A Discrete Time Run Shock Model

A system in discrete time periods is exposed to a sequence of shocks so that shocks occur randomly and independently in each period with a probability p. Considering k(≥1) as a critical level, we assume that the system does not fail when the number of successive shocks is less than k, the system fails with probability Ө, if the number of successive shocks is equal to k and the system completely fails as soon as the number of sequential shocks reaches k+1. Therefore, this model can be considered as a version of run shock model, in which the shocks occur in discrete periods of time, and the behavior of the system is not fixed when encountering k successive shocks. In this paper, we examine the characteristics of the system according to this model, especially the first and second-order moments of the system's lifetime, and also estimate its unknown parameters. Finally, a method is proposed to calculate the mean of the generalized geometric distribution.