Generalized Shock Model Based On The Frequency of Shocks: A Simple Approach

In a $\delta-$shock model, a system subject to randomly occurring shocks, the system fails when the time between two successive shocks lies below a threshold $\delta$. In this note, we study the generalization of this model where such $\delta-$shocks are accumulated and the system fails on the occurrence of $k^{th}$ such a $\delta-$shock. The probability distribution of the system failure time and the statistical characteristics are explicitly obtained. Normal approximation to the failure time distribution is proposed.