A shock and wear system with memory of the phase of failure

We study a system subject to shocks arriving following a Markovian arrival process. These shocks can be repairable or fatal. The non-fatal shocks cause damage to the system. The lifetimes between shocks and the repair times are governed by phase-type distributions. Repair is not as good as new, a geometric process is introduced for modeling the successive operational times. After a prefixed number of shocks, the system is replaced by a new and identical one. When the system is repaired, it returns to the operational phase in which the shock arrived, this is a model with memory of the phase of failure. This system is studied in a transient and stationary regime, the availability and the rates of occurrence of the different types of failures are calculated. This system extends others previously published in the literature. Numerical applications illustrate the calculations of the paper using the Poisson process, the PH-renewal process, and the Markov-modulated Poisson process as special cases of the arrival of shocks. These applications are performed presenting the results in an algorithmic form to be computationally implemented.