Maintaining a high or requirement-compliant level of reliability is a fundamental prerequisite for industries seeking to enhance production efficiency and reduce operational costs. This study investigates the reliability assessment of a retrial machine repair system with preventive maintenance and multiple imperfect coverage mechanisms. When the repair center is occupied with other tasks or engaged in repairing other failed machines, the failed machine is routed to a retrial orbit for later processing. The failure times of both operative and warm standby machines, as well as repair times, retrial times, and preventive maintenance times, are assumed to follow exponential distributions. We also assume that the coverage factor for failures of the operating machine is identical to that of the warm standby machine.

 In this study, we adopt a virtual cluster for training and operating large language models as the application scenario of the proposed queuing model. To align with the practical operating conditions of this scenario, we define Type I failures as the state in which all machines have failed, and Type II failures as the state in which all warm standby machines have failed. Based on the system assumptions and definitions, we first construct the state–transition rate diagram. Using this diagram, we formulate the differential–difference equations of the retrial system and subsequently transform them into their Laplace forms. A matrix-analytic method is then employed to obtain the steady-state probability solutions, from which the system reliability and the mean time to system failure (MTTF) are derived.

Comprehensive numerical analyses, supplemented by both graphical and numerical illustrations, are conducted to examine the effects of various system parameters on reliability and MTTF. The findings of this study provide a valuable reference for the design and maintenance of complex repairable systems, offering insights into how preventive maintenance and imperfect coverage mechanisms influence overall system reliability.