A new mixed δ-shock model and associated reliability properties

In reliability literature, shock models and multi-state systems have both been discussed extensively. Shock models have attracted a great deal of attention due to their important role in engineering systems. In the δ-shock model, a system fails if the interval time between two successive shocks is less than a pre-fixed threshold δ. In this study, a new version of mixed δ-shock models is defined by combining δ-shock and extreme shock models, thus causing the failure of a multi-state system in two ways: first, when k interarrival times between two consecutive shocks with magnitude larger than the critical threshold γ are in and second, when the interarrival time between two consecutive shocks is less than δ 1. The survival function is obtained for three cases: (i) system’s lifetime, (ii) when the system performs completely, and (iii) when the system performs partially. A Monte Carlo simulation study is carried out to validate the analytical results established here.