We consider the discrete-time 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.
"On the Discrete-Time Queues under -Policy with Single and Multiple Vacations." J. Appl. Math. 2013 1 - 6, 2013. https://doi.org/10.1155/2013/587163