Reliability and maintenance policies for a two-stage shock model with self-healing mechanism

In this paper, a two-stage shock model with self-healing mechanism is proposed as an extension of cumulative shock and delta-shock models. A change point is introduced to describe the two-stage failure process of a system and defined as the moment when the cumulative number of valid shocks reaches d. Before the change point, the system can heal the damage caused by a valid shock when the number of delta-invalid shocks reaches k in the trailing run of invalid shocks. Equivalently, the damage caused by previous i valid shocks can be healed when the number of delta-invalid shocks falls in [ik,(i+1)k) in the run of invalid shocks. The system loses self-healing ability when it reaches the change point, and then fails when the cumulative number of valid shocks reaches a prefixed value n (n > d). Based on the established model, the finite Markov chain imbedding approach is employed to obtain the probability mass function, the distribution function and the mean of shock length. Three preventive maintenance policies are proposed for the system under different monitoring conditions, and corresponding optimization models are constructed to determine the optimal quantities. Finally, numerical examples are given for the proposed model.