The Distribution of Maximum Flow with Applications to Multistate Reliability Systems

This paper describes an efficient Monte Carlo sampling plan for estimating the distribution of maximum flow in a directed network whose arcs have random capacities. Such a network can be used to represent a multistate system whose multistate components (capacities) are subject to random deterioration. The proposed sampling plan uses an easily computed a priori upper bound on the complementary distribution function to obtain an unbiased point estimator with smaller variance than the estimator obtained by crude Monte Carlo sampling. The paper also describes procedures for interval estimation and for assessing when the sampling experiment has achieved a specified accuracy. To facilitate sampling, we use cumulative processes to characterize deterioration, leading to the treatment of arc capacities as being multinormally distributed. We describe a technique for checking the appropriateness of this model with regard to lower and upper bounds on capacity. We also describe a procedure for deriving a confidence interval on the measure used to assess variance reduction. An example illustrates the sampling plan, and a concise summary gives all steps needed to implement the plan.