On the discrete-time Geo\(^X/G/1\) queues under \(N\)-policy with single and multiple vacations (Q1790017)

Summary: We consider the discrete-time Geo\(^X/G/1\) queue under \(N\)-policy with single and multiple vacations. In this queueing system, the server takes multiple vacations and a single vacation whenever the system becomes empty and begins to serve customers only if the queue length is at least a predetermined threshold value \(N\). Using the well-known property of stochastic decomposition, we derive the stationary queue-length distributions for both vacation models in a simple and unified manner. In addition, we derive their busy as well as idle-period distributions. Some classical vacation models are considered as special cases.