On a run-based δ-shock model with two critical levels

In reliability engineering, the δ-shock model is used to study shock-exposed systems that are sensitive to the length of the time distance between consecutive shocks. When the system failure depends on a certain number of consecutive shocks with an inter-arrival time within a critical range, we are dealing with a run-based δ-shock model. In this paper, a new run-based δ-shock model is introduced, under which the system fails when an inter-arrival time is less than a critical threshold δ₁ for the first time or k consecutive inter-arrival times fall in the interval (δ₁, δ₂), for 0 ≤ δ₁, < δ₂. We study the probability behavior of the system’s stopping time as well as the survival of the system under the proposed model. As an illustrative example, we examine the survival of the system when the arrival of shocks follows a Poisson process. Furthermore, an example of applications is provided to illustrate possible application aspects.