A Queue with Starter and a Queue with Vacations: Delay Analysis by Decomposition

Abstract

This paper analyzes both a queueing system that incurs a start-up delay whenever an idle period ends and one in which the server takes vacation periods. We show that the delay distribution in the queue with starter is composed of the direct sum of two independent variables: (1) the delay in the equivalent queue without starter, and (2) the additional delay suffered due to the starter's presence. Using this decomposition property, we easily derive the distribution of the delay suffered in the system with starter. This analysis is done for systems (both discrete and continuous time) whose interarrival times possess the memoryless property. Using this approach, we then analyze the M/G/1 system with vacation periods. First, we show that the M/G/1 with vacations can be considered as a special case of the M/G/1 with starter, so that the delay in the M/G/1 with vacations can be easily found by using the formula for the delay of the M/G/1 with starter. Second, using geometric arguments, we explain why the additional delay in the vacation system is distributed as the residual life of the vacation period.

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