resulting in blocking further connection requests. Blocked connection requests are not accounted for in the usage of wavelength resource. A poor placement of splitters may lead to poor wavelength resource utilization that in turn blocks a high percentage of the connection requests resulting in reduced usage of the wavelength resource. In order to isolate the effect of the splitter placement on the usage of wavelength resource from the restriction of the wavelength resource on each link, we assume that each link in the network has unlimited number of wavelengths. Let G = (V,E) represent an alloptical WDM network where V is the set of nodes and E is the set of links. If there is a link from node u to node v, there is also a link in the reversed direction. Let the number of splitters to be placed in G be denoted as k. A placement of splitters P = {p(v); v V } is defined as follows: { if a splitter is placed at node v, p(v) = () 0 if no splitter is placed at node v. The total number of splitters in the network is k. Thus, p(v) =k. (2) v V Let R = {r i, i =, 2,,m} be a sequence of m multicast requests. Let w i be the wavelength resource usage of the multicast request r i. If a multicast request requires one wavelength on one link, the wavelength resource usage of the request on the link is one unit of the wavelength resource. The total wavelength resource usage of a multicast request is the sum of the wavelength resource usage of the request on all links. Given a splitter placement P, the wavelength resource usage for a multicast request r i is calculated as follows. An IPlayer multicast tree is first constructed for the multicast request r i. Base on the locations of the splitters, P, a set of light trees T = {t,t 2,,t b } is constructed to realized the IPlayer multicast tree. The number of required light trees to realized an IPlayer multicast tree depends on the splitter placement P. Each of the light trees in the set T is assigned a different wavelength. Let light tree t j traverse through l(t j ) links. It is clear that the wavelength resource usage of light tree t j is l(t j ). The wavelength resource usage of multicast request r i is the sum of the wavelength resource usages of the light trees that realize its IPlayer multicast tree, i.e. b w i = l(t j ). (3) j= For example, Fig. (a) shows an IPlayer multicast tree for a multicast request rooted at source node. If node is the only splitter on the multicast tree, the multicast tree needs to be realized by a light forest consisting of three light trees as shown in Fig. (b). Each of the three light trees is assigned a wavelength. The wavelength resource usages of the three light trees are 6, 5, and 3 respectively from left to right. The wavelength resource usage of the multicast tree is the sum of IEEE Globecom /05/$ IEEE
3 (a) Fig.. Example of a light forest. The source is node. The rest of the nodes are destination nodes. Node is a multicast capable node, i.e. a splitter. The rest of the nodes are multicast incapable nodes. The multicast tree in (a) needs to be realized by a light forest consisting of three light trees in (b). the wavelength resource usages of the three light trees which is. Let W denote the total wavelength resource usage of all multicast requests in the sequence R = {r i, i =, 2,,m}. It is given as follows: m W = w i. () i= Let U be the per link average wavelength resource usage, which is calculated as m U = W E = w i i= E 2 6 (b), (5) where E is the number of links in the networks. The splitter placement problem is formulated as follows: Given an alloptical WDM network G =(V,E), a sequence of m multicast requests R = {r i, i =, 2,,m}, and k splitters, the objective is to find the splitter placement P that minimizes the per link average wavelength resource usage U. III. LOWER AND UPPER BOUND In this section, we derive the lower and upper bounds of the per link average wavelength resource usage U. The lower bound provides a reference value for the minimum required number of wavelengths per link in an alloptical WDM network for a given number of multicast connections. The upper bound provides a reference value for the maximum required number of wavelengths per link to supports a given number of multicast connections simultaneously. The lower bound is achieved when each of the multicast connections is actually a unicast from a source node to one of its neighbor nodes, whose wavelength resource usage is exactly one. Thus the lower bound of the per link average wavelength resource usage is simply m E, i.e. U m E. The upper bound of the per link average wavelength resource usage is given in the following theorem. Theorem : Given an alloptical WDM network G =(V,E), a sequence of m multicast requests R = r i, i =, 2,, m, and k splitters. The upper bound of the per link average wavelength resource usage is m V 2 E, i.e. Proof: Let ˆr be the multicast request among the m multicast requests that yields the maximum wavelength resource usage. Let the wavelength resource usage of multicast request ˆr be wˆr. Then the upper bound of the per link average wavelength resource usage is mwˆr E, i.e. U mwˆr. (7) E The maximum possible value of wˆr is derived as follows. Let ˆt be the IPlayer multicast tree corresponding to the multicast request ˆr. uppose that there are n nodes on the multicast tree ˆt. Thus, there are n links on the multicast tree ˆt. Recall that the wavelength resource usage wˆr is the sum of the wavelength resource usages of the light trees that realized the multicast tree ˆt. In other words, it is the sum of the wavelength resource usages of the multicast request ˆr on all the links on the multicast tree ˆt. uppose that there are k i links on which the wavelength usages of the multicast request ˆr are i units of wavelength resource, where i =, 2,,n. The wavelength resource usage of multicast request ˆr is calculated as follows: wˆr = k +2 k 2 + +(n ) k n. () We need to find the maximum possible value of wˆr for all possible combinations of k,k 2,,k n under the constraint that k + k k n = n, () since the total number of links on multicast tree ˆt is n. Now, consider the value of k, i.e. the number of links whose wavelength resource usages of multicast request ˆr are exactly one. If k =, multicast tree ˆt must be a tree consisting of two neighbor nodes and one link. By assumption, ˆr is the multicast request among the m multicast requests that yields the maximum wavelength resource usage. Then the network must be a network consisting of only two nodes connected by two links, one on each direction. The per link average wavelength resource usage is m 2 which is the same as its upper bound. Thus, the case where k =is proved. Next, consider the case where k 2. For notational convenience, let k = z. Consider the worst case where no splitter is on any one of the branch nodes of the IPlayer multicast tree ˆt. In this situation, the IPlayer multicast tree ˆt must only be realized solely by light paths instead of light trees. If there are k = z links on multicast tree ˆt whose wavelength resource usages of multicast request ˆr are exactly one, there must be no link whose wavelength resource usages of multicast request ˆr is more than z, i.e. k j =0, j = z +,z+2,,n. Thus, from equation (), we have k 2 + k k z = n z. (0) ubstituting into equation (), the wavelength resource usage of multicast request ˆr becomes wˆr = z +2k zk z. () U m V 2 E. (6) For a given value of z (or k ), the values of k 2,k 3,,k z that maximize wˆr are k 2 = k 3 = = k z =0 and IEEE Globecom /05/$ IEEE
4 k z =(n z). ubstituting the values of k 2,k 3,,k z into equation (), we obtain wˆr = z + z(n z) = nz z 2. (2) The value of z (or k ) that maximizes wˆr is n 2. ubstituting z = n 2 into equation (), we have wˆr = n2. (3) Finally, the maximum value of wˆr is obtained when the number of nodes on multicast tree ˆt equals to the number V 2 of nodes in the network V, i.e. n = V and wˆr =. ubstituting into inequality (7), the upper bound is obtained. This completes the proof of Theorem. IV. PLITTER PLACEMENT METHOD We propose two splitter placement methods, namely, kmaximum degree and kmaximum WR (wavelength reduction) methods, for alloptical WDM networks. The two methods are described in the following. A. The kmaximum degree method The idea of this method is that a node with more neighbor nodes is more likely to become a branch node of a multicast tree. Hence, placing a splitter on the node is expected to be effective in reducing the wavelength resource usage of the multicast connections. Therefore, this method sorts the nodes in the network in descending order according to the number of links connected to them and selects the first k nodes to place the splitters. The computation time of this method includes the time for checking the degree of each of the nodes and the time for sorting the nodes according the degrees of the nodes. The computational complexity of this method is O( V log V ). B. The kmaximum WR (wavelength reduction) method The idea of this method is as follow. If the placement of a splitter on a node yields more reduction on the wavelength resource usage, it is more beneficial to place a splitter at the node. The details are described in the following. Let s i be the shortest path spanning tree rooted at node i. Let h(s i ) denote the wavelength resource usage of s i when no splitter is placed in the network. Note that when no splitter is placed in the network, each multicast connection is realized by multiple light paths. The wavelength resource usage h(s i ) can be calculated by traversing the shortest path spanning tree s i in depthfirst manner. The computational complexity for traversing a shortest path spanning tree is O(β d ), where β is the maximum branching factor and d is the maximum depth of the shortest path spanning tree. Let h be the total wavelength usage of the set of all shortest path spanning trees rooted at each of the nodes when no splitter is placed in the network. Then h is calculated as follows: h = h(s i ). () i V The computational complexity for calculating h is V O(β d ). Let f(i, s j ) be the wavelength resource usage of s j when a splitter is placed at node i. Letf(i) denote the wavelength resource usage of the set of all shortest path spanning trees rooted at each of the nodes when a splitter is place at node i. Then f(i) is obtained as follows: f(i) = f(i, s j ). (5) j V The computational complexity for calculating each f(i), is also V O(β d ). The computational complexity for calculating all f(i), i V,is V 2 O(β d ). Let R(i) be the amount of reduction in wavelength resource usage when a splitter is placed at node i compared with no splitter in the network. Then R(i) is given as follows: R(i) =h f(i). (6) The kmaximum WR method sorts the values of R(i), i =, 2,, V, in descending order and selects the first k nodes to place the splitters. The overall computational complexity of the kmaximum WR method is dominated by the computational complexity for calculating all f(i), i V, which is V 2 O(β d ). V. IMULATION imulations are performed to study the performances of the proposed splitter placement methods. The performances of the proposed splitter placement methods are compared with that of a random placement method and that of the optimal placement. The random placement method randomly selects k nodes in the network and places the splitters at the k nodes. The optimal placement is obtained by exhaustive search. In addition, the effects of different algorithms for constructing light forests on the performance are also investigated. A. imulation Model Pure random graphs generated using the GTITM [0] tool are used to represent the networks. Two network sizes are considered in our simulations: 30nodes and 00nodes networks. For each simulation run, 00 random networks are generated. For each network, m multicast requests are generated. The source node of each multicast request is randomly selected from the nodes in the network. The set of destination nodes for a multicast request is selected in the following manner. First, a destination probability is chosen randomly between 0 and. Then the destination probability is used to determine whether each of the rest of the nodes in the network is included in the destination set. Therefore, the average number of destination nodes of the multicast connections is (N )/2. For each multicast request, the shortestpath multicast routing algorithm is used to construct a multicast tree. The reroutetosource algorithm proposed in [5] is used to construct a light forest to realize the multicast tree unless otherwise specified. For each network, the per link average wavelength resource usage U is first calculated. Then the average value over the 00 networks is calculated. Each data point in our graph is the average value of U over the 00 networks. IEEE Globecom /05/$ IEEE
5 Average wavelength resource usage per link (U) Random method k maximum degree method k maximum WR method Optimal placement Number of splitters Fig. 2. Average wavelength resource usage per link, N =30and m =60. Average wavelength resource usage per link (U) Random RerouteToource Degree RerouteToource WR RerouteToource Random MemberOnly Degree MemberOnly WR MemberOnly Number of splitters Fig. 3. Average wavelength resource usage per link, N = 00 and m = 200. B. imulation Results First, we compare the performances of different splitter placement methods for 30nodes networks. The number of multicast requests, m, generated for each network is 60. The number of splitters, k, to be placed in the network ranges from to 30. The results for the nodes networks are plotted in Fig. 2. From the figure, we can make the following observations: Both of the proposed splitter placement methods yield significant lower per link average wavelength resource usage than the random method. The kmaximum WR method produces lower per link average wavelength resource usage than the kmaximum degree method. The kmaximum WR method produces near optimal per link average wavelength resource usage. The maximum difference of the per link average wavelength resource usages produce by the kmaximum WR method and the optimal placement is.33%. Next, the performances of different splitter placement methods are compared for 00nodes networks. Optimal placement is not compared due to long computation time. The effects of different algorithms for constructing light forests on the per link average wavelength resource usage is also studied. Two algorithms for constructing light forests, namely, reroutetosource and memberonly, proposed in [5] are considered. Three splitter placement methods, namely, random, kmaximum degree, and kmaximum WR methods, are combined with the two algorithms for constructing light forests resulting in a total of six combinations. The number of multicast requests, m, generated for each network is 200. The number of splitters, k, to be placed in the network ranges from to 00. The results are plotted in Fig. 3. From the figure, in addition to the first and second observations made from Fig. 2, we can observe that a good algorithm for constructing light forests is able to further reduce the per link average wavelength resource usage. VI. CONCLUION Placing splitters at a subset of nodes instead of all nodes in an alloptical WDM network leads to significant cost reduction. A good splitter placement method is beneficial in efficient utilization of the wavelength resource and improving the performance of the network. The kmaximum WRmethod proposed in this paper is able to yield near optimal per link average wavelength resource usage. This method will be useful in the networkplanning phase for selecting suitable locations of the splitters. The kmaximum WRmethod combined with a good algorithm for constructing light forests for the multicast connection can further reduce the average wavelength resource usage. ACKNOWLEDGMENT This research was supported by the National cience Council, Taiwan, under grant NC223E , and under the Program for Promoting Academic Excellence of Universities, NC2752E PAE. REFERENCE [] C. A. Brackett,, Dense Wavelength Division Multiplexing Networks: Principles and Applications, IEEE Journal of elected Area on Communications, vol., no. 6, pp.6, August 0. [2] I. Chlamtac, A. Ganz, and G. Karmi, Lightpath Communications: An Approach to High Bandwidth Optical WAN s, IEEE Transactions on Communications, vol. 0, no. 7, pp.72, July 2. [3] C. iva Ram Murthy and M. Gurusamy. WDM Optical Networks: Concepts, Design, and Algorithms, PrenticeHall, [] L. H. ahasrabuddhe and B. Mukherjee, Light Trees: Optical Multicasting for Improved Performance in Wavelength Routed Networks, IEEE Communications Magazine vol. 37, no. 2, pp.6773, February. [5] X. Zhang, J. Y. Wei, and C. Qiao, Constrained Multicast Routing in WDM Networks with parse Light plitting, Journal of Lightwave Technology, vol., no. 2, pp. 727, December [6] W. Tseng and. Kuo, Alloptical Multicasting on WavelengthRouted WDM Networks with Partial Replication, Proceedings of IEEE ICOIN, 200, pp.3. [7] K. D. Wu, J. C. Wu and C.. Yang, Multicast Routing with Power Consideration in parse plitting WDM Networks, ICC 200, vol.2, pp [] M. Ali and J. Deogun, Allocation of Multicast Nodes in Wavelength Routed Networks, ICC 2000, vol.2, pp [] M. Ali, Optimization of plitting Node Placement in WavelengthRouted Optical Networks, IEEE Journal on elected Area in Communications, vol. 20, no., pp , October [0] E. W. Zegura, GTITM: Georgia Tech Internetwork Topology Models (software), itm.tar.gz, 6. Zegura/gtitm/gt IEEE Globecom /05/$ IEEE
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