Optimal replacement policies for a shock model with a change point

In this paper, a mixed shock model with a change point is proposed to fit the situation that the failure mechanism changes during the system operation. For the new model, the change point appears when a run of damaging shocks with a prefixed length occurs, which divides the failure process of the system into two stages. Before the change point, the system is assumed to be failure-free due to the system's own resistance to the shocks. After the change point, the system fails when the number of cumulative or consecutive damaging shocks reaches a corresponding critical value. A Markov chain and the one-step transition probability matrix are constructed based on the proposed model. Distributions of the time to the change point, the lifetime and the residual lifetime of the system are derived when the interarrival times between two consecutive shocks follow a general continuous phase-type distribution. For the new shock model, three replacement policies are designed to fit different monitoring situations of the system, and corresponding unconstrained optimization models are established to obtain the optimal replacement quantities. Numerical examples are presented to illustrate the proposed shock model and three replacement policies for the given form of the phase-type distribution. The proposed model can be applied to the actual railway track system and some new composite materials.