A reliability system under cumulative shocks governed by a BMAP

A system subject to simultaneous shocks is considered. The system fails when the cumulative quantity of shocks reaches a previously fixed threshold. The evolution of the system is studied using matrix-analytic methods, the model for the arrivals of shocks being a batch Markovian arrival process. Under this arrival model, the interarrival times between shocks are dependent. For this system the lifetime and the number of shocks while it is operational are calculated. Assuming that the system is replaced after failure, we study the renewal process associated with the replacements, and calculate the probability of the number of replacements. A numerical application is presented illustrating the calculations in the paper. This paper extends previously published literature.