The Markov discrete time δ-shock reliability model and a waiting time problem

Abstract

delta-shock model is one of the widely studied shock models in reliability theory and applied probability. In this model, the system fails due to the arrivals of two consecutive shocks which are too close to each other. That is, the system breaks down when the time between two successive shocks falls below a fixed threshold delta. In the literature, the delta-shock model has been mostly studied by assuming that the time between shocks have continuous distribution. In the present paper, the discrete time version of the model is considered. In particular, a proper waiting time random variable is defined based on a sequence of two-state Markov dependent binary trials and the problem of finding the distribution of the system's lifetime is linked with the distribution of the waiting time random variable, and we study the joint as well as the marginal distributions of the lifetime, the number of shocks and the number of failures associated with these binary trials.