Reliability of multi-state systems under Markov renewal shock models with multiple failure levels

摘要

We develop two reliability shock models with competing risks when the critical failure level of shock sizes varies with the system performance level. In the two models, the interarrival times and shock sizes are governed by an ergodic Markov chain whose state space is divided into two disjoint subsets and an absorbing Markov chain whose state space is partitioned into three disjoint subsets respectively. In previous research, critical failure levels of Markov renewal shock models are fixed constants, but it is common that the system tolerance or resistance to failure is reduced when withstanding shocks. In this case, under the first model, the system fails when the size of a single shock exceeds a critical level which is related to which subset the state belongs to just after the shock. In addition to the failure risk in the first model, the system under the second model has another failure risk that it fails when it enters the absorbing state. The theory of aggregated stochastic processes is applied to reduce the computational time by grouping states with similar performance level. Formulas of reliability indexes of systems under the two models are derived including probabilities of competing risks, the reliability functions and so on. Finally, a study case of a micro-engine system is conducted to illustrate the proposed model and the obtained results.