Analysis of a discrete-time GI/Geo/1 queue with single vacation

We analyse a discrete-time GI/Geo/1 queue with vacation in which the server takes exactly one Bernoulli vacation after each busy period based on exhaustive service. With the displacement operator method which is used to solve simultaneous non-homogeneous difference equations, we obtain the distributions of queue length at prearrival and arbitrary epochs, and waiting time for an arrival customer. We also explain the stochastic decomposition properties of queue length and waiting time in this system. Finally, some numerical results are presented. The model presented in this paper may be useful in polling systems where the trade-off between service and vacation times is adopted to capture processing and polling times accurately, and control the access to the communication media.