A shock model with two-type failures and optimal replacement policy

In this paper, a shock model for a repairable system with two-type failures is studied. Assume that two kinds of shock in a sequence of random shocks will make the system fail, one based on the inter-arrival time between two consecutive shocks less than a given positive value d and the other based on the shock magnitude of single shock more than a given positive value μ. Under this assumption, we obtain some reliability indices of the shock model such as the system reliability and the mean working time before system failure. Assume further that the system after repair is ‘as good as new’, but the consecutive repair times of the system form a stochastic increasing geometric process. On the basis of the above assumptions, we consider a replacement policy N based on the number of failure of the system. Our problem is to determine an optimal replacement policy N* such that the long-run average cost per unit time is minimised. The explicit expression of long-run average cost per unit time is derived, and the corresponding optimal replacement policy can be determined analytically or numerically. Finally, a numerical example is given.