Reliability modeling for multi-state systems with a protective device subject to degradation-shock competing failure

Industrial systems experience the dual effects of internal degradation and external shocks; incorporating protective devices effectively delays the failure process and extends operational lifespan. However, existing research lacks systematic modeling of coupling effects between the degradation-shock competing failure process and the full life-cycle dynamics of protective devices. To address this, this paper develops a multi-state competing failure model with a two-dimensional state space characterized by system degradation stages and protective device states. A piecewise gamma process characterizes the internal degradation, and a stage-dependent shock classification mechanism allows shock-type probabilities to vary with degradation stages. Furthermore, a competitive triggering strategy based on shared shock processes is proposed, and protective device failure is governed by the cumulative number of effective shocks. Degradation rate suppression and shock effect regulation are incorporated into a unified state-dependent framework. Expressions for critical transition times, state probabilities, and reliability are derived, and comparisons with Monte Carlo simulation support the validity of the relevant derivations. Finally, numerical results show that the protective device enhances reliability and reduces the proportion of hard failures, while most of the mean time to failure improvement is attributable to shock-effect regulation rather than degradation-rate suppression.