Reliability analysis of an extended shock model

Abstract

Suppose that a system is subject to a sequence of shocks which occur with probability p in any period of time $t=1,2,\dots$ , and suppose that ${\delta }_1$ and ${\delta }_2$ are two critical levels ( $1\le {\delta }_1<{\delta }_2$ ). The system fails when the time interval between two consecutive shocks is less than ${\delta }_1$ , and the time interval bigger than ${\delta }_2$ has no effect on the system activity. In addition, the system fails with a probability of, say, $\theta$ , when the time interval varies between ${\delta }_1$ and ${\delta }_2$ . Therefore, this model can be regarded as an extension of discrete time version of $\delta$ -shock model, and such an idea can be also applied in the extension of other shock models. The present study obtains the reliability function and the probability generating function of the system’s lifetime under this model. The present study offers some properties of the system and refers to a generalization of the new model. In addition, the mean time of the system’s failure is obtained under reduced efficiency which is created when the time between two consecutive shocks varies between ${\delta }_1$ and ${\delta }_2$ for the first time.