On the reliability and optimal maintenance of systems under a generalized mixed δ -shock model

Abstract

This article is a study on the reliability characteristics of a system under a failure model called the generalized mixed $\delta$ -shock model. We assume that the system is subject to shocks according to a stochastic process. Each shock may cause some damage to the system. The system fails either the magnitude of the damage caused by a shock exceeds a threshold ${\gamma }_2$ or the time between two consecutive shocks is less than a pre-specified threshold $\delta$ and simultaneously magnitude of the damage is bigger than a pre-specified critical threshold ${\gamma }_1$ ( ${\gamma }_1<{\gamma }_2$ ). The survival function and other characteristics of the system lifetime are investigated. By imposing a cost function, we arrive at an optimal replacement policy for the system based on the proposed failure model. Several examples are provided under which we illustrate the theoretical results numerically and graphically.