A Repairable N-Policy Queue with State-Dependent Arrivals and Bernoulli Interruption of Multiple Vacations

This paper studies an N-policy M(λ1,λ2)/G/1 repairable queue in which arrival rates depend on the server state and multiple vacations may be interrupted by a Bernoulli decision. When the queue first reaches N during a vacation, the server interrupts the current vacation with probability p and otherwise completes it. We use renewal arguments, total probability decomposition, and Laplace transforms to derive transforms of the transient queue length probabilities for arbitrary initial queue lengths. We then obtain the stationary queue length distribution, its probability-generating function, the mean queue length, a stochastic decomposition, and the server state probabilities. Discrete-event simulations validate the transient and stationary results. We also formulate a long-run average cost model subject to a mean waiting time constraint and jointly optimize the interruption probability p, threshold N, and vacation length T. For the multiple parameter groups considered, jointly optimizing (p,N,T) gives a lower cost than the separately optimized policies with p=0 and p=1, showing the benefit of including p as a control variable under these settings. With the effective arrival rate fixed, the optimized cost is lower than in the equal rate case when λ1>λ2 and higher when λ1<λ2 for the cases examined. The model can help decision makers select suitable operating policies and identify opportunities for cost reduction in service and maintenance systems.