D / G /1 Queues with Vacations

Many data switching systems have processors with arrival streams of regularly spaced tasks (e.g., individual bytes) requiring attention (e.g., directing the bytes to the appropriate outgoing line). In many of these systems the processor might also be required to handle other jobs (small maintenance routines or secondary tasks that await service in one or more other queues) as well. From the point of view of the primary queue of tasks, the processor ceases its service and takes a vacation. The performance of such a system is determined in part by the processor service schedule. In this paper, we define and analyze a model for investigating the waiting time probability distribution at the primary queue in terms of the primary task arrival rate, the service time distribution for the primary tasks, the probability distribution of the vacation duration, and the processor's service schedule. In addition, we discuss two characteristics of systems of this type that may have important design implications: (i) the steady-state waiting time probability distribution might be dependent upon the initial state of the system; and (ii) the waiting time might not decrease if the service time of the system decreases.