Reliability analysis of δ-shock models based on the Markovian arrival process

The Markovian arrival process (MAP) is a versatile counting process with dependent and non-identically distributed inter-arrival times following the phase-type distribution. In this article, we study the classical (Formula presented.) -shock model and a mixed (Formula presented.) -shock model by assuming the MAP of shocks. We derive explicit expressions for the reliability and the mean lifetime of the system. Further, we study an optimal replacement policy based on the MAP. We illustrate the developed results through several numerical examples.