Reliability assessment of system under a generalized cumulative shock model

Reliability assessment of system suffering from random shocks is attracting a great deal of attention in recent years. Excluding internal factors such as aging and wear-out, external shocks which lead to sudden changes in the system operation environment are also important causes of system failure. Therefore, efficiently modeling the reliability of such system is an important applied problem. A variety of shock models are developed to model the inter-arrival time between shocks and magnitude of shocks. In a cumulative shock model, the system fails when the cumulative magnitude of damage caused by shocks exceed a threshold. Nevertheless, in the existing literatures, only the magnitude is taken into consideration, while the source of shocks is usually neglected. Using the same distribution to model the magnitude of shocks from different sources is too critical in real practice. To this end, considering a system subject to random shocks from various sources with different probabilities, we develop a generalized cumulative shock model in this article. We use phase-type distribution to model the variables, which is highly versatile to be used for modeling quantitative features of random phenomenon. We will discuss the reliability characteristics of such system in some detail and give some clear expressions under the one-dimensional case. Numerical example for illustration is also provided along with a summary.