An extended extreme shock maintenance model for a deteriorating system

In this paper, we study a deteriorating system which is suffering random shocks from its environment. Assume that in the system's operating stage, whenever a shock arrives, it will do some damage to the system, but shocks with a ‘small’ level of damage are harmless for the system, while shocks with a ‘large’ level of damage may result in the system's failure. The system's deterioration is caused by both the external shocks and the internal load. In the external, the magnitude of the shock damage the system can bear is decreasing with respect to the number of repairs taken. In the internal, the consecutive repair time is increasing in the number of repairs taken. A replacement policy N, by which the system is replaced at the time of the Nth failure, is adopted. An explicit expression of the long run average cost per unit time is derived, and an optimal policy N^* for minimizing the long run average cost per unit time is determined analytically. A numerical example is also given.