A Discrete-Time Queue with Modified Vacation Policy

This paper examines a discrete-time Geo/Geo/1 queue, in which the server operates a randomized vacation policy with at most J vacations. After all messages are served in the queue, the server leaves for a vacation immediately. If there are some messages waiting in the queue upon the server returning from a vacation, it serves the waiting messages immediately. If no messages present in the queue, the server takes another vacation with probability p or goes into idle state with probability (1-p) until the next message arrives. By using the supplementary technique, the joint generating functions of the server state together with the system queue length and waiting time in the queue, vacation period, idle period, and busy period are derived. We also derive the distribution of the messages in the system at the busy initiation epoch and its expected length. Finally, a cost model developed to determine the optimum values of p and J at a minimum cost is also studied.