Analysis of system reliability with dependent repair modes

Abstract

This paper proposes an imperfect-repair model for repairable systems where two repair modes, perfect and minimal, occur in accordance with a Markov chain. It investigates the characteristics of the model and presents a statistical procedure for estimating the lifetime distribution of the system, based on consecutive inter-failure times. Under the Brown-Proschan imperfect repair model, the system is repaired to: good-as-new under perfect-repair, its condition prior to failure under minimal-repair. This imperfect-repair model generalizes the Brown-Proschan imperfect-repair model, by allowing first-order dependency between two consecutive repair modes. The model assumes that, at failure, the system undergoes either perfect repair (restore to like new) or minimal repair (restore to like just before failure), and the repair modes are subject to a Markov process. The estimation procedure is developed in a parametric framework for incomplete data where some repair modes are not recorded. The s-expectation-maximization principle is used to address the incomplete-data problem. Under the assumptions that the lifetime distribution belongs to a parametric family having aging property and explicit form of the survival function, an algorithm is developed for finding the MLE (maximum likelihood estimates) of the reliability parameters; the transition probabilities of the repair modes, as well as the distribution parameters. A Monte Carlo study shows the consistency, effect of aging rate, effect of transition types, and effect of missing data, for the estimates.