Shock models governed by the mixed poisson process with Pareto mixing variable

Abstract

In this paper, we first consider the mixed Poisson process with Pareto mixing variable (MPPP) studied by [Citation1] and explore its additional properties. We study the asymptotic behavior of the coefficient of variation and the weak positive dependence property of the MPPP. These properties make the MPPP suitable in real-life situations, and the same is explored in the application part of this paper. Next, we derive the distribution of the first-passage time, the first waiting time, the arrival time for the kth event, and the interarrival times for the MPPP. We also explore the application of the MPPP in shock modeling considering three different classes of shocks, namely the extreme shock model, the cumulative shock model, and the δ-shock model. The survival function, expected value, and variance of failure time are obtained for all the treated cases in the shock model. Further, we perform a sensitivity analysis to investigate the effect of model parameters on system reliability for the extreme shock model and δ-shock model.