Shock models based on renewal processes with matrix Mittag-Leffler distributed inter-arrival times

Some general shock models are considered under the assumption that shocks occur according to a renewal process with the matrix Mittag-Leffler distributed inter-arrival times. As the class of matrix Mittag-Leffler distributions is wide and well-suited for modeling the heavy tail phenomena, these shock models can be very useful for analysis of lifetimes of systems subject to random shocks with inter-arrival times having heavier tails. Some relevant stochastic properties of the introduced models are described. Finally, two applications, namely, the optimal replacement policy and the optimal mission duration are discussed.