Generalized mixed [formula omitted]-shock models with random interarrival times and magnitude of shocks

Abstract

Suppose that a system is affected by a sequence of shocks that occur randomly over time, and δ1, δ2, η1 and η2 are critical levels such that 0<δ1<δ2 and 0<η1<η2. In this paper, a new mixed δ-shock model is introduced for which the system fails with a probability, say θ1, when the time between two consecutive shocks is lying in [δ1,δ2], and the system fails with a probability, say θ2, when the magnitude of a shock is lying in [η1,η2]. The system fails with probability 1, as soon as the interarrival time between two successive shocks is less than δ1 or a shock with magnitude greater than η2 occurs. The corresponding survival function is derived under two scenarios of independence and dependence between the interarrival times and the magnitude of shocks. The first and second moments are also derived. To illustrate the behavior of the system’s lifetime, a simulation study is also conducted.