Matrix-analytic reliability modeling of k-out-of-n: G repairable systems with K-mixed redundancy and repairman’s multiple vacations

K-mixed redundancy, a recently developed redundancy technique, shows significant performance in improving system reliability. However, its application potential in complex repairable systems remains insufficiently explored. To address this research gap, this study constructs a comprehensive model for a k-out-of-n: G multi-state repairable system, which simultaneously integrates the K-mixed redundancy strategy and the repairman's multiple vacation policy. The operational components within the system may experience failure as a result of internal wear and external shocks. The lifetimes of components, the repair time of failed components, and the vacation time are regulated by different phase-type (PH) distributions. The advent of external shocks is dictated by a Markovian arrival process (MAP). The developed model of the repairable system is investigated in both stationary and transient states by the implementation of the matrix-analytic approach. By utilizing Kronecker operator theory and aggregated stochastic theory, reliability metrics have been derived. Ultimately, the validity of the model is ascertained via a numerical example, and a sensitivity analysis of the parameters is undertaken to further elucidate that the proposed model can augment the system's reliability.