Mixed shock model for the multi-state system with a two-phase degradation process under Markov environment

Reliability analysis of multi-state systems is crucial in engineering, particularly when dependent competing failure processes arise from both degradation and shocks. Many existing models do not fully capture the dependencies between shock processes and their influence on system deterioration. This study develops a mixed shock model under a Markov environment, integrating a two-phase degradation process. The shock processes are characterized using a Poisson phase-type process, which accounts for dependencies in damage increments. System states evolve through distinct functional levels: perfect function, degraded function, and severely degraded function. Soft failure occurs when cumulative degradation exceeds a predefined threshold, whereas hard failure results from extreme shocks or the accumulation of damage. By employing the finite Markov chain imbedding approach and phase-type distribution, explicit reliability functions are formulated. A case study on a spool valve validates the model, demonstrating its applicability in evaluating the reliability of multi-state systems. The proposed model provides an enhanced framework for assessing the reliability of complex engineering systems, addressing dependencies in degradation and shock processes.