Reliability and optimal triggering policy for multi-state systems subject to shocks and supported by a protective device

Failures of safety-critical systems could result in irretrievable economic loss and significant environmental damage. To mitigate the failure risk of safety-critical systems, a protective device is commonly designed to alleviate the damage caused by external shocks. In this paper, a multi-state system subject to shocks and supported by a protective device is considered. The protective device can be triggered to defense valid shocks when the system state is worse than a predetermined threshold. During the operating process of the protective device, the system is assumed to be failure-free due to the resistant effect of the protective device. After resisting a finite number of valid shocks for the system, the protective device loses the protective ability. A finite Markov chain imbedding method is employed to derive the system reliability characteristics. Based on the established reliability model, the state based triggering policy of the protective device is designed and an optimization model is constructed to determine the optimal triggering threshold. Finally, numerical examples are given to illustrate the proposed model and obtained results.