Survival probabilities for shock and wear models governed by phase-type distributions

A device submitted to shocks arriving randomly and causing damage is considered. The inter-arrival times follow continuous phase-type distributions. Lifetimes between shocks are affected by the number of cumulated shocks and they follow continuous phase-type distributions. Every shock can be fatal or not, with a probability that follows a discrete phase-type distribution. The device can support a maximum of N shocks. We calculate the distribution of the lifetime of the device in terms of the counting process of the number of shocks, and illustrate the calculations by means of a numerical application. Computational aspects are introduced. This model extends other previously considered in the literature.