Reliability evaluations for a multi-state k-out-of-n: F system with m subsystems supported by multiple protective devices

In engineering fields, the breakdown of a critical system can result in irreparable consequences, including task failures and even heavy casualties. Equipping systems with protective devices is an effective method to alleviate system failure risk in practical applications. However, existing research primarily focuses on single-component systems supported by a binary-state protective device. Moreover, existing studies assume that system deterioration is only caused by external shocks. For the first time, a multi-state k-out-of-n: F system containing m subsystems with multi-state protective devices is established in this study. In the proposed model, the deterioration of system and protective devices is caused not only by external shocks, but also by internal degradation. In the j-th subsystem, the corresponding protective device activates to work when the number of failed components reaches a threshold. During the operation of the protective device, the degradation rates of components, as well as the impacts and probabilities of adverse shocks on them decrease. The j-th subsystem breaks down once the number of failed components exceeds a critical value k_j. Based on the number of failed subsystems, the entire system is categorized into multiple states. Ultimately, the entire system fails when the number of failed subsystems reaches a predetermined value. A combination of the Markov process imbedding approach and finite Markov chain imbedding approach is employed to derive the probabilistic indices for the proposed system. Furthermore, a case study based on an aero-engine turbine system is presented to validate the proposed reliability model.