Dynamic Analysis of a 2-out-of-6:G Repairable System with Preventive Maintenance

This paper develops a 2-out-of-6:G repairable system with state-dependent preventive maintenance (PM) for functioning components. The 2-out-of-6:G configuration is adopted as a representative nontrivial case of the general k-out-of-n:G system, since it retains the essential features of component redundancy, multiple failure states, and a critical pre-failure state while remaining analytically tractable. When only two components remain functioning, PM is activated with probability β to reduce the risk of an immediate system breakdown caused by an additional failure. Using the supplementary variable method, the model is formulated as an abstract Cauchy problem. Within the C0-semigroup framework, the main operator is proved to generate a positive contraction C0-semigroup, ensuring the existence and uniqueness of a positive solution. By combining Greiner’s boundary perturbation method with spectral analysis, the strong convergence of the solution to the steady-state solution is established. Numerical simulations further illustrate the theoretical results and demonstrate the influence of preventive maintenance on system reliability performance.