A shock and wear model with dependence between the interarrival failures

The Markovian arrival process is a suitable model for the arrival of external failures, it has the relevant property of dependence between the consecutive interarrival times. In the present paper we consider a shock and wear system in which the interarrival of the internal failures are also dependent, and a Markovian arrival process is used for modeling the arrival of this type of failures. So, we present a system under two types of failure governed by two independent Markovian arrival processes. The failures can be repairable or not. Two types of repair are considered: minimal and perfect, the first one is an instantaneous repair and the repair time of the second one is governed by a phase-type distribution. Under these conditions, two systems are considered, depending on the finite number of minimal repairs before a replacement would be random or fixed. For these systems, the stationary distribution, the number of minimal and perfect repairs, and the number of replacements are calculated. Some special cases of the systems are presented, showing the generality of the models. A numerical application illustrates the results.