Static Network Reliability Estimation Under the Marshall-Olkin Copula

Abstract:

We consider a multicomponent system for which each component has a random state, and a monotone binary structure function of those states gives the operational status of the system: 0 if failed and 1 if operational. Although our methods apply more generally, in the talk we will focus on the special case of a network whose links have random binary states and which is operational when a certain subset of the nodes are connected by operational links. We are interested in estimating the unreliability (failure probability), u, of large networks when u is small. Both exact computation and standard Monte Carlo are then impractical. Various rare-event simulation methods have been designed for the situation where the components fail independently of each other, but this independence is arguably unrealistic. In this talk, we consider a dependence model based on a Mashall-Olkin copula, which captures the idea that some random events can take down certain groups of components simultaneously. We introduce and compare two types of rare-event simulation methods to estimate u in this setting. They are adapted from the independent case and they both extend the state of the network with auxiliary variables. One type is based on a conditional Monte Carlo idea called permutation Monte Carlo (PMC) and has a refinement named the turnip method. The other type uses generalized splitting (GS). Each type has several variants and the best one depends on the situation, as we shall illustrate. PMC and turnip give an estimator with bounded relative error in the limit when u converges to 0 for a fixed network, and they win when the network is not too large and u is very small. But when the network grows too large, they become ineffective and GS is the only viable method. We provide numerical illustration with networks of a few thousand links and withu < 10e-15.