Static Network Reliability Estimation via Generalized Splitting

Abstract

We propose a novel simulation-based method that exploits a generalized splitting (GS) algorithm to estimate the reliability of a graph (or network), defined here as the probability that a given set of nodes are connected, when each link of the graph is failed with a given (small) probability. For large graphs, in general, computing the exact reliability is an intractable problem and estimating it by standard Monte Carlo methods poses serious difficulties, because the unreliability (one minus the reliability) is often a rare-event probability. We show that the proposed GS algorithm can accurately estimate extremely small unreliabilities and we exhibit large examples where it performs much better than existing approaches. It is also flexible enough to dispense with the frequently made assumption of independent edge failures.

Dependent failures in highly reliable static networks

“…Good values for τ and {γ t } can be estimated by an (independent) adaptive pilot algorithm, as explained in the Appendix. In (Botev et al 2012) we argue that s = 2 is a good choice that yields satisfactory empirical results. Thus, we will use s = 2 for the rest of the paper.…”

“…To estimate the reliability of a network under the Marshall-Olkin copula we use the Generalized Splitting (GS) method described in (Botev et al 2012) (which is an adaptation of the splitting method of Kahn and Harris). Here we use the splitting procedure with the following important difference.…”

“…Here we use the splitting procedure with the following important difference. In (Botev et al 2012) we use the construction process, in which all components are initially failed and Y is a vector of repair times indicating the time at which each component of the network is repaired and becomes operational. The network is thus gradually "constructed" over time and we wish to estimate u = P(S(Y) > 1), where S(Y) is interpreted as the first time the network becomes operational, given the repair times Y.…”

“…, γ τ . This algorithm is an modification of the one given in (Botev et al 2012). We have tailored the algorithm to the destruction process, instead of the construction process.…”

“…Modeling dependence in static networks has been considered previously, but the existing proposals either do not offer a viable algorithm to estimate the reliability in the rare-event setting, or do not capture the dependence in a realistic way (Singpurwalla 2002, Ram andSingh 2009), (Walter et al 2009), (Botev et al 2012). For those proposals that consider the rare-event setting, the dependence is typically modeled using a Gaussian, Pareto, or Weibull copula, which do not easily account for rare shock events that can knock out a multitude of network components simultaneously.…”