On a discrete time shock model

In this paper‎, ‎we study a discrete time shock model which is defined based on the length of the time between successive shocks‎. ‎For a system that is exposed to a sequence of random shocks over time under this model‎, ‎if the interarrival time between two successive shocks is equal to a prefixed critical time point such as $\delta$‎, ‎the system fails‎, ‎and the system is not damaged otherwise‎. ‎We investigate the statistical behavior of the system s lifetime under the model‎, ‎therefore‎, ‎we obtain the probability generating function of the system s lifetime‎, ‎the mean time to failure and the variance of the system s lifetime‎, ‎the Laplace transform of the system s lifetime‎, ‎and some other related results‎. ‎We end the paper with an example‎.