Reliability analysis and device redundancy policy for the multi-state k-out-of-n: F system equipped with cold standby protective devices

In engineering practice, it is effective to equip protective devices (PDs) and utilize a redundancy strategy to enhance the reliability of safety–critical systems. However, existing studies mostly focus on the reliability of systems supported by a single PD. Motivated by engineering applications, this paper firstly explores a redundancy strategy of multi-state PDs featuring a standby mode and two power operational modes for the k-out-of-n: F system. The competing triggering mechanism of PDs is proposed, where they are activated when a certain number of damaging shocks accumulate or when the components in the system degrade to a critical level. The PDs are in the cold standby mode and do not provide protection for the system until their triggering conditions are met and they successfully connect to the system. Additionally, a generalized activation scenario of the PDs is considered, which includes an imperfect switching process. PDs provide different protection capabilities in different power modes, while running in high-power mode become more susceptible to failure due to valid shocks. The system reliability indexes are calculated by Markov process imbedding approach. A heuristic redundancy policy in a single PD is also designed and the comparative analysis with the proposed policy is conducted. An optimization model for the redundancy of PDs is established based on an engineering application. Numerical illustrations are provided to verify the model applicability and the advantage of the proposed redundancy policy of multiple standby PDs.