Well-Posedness and Asymptotic Behavior of the Standby Repairable System with Partial Failure and Priority

This paper investigates the well-posedness (WP) and asymptotic stability of a standby repairable system with partial failure and priority, wherein components in a partial failure state remain unrepaired during system operation. The governing integro-differential equations are reformulated as an abstract Cauchy problem (ACP) on an appropriately chosen Banach space via the introduction of a linear operator. Using C0− semigroup theory together with spectral analysis, we prove that the system operator generates a positive contraction C0− semigroup, thereby guaranteeing the existence of unique time-dependent solution (TDS). Moreover, through a detailed examination of the spectral distribution of the generator on the imaginary axis, we establish the strong convergence of the TDS to its steady-state solution (SSS) as time tends to infinity. Complementing these analytical results, a Monte Carlo simulation (MCS) is performed to track the transient evolution of reliability indices, providing numerical validation of the theoretical predictions. These results offer a theoretical basis for evaluating system reliability and inform maintenance strategies in continuous-production environments.