Estimating the s − t Reliability Function Using Importance and Stratified Sampling

This paper considers an undirected network G with a node set V and an arc set E = {1, …, n}. The nodes are perfect, but the arcs fail randomly and independently with known probabilities 1 − p₁, …, 1 − p_n. The system reliability g(p) is defined as the probability that two nodes, s and t ∈ V, are connected where p = (p₁, p₂), p₁ = (p₁, …, p_n′) and p₂ = (p_n′+1, …, p_n). This paper describes a highly efficient Monte Carlo sampling plan for estimating the sensitivity of g(p) as the component reliabilities in p₁ vary over a set of values P in the n′-dimensional unit hypercube. Sensitivity analysis becomes an important consideration when contemplating component replacement and alternative system designs, and when accounting for the effect of using sample estimates, based on historical failure data, for the true component reliabilities. The sampling plan is a major advance over most other Monte Carlo proposals which only estimate g(p) at a single point p. The method combines importance and stratified sampling techniques to gain its advantage. In addition to unbiased point estimates, the paper derives individual confidence intervals as well as simultaneous confidence intervals for all the points. It also describes the steps for implementation and illustrates how the plan works in practice.