A new δ -shock model for systems subject to multiple failure types and its optimal order-replacement policy

In this article, a generalized δ-shock model with multi-failure thresholds is studied. For the new model, the system fails depending on the interval times between two consecutive shocks which arrive according to a Poisson process. The shorter interval times may cause more serious failures and thus result in longer down times and more costs for repair. Assuming that the repair is imperfect, an order-replacement policy N is adopted. Explicitly, the spare system for replacement is ordered at the end of (N – 1)th repair and the aging system is replaced at the Nth failure or at an unrepairable failure, whichever occurs first. In addition, the system must meet the requirement of availability, that is, the long-run average operating time per unit time should not be lower than a certain level. The average cost rate C(N) and the stationary availability A(N) are derived analytically. Some convergence properties of A(N) and C(N) are also investigated. The optimal order-replacement policy N^* can be obtained numerically with the constraint of availability. Finally, an illustrative example is given and some sensitivity analyses are conducted to demonstrate the proposed shock model.