Maintenance of deteriorating systems subject to shocks

In this article we analyze a stochastic model for a maintained system which deteriorates as a result of the cumulative effects of random shocks. Each arriving shock results in a unit decrease in the state of the system, representing its residual life time. The system is repaired by a repairman arriving according to Poisson process. On arrival, if he finds the state of the system below a safety level α a preventive maintenance restores the system to the state β (>0); otherwise he does nothing. Initially the system is in a state β. An explicit expression for the probability mass function of the state of the system is derived.