Graduate School of Information, Production and Systems, Waseda University 9. Communication Network and LAN Design

9. Communication Network and LAN Design

9. Communication Network and LAN DesignCentralized Network Design1.1 Introduction of Centralized Network Design1.2 Capacitated Multipoint Network Design Problem

Backbone Network Design2.1 Introduction of Backbone Network Design2.2 A Standard GA for Backbone Network Design2.3 A Hybrid GA for Backbone Network Design

Bicriteria LAN Topological Design3.1 Formulation of b-LAN Topology Design3.2 B-LAN Topology Design based on GA3.3 Numerical Example Bi-level Hierarchical GA for Reliable Network Topological Design4.1 Introduction4.2 Problem Formulation4.3 Genetic Approach 4.4 Numerical Example

9. Communication Network and LAN DesignComputer Network Design ProblemsThe reason using computer networks:To share expensive hardware/software resourcesTo provide access to main system from distant locationsTo improve the reliability of computer systemsNetwork Design Problems:Shortest Path Routing ProblemNetwork Design Problems Considering ReliabilityNetwork Expansion Problems LAN Topology Design Problems, etc.

vBNS Backbone Network Maphttp://www.mci.com/index.jspvBNS: very high speed Backbone Network Services

NACSIS: National Center for Science Information System http://www.sinet.ad.jp/english/index.html

vBNS Logical Network Maphttp://www.mci.com/index.jsp

9. Communication Network and LAN DesignDescription of Computer Communication NetworkIllustration of a network with spanning tree structureConc.Conc.Mux.Mux.Mux.Conc.TerminalsConc.: concentratorMux.: multiplexerHost ComputerHost ComputerHost Computer. . ....

9. Communication Network and LAN DesignThe use of communication networks has increased significantly in the last decade due to the dramatic growth in the use of internet for business and personal use. As the society transforms itself to an information society the network becomes the primary source for information creation, storage, distribution and retrieval. The design and development of a reliable network to support the primary resource of an information society becomes a very critical activity. The reliability and service quality requirements of communication networks and the large investments in communication infrastructure have made it critical to design optimized networks that meet performance parameters. These factors have promoted researcher to develop new models and methodologies for network design.

9. Communication Network and LAN DesignA cost-effective structure for a large communication network is a multilevel hierarchical structure consisting of a backbone network (high level) and local access networks (low level). Boorstyn, R. R. and H. Frank: Large-scale network topological optimization, IEEE Trans. on Communication, Vol. COM-25, No.1, pp. 2947, 1977. Figure shows the hierarchical structure of communication networks. The backbone network, which connects local access networks to each other, is characterized by distributed traffic requirements and is generally implemented using packet switching techniques. : host: switching node: terminalLocalAccessNetworkLocalAccessNetworkBackboneNetwork

9. Communication Network and LAN DesignIn packet switching techniques, messages are broken into blocks of a certain size called packets; The packets, when they contain the destination address, can follow different routes toward their destination. The backbone network itself may be multilevel, incorporating, for example, terrestrial and satellite channels. Local access networks have in general centralized traffic patterns (most traffic is to and from the gateway backbone node) and are implemented with centralized techniques such as multiplexing, concentrating, and polling to share data coming from several terminals having lower bit rates on a single high capacity link. In special cases, the network may consist primarily of either centralized or distributed portions exclusively. Next page show vBNS (very high speed Backbone Network System) backbone network and its logical network map funded in part by NSF. vBNS is a research platform for advancement and development of high-speed scientific and engineering applications, data routing and data switching capabilities.

9. Communication Network and LAN DesignThe topological design problem for a large hierarchical network can be formulated as follows:Terminal and host locations (terminal-to-host and host-to-host),Traffic requirements (terminal-to-host and host-to-host)Candidate sites for backbone nodes, and cost elements (line tariff structures, nodal processor costs, hardware costs, etc.) Minimizing communication costs: Total communications costs = (backbone line costs) + (backbone node costs) + (local access line costs) + (local access hardware costs)

9. Communication Network and LAN DesignSuch that average delay and reliability requirements, which are index of quality of service, are met.Average packet delay in a network can be defined as the mean time taken by a packet to travel from a source to a destination node. Reliability is concerned with the ability of a network to be available the desired service to the end-users. Reliability of a network can be measured using deterministic or probabilistic connectivity measures.Colbourn, C. J.: The Combinatorics of Network Reliability, Oxford University Press, 1987. Kershenbaum, A.: Telecommunications Network Design Algorithms. McGraw-Hill: New York, 1993.

9. Communication Network and LAN DesignThe global design problem consists of two subproblems:The design of the backboneThe design of the local distribution networks.The two subproblems interact with each other through the following parameters:Backbone node number and locationsTerminal and host association to backbone nodesDelay requirements for backbone and local networksReliability requirement for backbone and local networks

9. Communication Network and LAN DesignOnce the above variables are specified, the subproblems can be solved independently.Boorstyn, R. R. and H. Frank: Large-scale network topological optimization, IEEE Transasctions on Communication, vol. COM-25, no 1, pp. 2947, 1977. Topological design of backbone and local access networks can be viewed as a search for topologies that minimize communication costs by taking into account delay and reliability constraints. This is NP hard problem which is usually solved by means of heuristic approaches. For example, branch exchange, cut saturation algorithms, concave branch elimination are well known greedy heuristic approaches for backbone network design. Gerla, M. and L. Kleinrock: On the topological design of distributed computer networks, IEEE Transactions on Communications, Vol. COM-25, No. 1, pp. 4860, 1977. Kershenbaum, A.: Telecommunications Network Design Algorithms. New York: McGraw-Hill, 1993. Tanenbaum, A.S.: Computer Networks, Prentice Hall, New Jersey, 1981. Easu and Williams algorithm, Sharma's algorithm and unified algorithm are known greedy heuristics for centralized network design. Easu, L.R. and K.C. Williams: On teleprocessing system design: a method for approximating the optimal network, IBM System Journal, 5, pp 142-147, 1966. Sharma, R. and M. El-Bardai: Suboptimal communications networks synthesis, Proceedings. of the IEEE International Conference on Communications, pp 19.11-19.16, 1970. Kershenbaum, A. and W. Chou: A unified algorithm for designing multidrop teleprocessing networks, IEEE Transactions on Communications, Vol. 22, pp. 1762-1772, 1974.

9. Communication Network and LAN DesignRecently, genetic algorithms has been successfully applied to network design problems. The studies for centralized network design are given as:

Abuali, F. N., R. L. Wainwright, and D. A. Schoenefeld: Determinant factorization: a new encoding scheme for spanning tree applied to the probability minimum spanning tree problem, Proceedings of 6th International Conference on Genetic Algorithms, pp. 470477, 1995. Elbaum, R. and M. Sidi: Topological design of local-area networks using genetic algorithms, IEEE/ACM Transactions on Networking, Vol. 4, No.5, pp. 766778, 1996.Kim, J.R.: Study on advanced genetic algorithms for reliable network design, Ph.D. Disertation, Ashikaga Institute of Technology, 2000. Kim, J.R. and M. Gen: Genetic algorithm for solving bicriteria network topology design problem, Proceedings. of the 1999 Congress on Evolutionary Computation, 2272-2279, 1999.Kim, J. R. and Gen, M.: A genetic algorithm for bicriteria communication network topology design, Engineering Valuation and Cost Analysis, Vol. 3, pp. 351-363, 2000. Lo, C.C. and W.H. Chang: A multiobjective hybrid genetic algorithm for the capacitated multipoint network design problem, IEEE Trans. on System, Man and Cybernetics-Part B, Vol. 30, No 3, pp 461-470, 2000.

9. Communication Network and LAN DesignThe studies for centralized network design are given as:

Palmer, C. C. and A. Kershenbaum: An approach to a problem in network design using genetic algorithms, Networks, Vol. 26, pp. 151-163, 1995. Zhou, G. and M. Gen: Approach to degree-constrained minimum spanning tree problem using genetic algorithm, Engineering Design and Automation, Vol. 3, No 2, pp. 157-165, 1997.Zhou, G. and M. Gen: A note on genetic algorithm approach to the degree-constrained minimum spanning tree problem, Networks, Vol. 30, pp. 105-109, 1997.Zhou, G. and M. Gen: A new tree encoding for the degree-constrained minimum spanning tree problem, in Gen, M. & R. Cheng, Genetic Algorithms & Engineering Optimization, John Wiley & Sons, New York, 2000.Zhou, G. and M. Gen: A genetic algorithm approach on tree-like telecommunication network design problem, Journal of Operational Research Society, Vol. 54, No 3, pp. 248-254, 2003.

9. Communication Network and LAN DesignThe studies for backbone design are given as follows:

Davis, L. and S. Coombs: Optimizing network link sizes with genetic algorithms, Modeling and Simulation Methodology, Knowledge Systems Paradigms. Amsterdam, The Netherlands: Elsevier, 1989. Kumar, A., R.M. Pathak, and Y.P. Gupta: Genetic algorithm - based reliability optimization for computer network expansion, IEEE Transactions on Reliability, Vol. 44, No 1, pp. 63 72, 1995.Kumar, A., R.M. Pathak, Y.P. Gupta, and H.R. Parsei: A genetic algorithm for distributed system topology design, Computers and Industrial Engineering, Vol. 28, No 3, pp. 659 670, 1995.Ko, K.T., K.S. Tang, C.Y. Chan, K.F. Man and S. Kwong: Using genetic algorithm to design mesh networks, IEEE Computer, pp. 56-58, 1997. Dengiz, B., F. Altiparmak, and A.E. Smith: Efficient optimization of all-terminal reliable networks, IEEE Transactions on Reliability, Vol. 41, No 1, pp. 18-26, 1997.Dengiz, B., F. Altiparmak, and A.E. Smith: Local search genetic algorithm for optimal design of reliable networks, IEEE Transactions on Evolutionary Computation, Vol. 1, No 3, pp. 179-188, 1997.Deeter, D.L., and A.E. Smith: Economic design of reliable networks, IIE Transactions, Vol. 30, pp. 1161-1174, 1998.

9. Communication Network and LAN DesignThe studies for backbone design are given as follows:

Pierre, S., and G. Legault: A genetic algorithm for designing distributed computer network topologies, IEEE Transactions on Systems, Man and Cybernetics-Part B: Cybernetics, Vol. 28, No 2, pp. 249-257, 1998. Cheng, S.T.: Topological optimization of a reliable communication network, IEEE Transactions on Reliability, Vol. 47, No 3, pp. 225-233, 1998.Konak, A., and A.E. Smith: A hybrid genetic algorithm approach for backbone design of communication networks, Proc. of the 1999 Congress on Evolutionary Computation, pp. 1817-1823, 1999.Liu, B. and K. Iwamura: Topological optimization models for communication network with multiple reliability goals, Computers & Mathematics with Applications, Vol. 39, pp. 59-69, 2000.Altiparmak, F., B. Dengiz, and A.E. Smith: Optimal design of reliable computer networks: a comparison of metaheuristics, Journal of Heuristics, Vol. 9, No 6, pp. 471-487, 2003.Altiparmak, F., M. Gen, B. Dengiz, and A.E. Smith: Topological optimization of communication networks with reliability constraint by an evolutionary approach, Proc. of International Workshop on Reliability and Its Applications, pp. 183-188, 2003.

9. Communication Network and LAN DesignApplication of Shortest Path Routing Problem

M.G.C. Resende: A Genetic Algorithm with Optimized Crossover for Routing Crossover for Routing on the Internet, INFORMS Annual Meeting, California, Nov., 2002.

OSPF (Open Shortest Path First)OSPF is a commonly used intra-domain routing protocol (IGP).Routers exchange routing information with all other routers in the autonomous system (AS).Complete network topology knowledge is available to all routers, i.e. state of all routers and links in the AS.OSPF routingAssign an integer weight w [1, wmax] to each link in AS.In general, wmax= 65535=2161.Each router computes tree of shortest weight paths to all other routers in the AS, with itself as the root, using Dijkstras algorithm.

9. Communication Network and LAN DesignOSPF weight setting OSPF weights are assigned by network operator.CISCO assigns, by default, a weight proportional to the inverse of the link bandwidth (Inv Cap).If all weights are unit, the weight of a path is the number of hops in the path.M.G.C. Resende propose a hybrid genetic algorithm to find good OSPF weights.Memetic algorithmMemetic Algorithms is a population-based approach for heuristic search in optimization problems. It combine local search heuristics with crossover operators. For this reason, some researchers have viewed them as Hybrid Genetic Algorithms.Genetic algorithm with optimized crossover

9. Communication Network and LAN Designgenerationcost[ Resende, 2002 ]

9. Communication Network and LAN Designcostdemand[ Resende, 2002 ]

9. Communication Network and LAN DesignCentralized Network Design1.1 Introduction of Centralized Network Design1.2 Capacitated Multipoint Network Design Problem

Backbone Network Design

Bicriteria LAN Topological Design

Bi-level Hierarchical GA for Reliable Network Topological Design

1.1 Introduction of Centralized Network DesignThe problem of effectively transmitting data in a network involves the design of communication subnetworks, i.e., Local access networks.

Local access networks are generally designed as centralized networks. A centralized network is a network where all communication is to and from a central site (backbone node). In such networks, terminals are connected directly to the central site. Sometimes multipoint lines are used, where groups of terminals share a tree to the center and each multipoint line is linked to the central site by one link only.

This means that optimal topology for this problem corresponds to a tree in a graph G =(V, E) with all but one of nodes in V corresponding to the terminals. The remaining node refers to the central site, and edges in E correspond to the feasible communication wiring. Each subtree rooted in the central site corresponds to a multipoint line.Usually, the central site can handle, at most, a given fixed amount of information in communication.

This, in turn, corresponds to restricting the maximum amount of information flowing in any link adjacent to the central site to that fixed amount.

1.1 Introduction of Centralized Network DesignIn the combinatorial optimization literature, this problem is known as the constrained Minimum Spanning Tree problem (c-MST). The mathematical model of c-MST is as follows:

1.1 Introduction of Centralized Network DesignWheren: the number of nodes in the networkTk: the kth multipoint link, and it may not exist for some k.W: given weightwij : the weight of the ith node to node j .T : the spanning treecij : the cost of connecting node i to node j, i.e., the cost of link (i,j);the cost matrix (cij) is symmetric. xij : the 0,1 decision variable; 1, if link (i,j) is selected, and 0, otherwise. Constraint:(2) guarantees that the links chosen for the topology do not include any cycles. (3) guarantees that enough links will be selected to connect the network (of n nodes). (4) guarantees that the total weight of the terminals on each multipoint line does not exceed the limit.

1.1 Introduction of Centralized Network DesignThe problem has been shown to be NP-hard by Papdimitriou.Papadimitriou, C. H.: The complexity of the capacitated tree problem, Networks, Vol. 8, pp. 217-230, 1978. Chandy and Lo, Kershenbaum and Chou, Elias and Ferguson had proposed heuristics approaches for this problem. Chandy, K. M. and T. Lo: The capacitated minimum spanning tree, Networks, Vol. 3, pp. 173-182, 1973.Kershenbaum, A. and W. Chou: A unified algorithm for designing multidrop teleprocessing networks, IEEE Transactions on Communications, Vol. 22, pp. 1762-1772, 1974. Elias, D. and M. Ferguson: Topological design of multipoint teleprocessing networks, IEEE Transactions on Communications, Vol. 22, pp. 1753-1762, 1974. Gavish also studied a new formulation and its several relaxation procedures for the capacitated minimum directed tree problem. Gavish, B.: Topological design of centralized computer networks: formulation and algorithms, Networks, Vol. 12, pp. 355-377, 1982.

1.1 Introduction of Centralized Network DesignRecently, GA has been successfully applied to solve this problem and its variants. Abuali, F. N., R. L. Wainwright, and D. A. Schoenefeld: Determinant factorization: a new encoding scheme for spanning tree applied to the probability minimum spanning tree problem, Proceedings of 6th International Conference on Genetic Algorithms, pp. 470477, 1995. Palmer, C. C. and A. Kershenbaum: An approach to a problem in network design using genetic algorithms, Networks, Vol. 26, pp. 151-163, 1995. Zhou, G. and M. Gen: Approach to degree-constrained minimum spanning tree problem using genetic algorithm, Engineering Design and Automation, Vol. 3, No 2, pp. 157-165, 1997.Zhou, G. and M. Gen: A note on genetic algorithm approach to the degree-constrained minimum spanning tree problem, Networks, Vol. 30, pp. 105-109, 1997. Lo, C.C. and W.H. Chang: A multiobjective hybrid genetic algorithm for the capacitated multipoint network design problem, IEEE Trans. on System, Man and Cybernetics-Part B, Vol. 30, No 3, pp 461-470, 2000. Zhou, G. and M. Gen: A new tree encoding for the degree-constrained minimum spanning tree problem, in Gen, M. & R. Cheng: Genetic Algorithms & Engineering Optimization, John Wiley & Sons, New York, 2000.Zhou, G. and M. Gen: A genetic algorithm approach on tree-like telecommunication network design problem, Journal of Operational Research Society, Vol. 54, No 3, pp. 248-254, 2003.

1.2 Capacitated Multipoint Network Design Problem Many important real-world decision problem involves multiple and conflicts objectives that need to be tackled while respecting various constraints, leading to overwhelming problem complexity. Lo and Chang propose a multiobjective hybrid GA (mo-hGA) for capacitated Multipoint Network Design (cMND) Problem to minimize total cost and delay. Lo, C.C. and W.H. Chang: A multiobjective hybrid genetic algorithm for the capacitated multipoint network design problem, IEEE Trans. on System, Man and Cybernetics-Part B, Vol. 30, No 3, pp 461-470, 2000. In their approach, the concept of subpopulation proposed by Schaffer had been used.Schaffer, J. D.: Multiple objective optimization with vector evaluated genetic algorithms, Proceedings of 1st International Conference on Genetic Algorithms, pp. 93100, 1985. After four subpopulations generated using different mechanism, population of next generation was obtained by mixing these subpopulations.

1.2 Capacitated Multipoint Network Design Problem When two objectives are considered, the mathematical model given above for the cMND problem is reformulated as a c-MST problem is as follows: Where, dij is the average delay on link (i,j), the delay matrix [dij] is symmetric.

1.2.1 Representation and InitializationThe genetic representation is a type of data structure that represents the candidate solutions of problems. Usually, different problems have different data structures or genetic representations. Gen, M. & R. Cheng: Genetic Algorithms and Engineering Optimization, John Wiley & Sons, New York, 2000.For a c-MST problem, there are three ways of encoding tree: Edge-based encoding,Vertex-based encoding,Edge and vertex-biased encoding.

1.2.1 Representation and InitializationEdge-based Encoding: A graph G = (V, E) is easily formed into a binary string which can be used as a chromosome for the GA. Each element of the chromosome represents a possible edge in the graph so there are n(n-1)/2 edges.where n is the number of vertices. The value of each element represents whether the specific edge connects with the pair of nodes or not. Consider an example with 6 vertices shown as follows. The table shows connectivity matrix. 132546Fig. 9.2 Six node tree. Table 9.1 Connectivity matrix of links.

1.2.1 Representation and InitializationEdge-based Encoding:

Note that there are (6x5)/2 = 15 possible edges for the example but only five are included.The other 10 are not connected.As it is seen that the index of chromosome is represented by following equation:

The chromosome representation is given as follows:

1.2.1 Representation and InitializationEdge-based Encoding:

This encoding method is really an intuitive representation for a tree.However, for an n-node graph, a spanning tree should be a connected subgraph with n 1 edges or non-loop subgraph with n 1 edges. But the edge based encoding can not preserve this property. If the bit string contains other than n 1 edges, it is not a tree. Even if the bit string contains n 1 edges, it unlikely represents a tree because there may exist loops. Thus it is quite likely that none of them would be trees in the initial population or after crossover and mutation operators in GA approach. Actually, the probability of obtaining a spanning tree with edge-based encoding is infinitesimally small as the number of vertices n increases. There are n(n-2) spanning trees in a fully connected graph with n vertices.While search space of the graph is 2(n(n-1)/2), the probability to randomly produce a spanning tree is only [n(n-2)]/ 2(n(n-1)/2)]. Hence, this encoding is a poor encoding for spanning tree because of the extremely low probability of obtaining a tree.

1.2.1 Representation and InitializationVertex-based Encoding:

One of the classic theorems in graphical enumeration is Cayley's theorem that there are n(n-2) distinct labeled trees on a complete graph with n vertices. Prfer provided a constructive proof of Cayley's theorem by establishing a one-to-one mapping between such trees and the set of all string of n-2 digits. This means that it is possible to use only n-2 digits permutation to uniquely represent a tree where each digit is an integer between 1 and n inclusive. This permutation is usually known as Prfer number.Prfer, H.: Neuer Beweis eines Satzes ber Permutationen, Archiv fuer Mathemtische und Physik, Vol. 27, pp. 742-744, 1918. In this study, Prfer number is used to represent a candidate tree and initial population is generated using complete random method.

1.2.1 Representation and InitializationEncoding Procedure of Prfer Numberprocedure: Encoding of Prfer Numberinput: a tree T output: Prfer number P

step 1: Let node i be the smallest labeled leaf node in a labeled tree T.step 2: Let j be the first digit in the encoding, as the code j incident to i is uniquely determined. The encoding is built by appending digits from left to right. step 3: Remove node i and the link from i to j, thus there is a tree with k-1 nodes.step 4: Repeat the steps above until one link is left. P is obtained.

1.2.1 Representation and Initialization132412596107811T = {(1, 3), (2, 5), (3, 4), (4, 8), (5, 8), (5, 9), (6, 9), (7, 8), (8, 11), (10, 12), (11, 12)}

1.2.1 Representation and InitializationDecoding Procedure of Prfer Numberprocedure: Decoding of Prfer Numberinput: Prfer number Poutput: Tree Tstep 1: Let P be the original Prfer number and let P' be the set of all nodes not included in P, which are designated as eligible nodes for consideration in building a tree T.step 2: Let i be the eligible node with the smallest label. Let j be the left most digit of P. Add the edge from i to j into the tree T. Remove i from P' and j from P. If j does not occur anywhere in P, put it into P'. Repeat the process until no digits are left in P.step 3: If no digits remain in P, there are exactly two nodes, r and s, still eligible for consideration. Add a link from r to s into tree and form a tree with k-1 links.

1.2.1 Representation and Initialization132412596107811T = {(1, 3), (2, 5), (3, 4), (4, 8), (5, 8), (5, 9), (6, 9), (7, 8), (8, 11), (10, 12), (11, 12)}

1.2.1 Representation and InitializationEdge and Vertex biased Encoding:

Palmer and Kershenbaum proposed the edge and vertex biased encoding method. Palmer, C. C. and A. Kershenbaum: An approach to a problem in network design using genetic algorithms, Networks, Vol. 26, pp. 151-163, 1995. In this method, a tree is encoded using a modified cost matrix. Based on the modified cost matrix, Prim algorithm is used to generate tree. For a graph with n vertices, the chromosome of the representation has biases including vertex bias bi and edge bias bij, for the n vertices and each of the n(n-1)/2 edges, for a total of [n(n+1)/2] biases. P1 and P2 are used as the multipliers of the maximum link cost Cmax. The cost matrix (Cmax) is biased by bi, bij, P1, P2 and Cmax using

1.2.1 Representation and InitializationEdge and Vertex biased Encoding:

Palmer and Kershenbaum claimed that this version of representation could encode any tree, given appropriate values of the bi, bj and bij. Palmer, C. C. and A. Kershenbaum: An approach to a problem in network design using genetic algorithms, Networks, Vol. 26, pp. 151-163, 1995. However, as pointed out by Gen and Cheng, this encoding has three major disadvantages: Gen, M. and Cheng, R.: Genetic Algorithms & Engineering Optimization, John Wiley & Sons, New York, 2000. It requires a very long encoding (memory cost);It needs a conventional minimum spanning tree algorithm to generate a tree from its encoding (computation cost);It contains no useful information such as degree, connection, etc, about a tree.

1.2.2 Subpopulation Generation MechanismsIn the study, four different subpopulation mechanisms have been used [Lo-Cheng, 2000]: Elitism reservation strategyStochastic universal samplingComplete random methodShifting Prfer vector

1.2.2 Subpopulation Generation Mechanismsa. Elitism Reservation Strategy:

In traditional GAs, a chromosome in the current generation is selected into the next generation with certain probability. The best chromosomes of the current generation may be lost due to mutation, crossover, or selection during the evolving process, and subsequently causes difficulty in reaching convergence. In other word, it takes more generations; i.e., running time, to get quality solutions. Tamaki et al.. proposed an elitism reservation strategy that permits chromosomes with the best fitness to survive and be carried into the next generation for multiobjective problems.Tamaki, H., H. Kita and S. Kobayahi: Multi-objective optimization by genetic algorithms: a review, Proceeding of 3rd International Conference on Evolutionary Computation, pp. 517-522, 1996.

1.2.2 Subpopulation Generation Mechanismsb. Shifting Prfer Vector:

The shifting Prfer vector is developed by Lo and Chang as a genetic operator. The operator replaces the leftmost element of a Prfer vector with a randomly selected nonleftmost element of the same vector. The purpose is to maintain maximum locality and realize local search. They had proofed that the new topology differs from the old one in at most two edges. The figure illustrates the new tree after the shifting Prfer vector is applied to the 7-node tree. It is seen from figures that the new tree and the old one differ in only one edge. 132654132654Fig. 9.3 New tree after applying the shifting Prfer vectorP = [3, 3, 5, 6]P = [1, 3, 5, 6]

1.2.2 Subpopulation Generation Mechanismsc. Stochastic Universal Sampling:

A simple way to perform sampling is to spin a roulette wheel.Unfortunately, this sampling method does not guarantee that any particular sample will actually be chosen in any given generation. This is a well-known problem of the roulette wheel selection method. Baker suggested the stochastic universal sampling method. Baker, J.: Adaptive selection methods for genetic algorithms, Proceedings of 2th International Conference on Genetic Algorithms, pp. 100-111, 1987. Bakers algorithm completes the whole sampling in a single pass, and requires only one random number.A wheel spin, whose size is equal to the population size, is divided into a number of equally spaced markers. A single spin is used to generate the random number.The expected value ek for chromosome k is expressed as ek = popSize x pk, where popSize represents population size and pk represents selection probability.

1.2.2 Subpopulation Generation Mechanismsd. Complete Random Method:

Population is generated according to random number and random position. The major reason for using the complete random method is to maintain the diversity of the population.

Mixing Method:

Firstly, four subpopulations are generated according to The elitism reservation strategyThe shifting Prfer vectorThe stochastic universal samplingThe complete random method, Then, mixing these four subpopulations produces the population of next generation.

1.2.3 Mixing MethodLo and Chang [IEEE-SMC, 2000] had explained the main idea on mixing method as follows:

There are two competing factors in the selection procedure of a genetic search. They are selection pressure and population diversity. To increase of selection pressure decreases the diversity of the population.Michalewicz, Z.: Genetic Algorithms + Data Structure = Evolution Programs, 3nd ed., New York: Springer-Verlag, 1996. The stochastic universal sampling method increases selection pressure. However, it may cause the premature convergence of a genetic search. To decrease selection pressure, the complete random method can be used in conjunction with the stochastic universal sampling method. Nevertheless, the best chromosomes of the current generation may be lost due to crossover and mutation. In order to find globally optimal solutions, the shifting Prfer vector is added to the selection procedure.

1.2.4 mo-hGA procedure for Capacitated mo-NDPprocedure: mo-hGA for Capacitated mo-NDP (Lo-Cheng 2000)input: network data (V, A, C, D, W), GA parametersoutput: Pareto optimal solutions E(P)begint 0;initialize P(t) by Prfer number encoding; objectives z1(P), z2(P) by Prfer number decoding;create Pareto E(P);fitness eval(P);while (not termination condition) dogenerate subpopulation, C1(t) by elitism reservation strategy;generate subpopulation, C2(t) by shifting Prfer vector;generate subpopulation, C3(t) by stochastic universal sampling;generate subpopulation, C4(t) by complete random method; objectives z1(Ci), z2(Ci), i=1, 2, 3, 4 by Prfer number decoding;fitness eval(C) by Prfer number decoding;select P(t+1) from Ci(t), i=1, 2, 3, 4 by mixing routine;t t + 1;endoutput Pareto optimal solutions E(P);end

1.2.5 Experimental Results In order to evaluate the solutions of Capacitated mo-NDP obtained by mo-hGA, Lo and Chang examined a set of problems, with 7, 14, 28, and 56 nodes, respectively. While they took population size as 100 for each problem, crossover and mutation probabilities and maximum number of generations had been changed according problem size.

mo-hGA was coded in the C++ language and run on an Intel Pentium-66 MHz PC with 64 MB RAM. Table 9.2 Parameters of GA

Number of NodesCrossover probabilityMutation probabilityMaximum Generations 70.50.9 200140.40.9 1000280.4 0.99 1000560.4 0.99 10000

1.2.5 Experimental ResultsTo investigate performance of the mo-hGA according to solution quality and solution time, veGA (vector evaluated GA) developed by Schaffer and rwGA (single objective GA) had been used. In rwGA, weights are assigned to each objective function and the weighted objectives are combined into single objective function. Schaffer, J. D.: Multiple objective optimization with vector evaluated genetic algorithms, Proc. of 1st Inter. Conf. on GAs, pp. 93100, 1985. In this study, based on the preliminary studies, weights w1(cost) and w2(delay) were set to 0.5.

1.2.5 Experimental ResultsProblem 1 [Lo-Cheng, 2000]: For the 7-node network, nodes are randomly distributed. Cost, delay and constrained weight matrices are given as follows.

Cost Matrix [cij]Constrained Weight Matrix [wij]Delay Matrix [dij]Note that: - represents infinityNote that: - represents 01 2 3 4 5 6 712345671 2 3 4 5 6 7123456715347264, 12, 32, 12, 33, 13, 34, 14, 31, 21, 32, 21, 34, 14, 32, 12, 33, 13, 34, 14, 35, 15, 3Note that: - represents infinity

1.2.5 Experimental ResultsSince the problem size is small, all the nondominated solutions; i.e., all possible spanning trees, are enumerated to compare the quality of the solutions obtained by mo-hGA, veGA and rwGA. The pair (total cost, total delay) represents a solution. By enumeration, six nondominated solutions are found for the problem:

(13,92), (14,91), (15,85), (16,84), (18,78) and (19,77).

With ten runs, nondominated solutions obtained by mo-hGA can be summarized as follows:

set 1: {(13,92), (14,91), (15,85), (16,84), (18,78), (19,77)}set 2: {(13,92), (15,85), (16,84), (18,78), (19,77)}set 3: {(14,91), (15,85), (16,84), (18,78), (19,77)}

1.2.5 Experimental ResultsSince sets 1, 2, and 3 are obtained 4, 4, and 2 times, respectively, Lo and Chang showed that the mo-hGA finds 90% of all nondominated solutions. Computational results are given as follows: Lo, C.C. and W.H. Chang: A multiobjective hybrid genetic algorithm for the capacitated multipoint network design problem, IEEE Trans. on System, Man and Cybernetics-Part B, Vol. 30, No 3, pp 461-470, 2000.

AlgorithmCPU time(Sec)Average number of non-dominatedSolutions obtained (A)Ratio(A/B)Cost Delay mo-hGA2.2785.40.9001377rwGA2.1602.00.3331378veGA2.3651.50.2501578 where A: The number of non-dominated solutions obtained by the algorithm. B: The number of all non-dominated solutions, which is 6 in this problem.

1.2.5 Experimental ResultsProblem 2, 3 and 4 [ Lo-Cheng, 2000 ]. In these problems, the networks have 14, 28, and 56 nodes and the nodes are randomly distributed. Computational results are given as follows:28-node14-node56-node

AlgorithmCPU time(Sec)Average number of non-dominatedSolutions obtained (A)Cost z1Delay z2 mo-hGA37.9009.563160rwGA28.8302.566168veGA45.0402.565162

AlgorithmCPU time(Sec)Average number of non-dominatedSolutions obtained (A)Cost z1Delay z2mo-hGA56.79213.0585628rwGA76.192 1.5866891veGA83.400 1.5854912

AlgorithmCPU time(Sec)Average number of non-dominatedSolutions obtained (A)Cost z1Delay z2mo-hGA2167.58015.015831618rwGA3780.462 1.316781620veGA4138.066 1.316751685

1.2.5 Experimental ResultsAs the results, mo-hGA always Finds more nondominated solutions than veGA and rwGA. Also finds the best solution in terms of cost and the best solution in terms of delay. In addition, for small size problems with 7 and 14 nodes:The mo-hGA takes more time than the rwGA, but less time than the veGA and also it generates better solutions than rwGA and veGA. For large size problems with 28 and 56 nodes:The mo-hGA takes the least time to generate nondominated solutions.

1.2.5 Experimental ResultsTo evaluate the correctness of the mix method [Lo-Cheng, 2000] :Lo and Chang evaluated four algorithms.

Algorithm 1, only the stochastic universal sampling method was used in the selection procedure. Algorithm 2 was obtained by adding the complete random method to Algorithm 1. Algorithm 3 was obtained by adding the elitism reservation strategy to Algorithm 2. Algorithm 4 (the mo-hGA) was obtained by adding the shifting Prfer vector to Algorithm 3.

Network nodes of 7, 14, and 28 were considered.

1.2.5 Experimental ResultsComputational results are given as follows [Lo-Cheng, 2000] :Lo, C.C. and W.H. Chang, A multiobjective hybrid genetic algorithm for the capacitated multipoint network design problem, IEEE Trans. on System, Man and Cybernetics-Part B, Vol. 30, No 3, pp. 461-470, 2000.

Number of NodesAlgorithmsCPU time(Sec)Average number of non-dominatedSolutions obtained 7Algorithm 1 3.630 1.45Algorithm 2 3.180 1.50Algorithm 3 3.020 3.20Algorithm 4 2.278 5.4014Algorithm 148.110 1.50Algorithm 243.110 2.00Algorithm 342.150 5.20Algorithm 437.900 9.5028Algorithm 196.560 1.50Algorithm 286.860 2.00Algorithm 370.720 6.10Algorithm 456.79213.00

1.2.5 Experimental ResultsBased on the results, Lo and Chang demonstrated that Algorithm 4 not only found more nondominated solutions but also obtained them faster than the other three algorithms.

Another analysis had been made on running time of rwGA, veGA, and mo-hGA in terms of the number of operations executed. For the three GAs, eight operations had been identified.

Weighted sumSubgroup selectionShuffleElitism reservationShifting Prfer vectorComplete randomCrossover & MutationPrfer decoding

1.2.5 Experimental ResultsThey assumed that every operation took the same amount of CPU time. Operational analysis is detailed as follows:

After this analysis, they showed that the mo-hGA required the least number of operations.

rwGAveGAmo-hGA [Lo-Cheng, 2000]Weighted Sum (a)M*popSizeN/AN/ASubgroup Selection (b)N/AM*popSizeN/AShuffle (c)N/AM*popSizeN/AElitism Reservation (d)N/AN/AM*popSize/4Shifting Prufer Vector (e)N/AN/AM*popSize/4Complete Random (f)N/AM/AM*popSize/4Crossover & Mutation (g)M*(N-3)*popSize +M*popSizeM*(N-3)*popSize +M*popSize(M*(N-3)*popSize +M*popSize)/4Prufer Decoding (h)M*NlogN*popSizeM*NlogN*popSizeM*NlogN*popSizeTotal(a)+(b)+(c)+(d)+(e)+ (f)+(g)+(h)M*(N+NlogN-1)* popSizeM*(N+NlogN)*popSizeM*((N+1)/4+NlogN)* popSizewhere: M represents the maximum generations, popSize represents the population size, N represents the number of nodes, and N/A means Not Applicable

9. Communication Network and LAN DesignCentralized Network Design

Backbone Network Design2.1 Introduction of Backbone Network Design2.2 A Standard GA for Backbone Network Design2.3 A Hybrid GA for Backbone Network Design



2.1 Introduction of Backbone Network DesignAs explained in beginning of the chapter, for cost-effectiveness, communication networks are designed in a multilevel hierarchical structure consisting of backbone networks and local access networks (LAN).A simple model for a backbone network is an undirected graph G = (V, E) with node set V and edge set E. In this model, the nodes represent the connection points where LANs are hooked up to the backbone network via gateways. In addition to being connection points, the nodes are the processing units that carry out traffic management on the network by forwarding data packets to the nodes along their destinations (i.e., known as routing). The edges represent the high capacity multiplexed lines on which data packets are transmitted bi-directionally between the node pairs. Each network link is characterized by a set of attributes which principally are the flow and the capacity.

2.1 Introduction of Backbone Network DesignFor a given link between a node pair (i,j), the flow fij is defined as the effective quantity of information transported by this link, while its capacity Cij is a measure of the maximal quantity of information that it can transmit. Flow and capacity are both expressed in bits/s (bps). The traffic ij between a node pair (i,j) represents the average number of packets/s sent from source i to destination j. The flow of each link that composes the topological configuration depends on the traffic matrix.Indeed, this matrix varies according to the time, the day and applications used.Kershenbaum, A.: Telecommunications Network Design Algorithms, New York: McGraw-Hill, 1993. Although some designs are based on the rush hour traffic between two nodes as in the study of Dutta and Mitra, generally the average traffic is considered in designs.Dutta, A. and S. Mitra: Integrating heuristic knowledge and optimization models for communication network design, IEEE Transactions on Knowledge Data Engineering, Vol. 5, No. 6, pp. 9991017, 1993. Boorstyn, R. R. and H. Frank: Large-scale network topological optimization, IEEE Transasctions on Communication, Vol. COM-25, No.1, pp. 2947, 1977. Gerla, M. and L. Kleinrock: On the topological design of distributed computer networks, IEEE Transactions on Communications, Vol. COM-25, No. 1, pp. 4860, 1977. Kershenbaum, A.: Telecommunications Network Design Algorithms. New York: McGraw-Hill, 1993.

2.1 Introduction of Backbone Network DesignThe topological design of backbone networks can be formulated as follow: Given the geographical location of nodes, the traffic matrix M = (ij), the capacity options and their cost, the maximum average delay Tmax, minimize the total cost over topological configuration together with flow and capacity assignment, subject to delay and reliability constraints. In addition to this formulation, some formulations associate a cost with the waiting time of a packet in the network, or seek to optimize the cost and the average delay by making abstraction of the reliability constraint. Gavish, B.: Topological design of computer communication network - The overall design problems, European Journal of Operational Research, vol. 58, pp. 149172, 1992. Others seek to minimize the cost under delay and flow constraints.Tomy, M. J. and D. B. Hoang: Joint optimization of capacity and flow assignment in a packet-switched communications network, IEEE Transactions on Communications, Vol. COM-35, No. 2, pp. 202209, 1987. The delay constraint is expressed as follows: T < Tmax. The flow and capacity assignments depend on the traffic and the average length of packets. The optimum capacity assignment can be obtained by minimizing the average delay. Schwartz, M. and T. E. Stern: Routing techniques used in computer communication networks, IEEE Transactions on Communications, Vol. COM-28, No. 4, pp. 539552, 1980.

The average packet delay (T) of a network is given as

2.1 Introduction of Backbone Network DesignThe reliability is concerned with the ability of a network to be available the desired service to the end-users. In literature, reliability measures to define connectivity level of a network are classified as probabilistic and deterministic.Probabilistic measures give the probability of a network being connected.Colbourn, C. J.: The Combinatorics of Network Reliability, Oxford University Press, 1987. Deterministic measures are related with the vulnerability of a network such as k-connectivity, (k-node disjoint paths between each pair of node), articulation, cohesion, etc. Newport, K. T. and P.K. Varshney: Design of survival communications networks under performance constraints, IEEE Transactions on Reliability, Vol. 40, No 4, pp. 443-440, 1991. Generally, deterministic measures are considered as network reliability constraint, because they are computationally more tractable than the probabilistic measures and ensure minimum reliability.

2.1 Introduction of Backbone Network Design

Since topological design of communication networks is NP-hard problem, heuristics techniques has been applied such as branch exchange method (BXC), concave branch elimination, cut-saturation algorithm (CSA). Boorstyn, R. R. and H. Frank: Large-scale network topological optimization, IEEE Transactions on Communication, Vol. COM-25, No.1, pp. 2947, 1977. Gerla, M. and L. Kleinrock: On the topological design of distributed computer networks, IEEE Transactions on Communications, Vol. COM-25, No. 1, pp. 4860, 1977. Kershenbaum, A.: Telecommunications Network Design Algorithms. New York: McGraw-Hill, 1993.Tanenbaum, A.S.: Computer Networks, Prentice Hall, New Jersey, 1981.

Newport and Varshney modified CSA considering network survivability into design process. Newport, K. T. and P.K. Varshney: Design of survival communications networks under performance constraints, IEEE Transactions on Reliability, Vol. 40, No. 4, pp. 443-440, 1991.

2.1 Introduction of Backbone Network DesignKershenbaum et al. proposed another heuristic called as MENTOR. Kershenbaum, A., P. Kermani, and G. A. Grover: MENTOR: An algorithm for mesh network topological optimization and routing, IEEE Transactions on Communications, Vol. 39, No. 4, pp. 503513, 1991.

Gavish formulated an overall deign problem, which includes both local access network and backbone design together, as a nonlinear optimization problem. Gavish, B.: Topological design of computer communication network - The overall design problems, European Journal of Operational Research, Vol. 58, pp. 149172, 1992. The solution methodology depends on Lagrangean relaxation and subgradient optimization procedure.

Dutta & Mitra and Sykes & White have proposed heuristic approaches introducing the artificial intelligence concept of knowledge-based system for the problem.Dutta, A. and S. Mitra: Integrating heuristic knowledge and optimization models for communication network design, IEEE Transactions on Knowledge Data Engineering, Vol. 5, No. 6, pp. 9991017, 1993. Sykes, E. A. and C. C. White: Specifications of a knowledge system for packet-switched data network topological design, Proceedings of Expert Systems Government Symposium, Mc Lean, VA, pp. 102110, 1985.

2.1 Introduction of Backbone Network DesignRecently, genetic algorithms have also been applied successfully to design of backbone networks.

Davis, L. and S. Coombs: Optimizing network link sizes with genetic algorithms, in Modeling and Simulation Methodology, Knowledge Systems Paradigms. Amsterdam, The Netherlands: Elsevier, 1989. Kumar, A., R.M. Pathak, Y.P. Gupta, and H.R. Parsei: A genetic algorithm for distributed system topology design, Computers and Industrial Engineering, Vol. 28, No. 3, pp. 659 670, 1995.Ko, K.T., K.S. Tang, C.Y. Chan, K.F. Man and S. Kwong: Using genetic algorithm to design mesh networks, IEEE Computer, pp. 56-58, 1997. Konak, A., and A.E. Smith: A hybrid genetic algorithm approach for backbone design of communication networks, Proc. of the 1999 Congress on Evolutionary Computation, pp. 1817-1823, 1999. Pierre, S., and G. Legault: A genetic algorithm for designing distributed computer network topologies, IEEE Transactions on Systems, Man and Cybernetics-Part B: Cybernetics, Vol. 28, No 2, pp. 249-257, 1998.

2.2 A Standard GA for Backbone Network Design Pierre and Legault developed a standard GA for designing backbone network considering a deterministic reliability measure. Pierre, S., and G. Legault: A genetic algorithm for designing distributed computer network topologies, IEEE Transactions on Systems, Man and Cybernetics-Part B: Cybernetics, vol. 28, no 2, pp. 249-257, 1998.

2.2.1 RepresentationBinary encoding had been used to represent a network. In a chromosome, 1s and 0s symbolize the existence or nonexistence of a link. The length of chromosomes is directly linked to the number of nodes in the network. Thus, a network composed of n nodes will be represented by chromosomes of length n(n-1)/2, that is the maximum number of links that compose a network of n nodes.

2.2.1 RepresentationA 3-node-connected topology, with its associated chromosome, for a network. Consider the example with 8 nodes shown as follows:

An undirected graph G = (V, E) is 3-node-connected if and only if G can be obtained from the complete graph on 4 nodes by the following two operations:(i) Add a new edge(ii) Take a node z and replace it by two nodes z1, z2, put an edge between them, and put edges incident to z1, z2 such that the union of the neighbors of them are exactly the original neighbors of z and there are at least two neighbors of zi for i = 1, 2.

1 2 3 4 5 6 7 812345678

2.2.1 RepresentationNote that there are (8x7)/2 = 28 possible links for this example but only 13 are included; the other 15 are not in the topology. The chromosome representation of the topology (x) is shown as follows:

It is possible to see that the index of the topology is represented by the following equation:

2.2.2 InitializationSince network reliability constraint is k-node-connectivity, the initial population is generated in two phases. In a first phase, an initial topology of degree k is produced Then other chromosomes are created by modifying the initial topology, i.e, chromosomes are created with adding new links between 1 and . The creation of the initial topology is done deterministically.

The link selection criterion is based on Euclidean distances between nodes. Firstly, the links, whose distance between nodes is the shortest, are added to chromosome and chromosome is checked whether it is k-connected, i.e, all nodes have at least k incident links, or not. If not, other links are added to chromosome until the resulting topology is k-connected.

2.2.2 Initializationprocedure: Initial Populationinput: number of possible links output: chromosome vstep 1: Sequence the links in increasing order of their distancestep 2: Create the initial topology 2.1 Add the first shortest links to topology.2.2 Check its connectivity. If it is not k-connected, add one or more shortest links from remaining link set until satisfy k-connectivity. step 3: Repeat following steps until reach to population size3.1 Determine the number of links (l) between 1 and 3.2 Obtain new topology with adding l shortest links from remaining link set to initial topology. GA Procedure of Initial Population

2.2.3 EvaluationThe problem is to design of a backbone network for minimizing total communication cost under maximum allowable average delay and k-node-connectivity constraints. Fitness function is the objective function of the problem. Once a candidate network with k-node-connectivity is obtained:Firstly capacity assignment of its links is realized considering capacity options.Then average delay time

and cost are calculated. Link cost is function of its capacity and distance. Therefore, the cost of a link comprises two components: a permanent cost related to the capacity of link. a variable cost related to the physical length of this link. Let dij(lij, cij) the leasing cost of a link between nodes i and j, of capacity cij and of length lij, the total link cost CL of a candidate network is calculated as follows:

2.2.3 EvaluationIn the study, six different capacity options had been considered. And the leasing costs of different types of link is given as follows:

Pierre, S., and G. Legault: A genetic algorithm for designing distributed computer network topologies, IEEE Transactions on Systems, Man and Cybernetics-Part B: Cybernetics, Vol. 28, No 2, pp. 249-257, 1998.Table 9.3 Capacity options and their costs

capacity(Kbps)variable cost(($/month)/km)fixed cost($/month) 9.6 3.010.0 19.2 5.012.0 56.010.015.0100.015.020.0200.025.025.0560.090.060.0

2.2.3 EvaluationThe capacity of each link between node pairs for a topology had been assigned considering related link flow and capacity options as follows:Table 9.4 Link flows and capacities associated with the network

link (i, j)flow fij (Kbps)capacity cij (Kbps)(1, 2) 20.00 56.00(1,3) 40.00 56.00(1, 6) 10.00 19.20(2, 3) 40.00 56.00(2, 4) 30.00 56.00(3, 5)100.00200.00(3, 6) 50.00 56.00(4, 5) 40.00 56.00(4, 8) 20.00 56.00(5, 7) 40.00 56.00(5, 8) 30.00 56.00(6, 7) 20.00 56.00(7, 8) 20.00 56.00

2.2.4 Genetic OperatorsOne-cut point crossover, bit-flip mutation and inversion are used as genetic operators. Crossover Operator:In crossover operator, chromosomes selected with roulette wheel selection mechanism are mated the best chromosome of the current generation to obtain offspring. Inversion Operator:As a second phase of crossover operator, inversion is applied to offspring, i.e, the gene at position 1 is transferred to position l (length of the chromosome), while the gene at position l replaces gene 1; the gene at position 2 is transferred to position l - 1, and the gene in position l - 1 is transferred to position 2, and so on.

2.2.5 Computational Results To measure the efficiency of the GA and the quality of its solutions, different networks sizes between 6 and 30 nodes were considered.

The experiments had been performed on a 33 MHz 486 IBM PC.

Population size, crossover probability, mutation probability, inversion probability and number of generations were set to 80, 0.95, 0.12, 0.23 and 40, respectively.

While average packet size was taken as 1000 bits, uniform traffic was set to 5 packets/second. Pierre, S., and G. Legault, A genetic algorithm for designing distributed computer network topologies, IEEE Transactions on Systems, Man and Cybernetics-Part B: Cybernetics, vol. 28, no 2, pp. 249-257, 1998.

2.2.5 Computational ResultsFirstly, Pierre and Legault had examined the GA performance according to initial topology considering 3-node-connectivity. They showed that while GA improved the solutions only very slightly for the networks with less than 15 nodes, after that it realized at least 15% improvement on the initial topology. Table 9.5 Cost versus size of the network Pierre, S. and G. Legault: A genetic algorithm for designing distributed computer network topologies, IEEE Transactions on Systems, Man and Cybernetics-Part B: Cybernetics, vol. 28, no 2, pp. 249-257, 1998.

No.nCL ($/month)InitialtopologyCL ($/month)GeneticalgorithmStandarddeviation% ofimprovement162163220733858.255 4%2103533035462671.501-31247810436424397.09 9%41589856728732954.0719%52532013427253916095.8115%6304086843465291046.4515%

2.2.5 Computational ResultsThe performance of GA had been also compared with Simulated Annealing (SA) and Cat Saturation (CS) algorithms. In the analysis, number of generations, probabilities of crossover, mutation and inversion had been taken as 25, 0.80, 0.10 and 0.20, respectively. Results are given the GA offers better solution than SA in terms of network cost (CL) and average delay (T).Table 9.6 Comparative Results SA and GA

No.NKSAGA% CL%TCL ($/month)T(ms)CL ($/month)T(ms)Tmax(ms) 1153 74100123.02 7019886.88125 6%42% 2183138805114.0410184691.5511536%25% 3184119238 86.9711003084.23 90 8% 3% 4185111725103.4211228789.21105 -1%16% 5203189570105.9012852484.5810647%25% 620414086379.2813342775.90 80 6% 4% 720515687299.8313444194.7310017% 5% 8253284797100.6022919867.9110124%48% 9254323333105.6220117495.4110661%11%10255284797100.6020677583.1410138%21%

2.2.5 Computational ResultsWhen GA compared with CS, maximum delay had been set to 120 ms. Comparative results have showed that the GA appears to be more effective when network size increases. In addition, Pierre and Legault noted that CS generally took less running time than GA for converging to a good solution. Table 9.7 Comparative Results CS and GA

No.NCSGA% CLCL ($/month)T (ms)CL ($/month)T (ms) 110 9690876.910272384.3-6.0% 21211242777.211480486.7-2.1% 31412421582.612319892.40.8% 41513078882.8123987100.35.2% 51613088476.4122226102.46.6% 61813822892.412873098.76.9% 72014264386.8129436109.79.3% 82214720892.3132842108.59.8% 92414824295.4135708117.48.5%102515543097.2139315115.810.4%

2.3 A Hybrid GA for Backbone Network Design Konak and Smith proposed a hybrid GA (hGA) for this problem. Konak, A., and A.E. Smith: A hybrid genetic algorithm approach for backbone design of communication networks, Proc. of Congress on Evolutionary Computation, pp. 1817-1823, 1999. Although GA performs search by exploiting information sampled from different regions of the solution space, crossover and mutation are generally blind operators, i.e., they do not use any problem specific information. This property causes the GA slow convergence before finding an accurate solution and failure to exploit local information. Gen, M. and Cheng, R.: Genetic Algorithms & Engineering Optimization, John Wiley & Sons, New York, 2000. In this study, Cat Saturation (CS), a well known heuristic method for design problem, was combined with mutation operator to improve candidate networks in search process.

2.3.1 Representation and InitializationAdjacency matrix had been used to represent a network design. In this representation, a network is stored n x n matrix A = {aij}, where n = |V| and aij = 1 if link (i, j)E' (E' is set of links in a topology), otherwise aij = 0. Since there are link alternatives having different capacities to assign between two nodes in the network, Konak and Smith made a slight modification on adjacency matrix to handle link alternatives. According to this modification, if there is a link (i, j), aij equals to integer values corresponding to link alternatives, otherwise aij equals to 0. For example, a12 = 2 means that link (1, 2) exists and it is a type 2 link. Since links are bidirectional only upper triangular of A is needed for representation. Initial population contains randomly generated solutions that satisfy k-node-connectivity.

2.3.1 Representation and InitializationLet number of link alternatives be 3. A 3-connected network is given as follows [Konak-Smith,1999]:The number on the link represents link type.

Its chromosome representation is given as follows:

Chromosome representation Sample network topology

2.3.2 Genetic OperatorsUniform crossover operator with k-node-connectivity repair algorithm was used to obtain offspring. For crossover, two parents are selected from population with q-tournament selection with q=2. The selected parents produce only one child. If both parents possess a link and types of link are same in both parents, the child will have that link. If the type of link is different in both parents, the type of link in child is selected from parents with 0.5 probability. If both parents do not possess the link, the child will not. If one parent has the link and the other has not, the child will have the link with 0.5 probability. This crossover operator ensures that the child inherits the common topological properties of the parents. After crossover, if the child is not k-node-connected, it is brought to k-node-connectivity by directly connecting the nodes violating k-node-connectivity.

2.3.2 Genetic OperatorsUniform crossover operatorRepresents the uniform crossover operator as follows:Parent 1 Parent 2 23632321231524Offspring23564323212212

1234561234561-12100-301202-3001-03013-221-2324-10-125-2-06--

1234561-321002-03013-2224-125-06-

2.3.2 Genetic OperatorsThe mutation operator is a modified version of Cat Saturation (CS) and serves as a local search operator to improve solutions in the population. But it is computationally expensive procedure in hGA. Therefore, it is only applied to good solutions, which are most likely to improve upon the best feasible solution. Since the parents in a population are the best solutions of the last population; mutation operator is applied to them. After mutation, the original network is replaced with the mutated one.

2.3.3 EvaluationThe problem considered in this study is to design communication backbone network for minimizing cost subject to maximum allowable average delay and k-node-connectivity. The evaluation of candidate networks depends on the routing and the capacity assignment algorithm. In the study, the shortest path policy for static routing had been used. Where routing policy is determined once and is not change with changing traffic patterns.In this policy, the packets between two nodes are sent via the shortest path between them. In hGA, the shortest paths are determined by Dijkstra's algorithm and k-node-connectivity is tested by using the shortest path augmenting algorithm. Based on the these explanations, the general procedure for the evaluation of the networks is as follows:

step 1: Determine the routing schema;step 2: Calculate the average flows on the links; step 3: Determine the capacity of the links; step 4: Calculate average delay of the network and test k-node-connectivity; step 5: Calculate cost of the network.

2.3.4 Selection Mechanism After mutation operator, it is possible to obtain infeasible networks which do not meet k-node connectivity and maximum delay requirements. Discarding these solutions is not desirable, because an infeasible solution an infeasible solution might be closer to the optimum than many feasible solutions. In the selection routine:first, the offspring and mutated solutions are sorted according to the following two rules: Between two feasible or two infeasible solutions with respect to the delay constraint only, the one with the lower cost is superior to other one.Between one feasible and one infeasible solution: if Tinfeasible < (1+) Tmax, where is a small number less than 1, the infeasible solution is treated as if it was feasible and the first rule is applied. Otherwise, the feasible solution is superior to the infeasible one, regardless of cost.

2.3.5 hGA procedure for Backbone Network Design procedure: hGA for Backbone Network Design (Konak-Smith,1999)input: network data (V, A, F, C), GA parametersoutput: best total link cost begint 0;initialize P(t) by adjacency matrix; fitness eval(P);while (not termination condition) docrossover P(t) to yield C(t) by uniform crossover operator;mutation P(t) to yield C(t) by cat saturation algorithm;fitness eval(C); select P(t+1) from P(t) and C(t);t t+1;end output best total link cost; end

2.3.6 Computational Results To investigate the performance of hGA, the problem set given in Pierre and Elgibaoui [54] was used. Pierre, S. and A. Elgibaoui: A tabu search approach for designing computer-network topologies with unreliable components, IEEE Transactions on Reliability, vol.46, no 2, pp. 350-359, 1997.

In this reference, Tabu Search (TS) had been employed to solve this problem. Konak and Smith compared the average results of the hGA with the best results of TS by using a paired t-test. The p value indicates the significance of the hypothesis that hGA generates lower cost than the results of TS. Based on the statistical analysis, they showed that hGA managed to improve on the results of the TS with reasonable CPU times in almost all cases. In addition, paired t-test was applied to compare with the best results of hGA and TS and it was demonstrated that the best results of hGA excluding problem 9 were statistically better than the results of TS at a significance level of 0.043.

2.3.6 Computational Results

ProblemhGAparametersReference ResulthGA AveragehGA WorsthGA BesthGA vs. TS#{|v|,K Tmax}{ , NG, }CostTCostTCPUCostTCostT%Costp1{10, 2, 100}{10, 100, 0.1} 25272 6924923940.78 2523097 24171 89 1.30.0762{10, 2, 100}{10, 200, 0.1} 25272 6924901951.5 2564193 24507 99 1.40.0693{10, 2, 100}{15, 200, 0.1} 25272 6924646982.3 2488399 24137 99 2.40.004*4{15, 2, 150}{15, 100, 0.1} 6516114255985922.9 59138 102 55008 1114.00.000*5{15, 2, 100}{15, 100, 0.1}N/A55510913.6 5609290 54838 986{15, 2, 100}{15, 100, 0.0}N/A56369793.6 5782580 55112 837{15, 3, 200}{10, 100, 0.1} 60907 7957946922.8 5842384 57239 79 4.80.000*8{15, 4, 200}{10, 100, 0.1} 60013 93592931013.7 6113492 57334122 1.20.1629{15, 4, 200}{10, 100, 0.1} 41626 90592931013.7 6113492 57334122-29.710{20, 2, 150}{15, 100, 0.1}144131104102272798.410528460 98017 7429.00.00011{20, 3, 250}{15, 100, 0.1}1054951121014398512.310371488 99759102 3.80.001*12{20, 4, 250}{15, 100, 0.1}103524 881011669016.110226479100054101 2.20.002*13{20, 5, 250}{15, 100, 0.1}10612283.11046658310.910624072102617 96 1.30.07414{20, 3, 50}{15, 100, 0.1}N/A107846497.810973948106413 4915{20, 3, 50}{15, 100, 0.0}N/A108454499.911102650105337 4916{23, 2, 80}{15, 100, 0.1}210070 581399256412.814446066137270 7034.60.000*17{23, 3, 190}{15, 100, 0.1}148340 911327747520.413456265131096 8610.50.000*18{23, 4, 190}{15, 100, 0.1}163129 901336378123.013446985132002 6718.00.000*19{23, 5, 190}{15, 100, 0.1}139154 861347418926.013560690133411 80 2.50.000**Significant results at a level of 0.05



Bicriteria LAN Topological Design3.1 Formulation of b-LAN Topology Design3.2 B-LAN Topology Design based on GA3.3 Numerical Example


3.1 Formulation of b-LAN Topology DesignWe define the following notations in order to formulate the bicriteria LAN topology design considering the message delay and connected cost.

3.1 Formulation of b-LAN Topology DesignM/M/1 model is used to describe a single cluster (LAN segment) behavior.Then we can formulate the bicriteria LAN topology design problem as the following nonlinear 0-1 programming model:

3.1 Formulation of b-LAN Topology Design

3.2 b-LAN Topology Design based on GAInitialization

From the previous example, we can see that all digits of the Prfer number are the figures of the centers displayed by box.So, the initialization of a chromosome(i.e., a Prfer number) is performed from that randomly generated n+m-2 digits in range [1, n].

3.2 b-LAN Topology Design based on GADefinition: Pareto optimal solutions

3.2 b-LAN Topology Design based on GAEvaluationinput: chromosome v, network data (V, A, C)output: fitness eval(v)step 1: Convert chromosome represented by Prfer number to a spanning matrix Xstep 2: Calculate the objective values ( )step 3: Choose the solution points, and compare with the stored solution points at the previous generation and select the best points to save again, where is the maximum (minimum) value of objective function q at generation t.

3.2 b-LAN Topology Design based on GA

3.2 b-LAN Topology Design based on GASelectioninput: chromosomes P(t), C(t)output: chromosomes P(t+1)

3.2 b-LAN Topology Design based on GACrossover (multi-point or uniform)

Mutation (swap)selected positions

parent 1:17234658

parent 2:46357182

offspring:47354688

3.2 b-LAN Topology Design based on GAModification of the ChromosomeBecause of the existence of the maximum number which is capable of connecting on each center, the chromosomes generated randomly in the initial population and the offspring produced by crossover may be illegal in the sense of violating the maximum number of connection for each center.Let be the set of centers whose the maximum number of connection has not been checked and modified in a chromosome.If a center i violates the constraint with the maximum number gi of connection for center i, this means that the number of this center in the chromosome is more than gi-1. Then decrease the number of the center by checking the extra center and randomly replace it with another center from .

3.3 Numerical Example

3.3 Numerical ExampleMethod of TOPSISFor each Pareto solutionconstruct the normalized values:calculate the separation measures:calculate the Euclidean distance , where the solution closest to 1 is selected.

3.3 Numerical ExampleUsing the TOPSIS method, the chromosome of the best compromise solution in the previous figure is as follows (16 14 connection cost and 0.09935 message delay)chromosome v: [ 6 3 5 5 6 6 4 1 5 2 3 3 4 1 5 2 5 3 2 2 4 2 1 1 2 6 4 6 5 5 6 1 2 4 ]




Bi-level Hierarchical GA for Reliable Network Topological Design4.1 Introduction4.2 Problem Formulation4.3 Genetic Approach 4.4 Numerical Example

4. Bi-level Hierarchical GA for Reliable Network Topological DesignSpanning Tree Topology

Topological design problems for broad band communication networks have received attention by many related researchers.

The topological structure of these networks can be based on service centers, terminals (users), and connection cables.

Lately, these network systems are designed with fiber optic cable, since the requirements from users increased.

4.1 IntroductionFiber-optic cable is used because of its capabilities: Huge bandwidth (nearly 50 Tbps)Low signal attenuation (as low as 0.2 dB/km)Low signal distortionLow power requirementLow material usageSmall space requirementConsidering the high cost of the fiber optic cable, it is desirable that the network architecture is composed of a spanning tree.As designing the network systems, an important step is to find the best layout of components to optimize the performance criteria, such as cost, message delay, traffic, reliability, and so on.

4.1 IntroductionWe introduce a bi-level hierarchical GA for designing the reliable network topology with a structure that is comprised of the spanning tree of the centers and users using fiber-optic cables.

Generally, anticipating future requirements of network, centers are set up by government, public enterprise, or some network providers.

Then, users in each house connect these centers with some criteria such as connection cost, average message delay, and so on.

We can see the special structure that has two kinds of decision maker, i.e., government, public enterprise, or some network providers, and users.

4.1 IntroductionWe can also see two hierarchical levels exists between decision makers.

So, network design problems can be represented by hierarchical optimization or bi-level programming problems, which are extensions of Stackelberg games and decentralized planning problems with multiple decision makers in a hierarchical organization.

Thus, we propose a bi-level hierarchical genetic algorithm for solving the network design problems with network reliability constraints, which can be represented by hierarchical optimization problems.

4.1 IntroductionWe define the following notations in order to formulate the bicriteria network topology design:n: the number of service centersm: the number of users to be clusteredX: nn center topology matrix

Y: n m center topology matrix

S: n(n+m) spanning tree matrix [X Y]gi: the maximum number of users capable of connecting to center iR(S): the network reliability of a candidate network design X

4.1 IntroductionWe define the following notations in order to formulate the bicriteria network topology design:w1ij : the cost of the link between centers i and j w2ij : the cost of the link between center i and user j Ci : the traffic capacity of center iij : the delay per bit due to the link between centers i and j U : mm users traffic matrix

4.2 Problem FormulationWe can formulate the bicriteria network topology design problem with an connection cost and an average message delay objective functions, considering network reliability:A M/M/1 model is used to describe a single cluster (LAN segment) behavior.

4.2 Problem FormulationBi-level Programming ProblemThe above mathematical model is formulated as bi-level programming problem that is a simple example of multi-level programming problems to solve the decentralized planning problems with multiple decision makers in a hierarchical organization.In these cases, the Stackelberg strategy (solution) has been usually employed as solution concept, based on Stackelberg game theory. A hierarchical organization has the following common features: Interactive decision-making units exist within a predominantly hierarchical structure; The execution of decisions is sequential, from higher (leader or upper) to lower (follower) levels;

4.2 Problem FormulationEach unit independently maximizes its own benefits, but is affected by the actions of other units through externalities; The external effect on a decision maker's problem can be reflected in both his/her objective function and the set of feasible decisions.In the basic concept of hierarchical optimization techniques, The leader sets his/her goal or decisions and then asks followers for their optima calculated in isolation;The follower's decisions are submitted to and then modified by the leader with consideration of the overall benefit for the organization; The process is continued until a solution is reached.

4.3 Procedure of Tree-based Reliability CalculationNetwork Reliability: the probability of all operative nodes being connected, i.e., the probability of all operating nodes communicating. We have two subtrees, one with root i and the other having as its root j.

We assume the reliability of subtree with root i and subtree with root j are known.

We wish to compute the reliability of new- subtree with root j obtained by joining subtree with root i into subtree with root j using the link (i, j).

Then, we join the rooted subtrees into larger and larger rooted subtrees using these recursion relations until the reliability of entire network is obtained.Fig 9.4 Recurrence Relation of Treeroot jroot i

4.3 Procedure of Tree-based Reliability CalculationFor the root node 1 and n nodes, we can calculate the reliability of tree networks as follows:input: network data (V, A, C, D), chromosomeoutput: probability of all operating nodes communicating

4.4 Genetic ApproachRepresentation

InitializationWe employ the following representation that has two kinds of genotype: The service centers are represented by Prfer numberThe users are described by clustering string. Therefore, we randomly generate the chromosome in the initialization process shown in the previous figure, i.e..Service centers are composed of n-2 digits (Prfer number) randomly generated in the range [1, n]Users are make up of m digits (clustering string) randomly generated in the range [1, n], which mean how to allocate the users to service centers so that each user belongs to a specific service center.

1234Prfer number:1122

4.4 Genetic Approach InitializationCenters:Users:

67891011121314151617181920212223

4.4 Genetic Approach For upper-level, we evaluate as follows:

For lower-level, we evaluate as follows: Evaluation Selectioninput: chromosomes v(t), v(t)output: chromosomes v(t+1)

4.4 Genetic ApproachCrossover (multi-point or uniform)

Mutation (swap)selected positions

parent 1:17234658

parent 2:46357182

offspring:47354688

4.4 Genetic Approachbi-level GA for reliable network design problems formulated by hierarchical optimization problem.

procedure: bi-level GA for RND problems input: network data (V, A, C, D), GA parametersoutput: best total link cost begint 0;initialize population of upper-level PL(t); while (not termination condition) dot 0;while (not termination condition of lower-level) doinitialize population of lower-level PF(t);crossover PF (t) to yield CF (t) by multi-point or uniform crossover;mutation PF (t) to yield CF (t) by swap mutation;fitness eval(CF); select PF (t+1) from PF(t) and CF(t);t t+1;endcrossover PL (t) to yield CL (t) by multi-point or uniform crossover;mutation PL (t) to yield CL (t) by swap mutation;fitness eval(CL);select PL(t+1) from PL(t) and CL(t); t t+1;end output best total link cost; end

4.5 Numerical Example6 service centers (n = 6), 30 users (m = 30) The cost matrix of links between centers i and j (cij) is randomly generated from [100, 250]The cost matrix of links between center i and user j (cij) is randomly generated from [1, 100]The traffic capacity of center i (Ci) is 300Also we set the operative probability of centers as 0.95, the operative probability of users as 0.9, the operative probability between centers as 0.9, the operative probability between center and user as 0.85, and Rlim is 0.9.We adopt the users traffic matrix U described in the paper proposed Elbaum and Sidi (IEEE/ACM Trans. on Net., 1996).The parameters for genetic algorithm are set as follows:popSizeL =100, popSizeF =200maxGenL =200, maxGenF =50, pC =0.4, pM =0.6experimented by 20 times.

4.5 Numerical ExampleThe best result is as follows (cost is 1377, delay is 0.020446, reliability is 0.920843): Centers: 3 2 3 1Users : 5 4 4 6 4 5 3 3 1 6 5 1 6 2 4 2 1 2 3 3 2 1 1 3 1 5 5 5 5 2

ConclusionThe reliability and service quality requirements of communication networks and the large investments in communication infrastructure have made it critical to design optimized networks that meet performance parameters. These factors have promoted researcher to develop new models and methodologies for network design.In this chapter, We introduced Centralized Network DesignBackbone Network DesignLAN Topological DesignReliable Network Topological DesignWe proposed several genetic algorithms for solving communication network design problems. In numerical experiments, we can see that the genetic algorithms have effectiveness

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Graduate School of Information, Production and Systems, Waseda University 9. Communication Network and LAN Design