A Genetic Algorithm Approach to Optimal Topological Design of All Terminal Networks

In the design of communication networks, one of the fundamental considerations is the reliability and availability of communication paths between all terminals. Together, these form the network system reliability. The other important aspect is the layout of paths to minimize cost while meeting a reliability criterion. In this paper, a new heuristic search algorithm based on Genetic Algorithms (GA) is presented to optimize the design of large scale network topologies subject to a reliability constraint. The search works with an improved Monte Carlo simulation technique to estimate the system reliability of a network topology. 1 Corresponding author. INTRODUCTION An important part of network design is to find the best way to layout the components (nodes and arcs) to minimize cost while meeting a performance criterion such as transmission delay, throughput or reliability. This design stage is called “Network Topological Optimization”. In a topological network design problem, a main concern is to design networks which operate effectively and without interruption in the presence of component failures. Reliability is concerned with the ability of a network to carry out desired network operations. Generally, a large scale network has a multilevel, hierarchical structure consisting of a backbone network and several local access networks (Boorstyn et al, 1977). Therefore, designing the topology of a large scale network can be divided into two problems, the backbone network design and the local network design. This study is mainly interested in large scale backbone network design. For backbone network design, an important connectivity measure is reliability. In a communication network, all terminal network reliability can be defined as the probability that every pair of nodes can communicate with each other (Jan et al, 1993; Colbourn, 1987). Many studies have considered topological optimization with a network reliability criterion. For example, Jan et al (1993) used a decomposition method based on branch and bound to minimize total network cost under a system reliability constraint. Their method can only solve small networks because as the number of arcs increases, the number of possible layouts grows faster than exponentially. Because of this complexity, other existing methods are not computationally feasible for designing large scale network topologies with very confining assumptions. Therefore, a heuristic search algorithm based on Genetic Algorithms (GAs) is developed to find a network topology which has minimum cost, subject to a system reliability constraint. A computer communication network can be modeled by a probabilistic graph G = (N, L, p), in which N and L are the set of nodes and arcs that corresponds to the computer sites and communication links, respectively. The networks considered in this paper are assumed to have bi-directional links and therefore are modeled by graphs with non-oriented edges. We further assume that the graph under discussion has no redundant arcs. Any graph G = (N, L) is said to be connected if there is at least one path between every pair of nodes. A sub-graph G1 of G is a graph, of which all nodes and arcs are contained in G. i.e., G1 = (N1, L1) where N1 ⊆ N and L1 ⊆ L. If N1 = N, the sub-graph G1 is called a “spanning sub-graph”. In a connected graph G of arcs and n nodes, a tree T is a spanning tree consisting of n-1 arcs. The deletion of any edge from a tree results in a disconnected graph. Therefore a connected graph should be at least a spanning tree with n-1 edges. A communication network topology should be at least a spanning tree and communication network reliability must be greater than the required system reliability value, p0. In addition to a simple network connectivity check (i.e., does a minimum spanning tree exist in the network), Roberts and Wessler (1970) proposed a “2connectivity” measure in the design of communication network topologies. “2connectivity” means that there are at least two paths between each pair of nodes, rather than one. In the literature, many studies considered this measure to be a reasonable constraint of reliability in the design of network topology. In our study, it is used to establish the initial population and to constrain subsequent populations. Therefore, the final network design will meet the system reliability constraint and contain at least two different paths between all pairs of nodes. Under the following assumptions: (1) the location of each network node is fixed and given, (2) each cij and the p are fixed and known, where cij is the cost of link in the network between nodes i and j, and p,q are link reliability and unreliability (p + q = 1), (3) each link is bi-directional, (4) there are no redundant links in the network, and (5) the probability of failure of a link is independent of the states of the other links, the main problem can be stated mathematically as follows: Minimize z = ∑ cij xij Subject to : f(x) ≥ p0 { } x 0,1 ij ∈ are the decision variables and f(x) is the network reliability. The all terminal system reliability of a network is defined to be the probability that every pair of nodes can communicate with each other. At any instant of time, only some arcs of G may be operational. A state of G is a sub-graph (N,L′) with L′∈L, where L′ is the set of operational arcs. An operational state is generally called a pathset, and a minimal operational state is a min-path. A failed state L′ is called L \ L′ (a cutset) and when L′ is a maximal failed state, L \ L′ is a min-cut (Colbourn et al, 1991). The reliability of G, RelK(G), is the k-terminal reliability: If K=N, this is the all terminal reliability, Rel(G). It is easy to formulate a network in state L′ ⊆ L, with reliability as follows: ∏ pe ∏ qe where L′ is the set of operational arcs. e ∈ L’ e ∈ ( L \ L’ ) Summing this state occurrence probability over all operational states gives the network system reliability. There are basically two approaches to network system reliability calculation; simulation and analytic. All known analytic methods have worst case computation times which grow exponentially with the size of the network (Aggarwal et al, 1982; Nakazawa, 1981; Cavers, 1975; Rai, 1982; Aggarwal and Rai, 1981). Monte Carlo simulation methods, for which computation time grows only slightly faster than linear with network size, have been the method of choice for more than trivial sized networks. In this research, we used a Monte Carlo simulation technique to predict the network reliability which substantially reduces the variance of the estimator when compared to “crude” Monte Carlo (Yeh et al, 1994). This reduced variance Monte Carlo is based on a two tiered hierarchical approach to sampling which makes use of how many arcs fail during a given simulation. SOLUTION APPROACH BASED ON GENETIC ALGORITHMS A GA is developed as a solution methodology for network topological optimization with a reliability constraint. GAs were pioneered by Holland (1975) and Goldberg (1989) for continuous non-linear optimization, and later extended (Cohoon et al, 1991; Biegel et al, 1990; Muhlenbein et al, 1988) for combinatorial problems. In GA, the search space is composed of possible solutions to the problem, each represented as some convenient data structure, referred to as the chromosome. Each chromosome has an associated objective function value, called the fitness value. A good chromosome is the one that has a high fitness value. A set of chromosomes together with their associated fitness values is called the population. This population, at a given stage of the GA, is referred to as a generation. In a conventional GA, candidate solutions are represented by strings of numbers using a binary or non-binary alphabet. The present algorithm uses a binary coding structure for representing candidate solutions. A binary set is used to represent arcs, where the maximum number of non-redundant, undirected arcs for a network of n nodes is given by (n-1)n/2. For example, a simple network whose base graph consists of 5 nodes and 10 possible links can be represented by: [ 1 1 0 1 1 0 1 1 0 1 ] [ x12 , x13 , x14 , x15 , x23 , x24 , x25 , x34 , x35 , x45 ] where xij represents a link connecting two nodes i and j. If xij is equal to 1, there is a connection between these two nodes. If xij is equal to 0, then there is no connection. The initial population which consists of a set of feasible solutions (2connected networks) is generated in a random fashion. For determining this initial population, a number of experiments were carried out. A candidate network consists of some randomly selected arcs between nodes. The selection of the probability values which are used in deciding whether an arc exists or not was an important step to generate the initial population. In an experimental design with 10, 20 and 30 nodes, the following characteristics were systematically controlled. i) Arc probabilities between [0,1], which determines the existence of an arc between nodes, are selected, ii) The system reliability value of each connected network is estimated using Monte Carlo simulation, iii) The probability values of the existence of arcs and the corresponding network reliability values are compiled. The aim was to determine the intervals of the probability values which result highly reliable networks. Any initial population can then be generated by using probabilities within these intervals. Table 1 shows the resulting probability intervals from the experiments described above which were used for the initial populations. Table 1. Probability values used to generate the initial population. Number of Nodes (n) Probability of an Arc