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Abstract:
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A system is experiencing both sporadic shocks that cause damage and positive interventions that induce partial healing. The accumulated effect of k positive interventions nullify the damaging effect of one shock. The system is said to be in Stage 1, when it can possibly heal, until the net count of shocks (shocks arrived minus shocks nullified) reaches a threshold N_1. Thereafter, the system enters Stage 2, when no more healing is possible. The system fails when the net count of shocks reaches another threshold N_2 > N_1. The inter-arrival times between shocks and those between positive interventions follow arbitrary distributions F and G respectively, thereby removing the restrictive assumption of exponential times found in the literature. We find the distributions of the sojourn time in Stage 1 and the failure time of the system. Finally, we find the optimal time to replace the system to minimize the maintenance cost per unit of time.
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