Reliability Assessment for Censored ${\boldsymbol{\delta}}$-Shock Models

Abstract

This paper is devoted to study censored $\delta$-shock models for both cases when the intershock times have discrete and continuous distributions. In particular, the distribution and moments of the system’s lifetime are studied via probability generating functions and Laplace transforms. For discrete intershock time distributions, several recursions for evaluating the probability mass function, the survival function and the moments of the system’s lifetime are given. As it is shown for the discrete case, the distribution of the system’s lifetime is directly linked with matrix-geometric distributions for particular classes of intershock time distributions, such as phase-type distributions. Thus, matrix-based expressions are readily obtained for the exact distribution of the system’s lifetime under discrete setup. Also, discrete uniform intershock time distributions are examined. For the case of continuous intershock time distributions, it is shown that the shifted lifetime has a compound geometric distribution, and based on this, the distribution of the system’s lifetime is approximated via discrete mixture distributions having a mass at $\delta$ and matrix-exponential distributions for the continuous part. Both for the discrete and the continuous case, Lundberg-type bounds and asymptotics for the survival function of system’s lifetime are given. To illustrate the results, some numerical examples, both for the discrete and the continuous case, are also given.