Numerical Solution and Inference for Interval-Reliability of Repairable Components

The renewal and interval-reliability functions for repairable components are solved for given general probability distribution of renewal interarrival times. Two powerful numerical methods are 1) A finite difference approach wherein the renewal equation is written in discrete form and then the resulting system of algebraic equations is solved recursively. 2) Transforms of the renewal equation are treated and Fourier series expansions are used. The Fourier series approach has a wider range of applicability than finite difference in that it can be used to solve the complete renewal problem. A new equivalent form of the interval-reliability integral equation leads to a computationally faster scheme (by a factor of 10) and a simplified approximate solution for high reliability components. A numerical solution for confidence intervals has also been generated for the average interval-reliability of a component within a system, given component failure data, using a pseudo-Bayesian approach. The goal is to choose priors that lead to classical limits and not the usual Bayesian limits. The intervals yield close-to-exact frequency limits depending on sample size, Weibull shape parameter and the true reliability.