Reliability assessment of a multi-state mixed $$\delta $$ -shock model based on discrete PH distribution

Both the extreme shock model and the $$\delta $$ -shock model have been extensively studied and applied in various fields. Nevertheless, the research of these shock models under discrete-time conditions remains limited. Under discrete-time conditions, this paper develops a multi-state mixed $$\delta $$ -shock model that incorporates both a generalized $$\delta $$ -shock and an extreme shock. The research has the following two main features: on the one hand, the arrival process of the shocks is generalized by assuming that the shock intervals follow a discrete PH distribution; on the other hand, by constructing a discrete-time Markov chain, the discrete PH distribution representations of the lifetime, operating time in perfect state and operating time in inferior state of the system are given, and the expected values and probability mass functions of the corresponding measures of the system are derived through these discrete PH distributions. Additionally, by means of the numerical experiment, the relationship between various measures and parameters of the system is elucidated. The proposed mixed $$\delta $$ -shock model can degenerate into the extreme shock model or the $$\delta $$ -shock model under certain conditions, thereby providing a new perspective and approach for comprehending and utilizing these shock models.