Bounds on expected performance of networks with links subject to failure

Upper and lower bounds on the expected maximum flows and benefits in capacitated networks with links subject to random failure are presented. Such bounds are of interest both because of the high or prohibitive cost of computing the exact expected values, and because the exact values will depend on the extent to which management avails itself of the opportunities for rerouting of flows in response to link failures. Algorithms for computing these bounds are discussed. These algorithms are sufficiently efficient to be applied to large networks having numerous possible failure states. An application to water-network seismic risk analysis is described. I. INTRODUCTION Networks subjected to extreme events such as earthquakes or severe weather are apt to experience failures of individual links. This paper presents computable bounds on network performance for these cases, the bounds being calculated by solving minimum-cost flow routing problems. These bounds were developed to assess network performance for numerous possible situations, for nonlinear performance measures, and for large networks. Two measures of network performance are used here: (i) the expected flow to all demand nodes and (ii) the expected benefits due to flows, where benefits are a function of flows from supply nodes to demand nodes. In some cases, the bounds proposed here will equal the expected network performance. In particular, a bound is exact for cases in which a manager is unable or unwilling to reroute flows after link failures occur. Calculation of bounds on network performance is proposed because the exact determination of network performance, subject to probabilisitic failure, is excessively burdensome for large networks. For a network with n links subject to failure there are 2" failure states to be enumerated. Even calculation of the probability that a desired flow pattern may be achieved involves substantial computation; Willie [8] presents one algorithm to calculate this probability, based upon enumeration of link sets whch either ensure (in the case of successful operation of all members) or prevent (in the case of failure of all members) successful operation of the network. Partial enumeration of such sets reduces computation time, but introduces a possible bias in the resulting probability estimate. Onaga [7] considered the problem of calculating bounds for the expected value of