Analysis of a k-out-of-n:G System with External Customer Service, Preemptive Priority and Working Vacation under N-Policy

This paper analyzes a k-out-of-n:G repairable system with a single server that repairs failed components and serves external customers during idle periods. The server gives priority to internal repairs whenever the number of failed components reaches a threshold N. When no internal or external jobs are present, the server takes multiple vacations with phase-type distributed durations while continuing to serve external customers at a reduced rate. Component failures are exponentially distributed, external customers arrive according to a Poisson process, repair times follow a phase-type distribution, and external service times are exponentially distributed. The Matrix-Geometric Method is employed to obtain stability conditions and steady-state distributions. Numerical studies are conducted to examine the effects of system parameters on reliability measures and to determine the optimal N-policy threshold by minimizing a cost function.