Reliability assessment for a k-out-of-n: F system supported by a multi-state protective device in a shock environment

Protective devices can reduce system failure risks by protecting systems from external shocks and therefore play a critical role in the stable operations of safety–critical systems. Because of the importance of protective devices in real engineering, their performances attract extensive attentions, but existing research on protective devices are quite limited. Previous studies mainly focused on evaluating the reliability of single-unit systems equipped with binary-state protective devices by considering triggering mechanisms based on system state or shock numbers. To fill the research gaps, this paper proposes a k-out-of-n: F system with a multi-state protective device subject to external shocks, where the triggering of the protective device depends on the number of failed components in the system. Furthermore, the protective capacity of the multi-state protective device becomes weaker as the device state deteriorates due to valid shocks. After suffering a certain number of valid shocks, the protective device may transit into a lower state directly, or it can resist some additional valid shocks in the current state because of its resistance against shocks. A combination of the finite Markov chain imbedding approach and phase-type distributions is employed to analyze a series of probabilistic indices both for the system and the protective device. Finally, a case study based on a multi-cylinder diesel engine supported by a cooling system is provided to demonstrate the applicability of the proposed model.