Characterizing the idle-period distribution of GI/G/1 queues

Abstract

A variety of performance measures of a GI/G/1 queue are explicitly related to the idle-period distribution of the queue, suggesting that the system analysis can be accomplished by the analysis of the idle period. However, the ‘stand-alone' relationship for the idle-period distribution of the GI/G/1 queue (i.e. the counterpart of Lindley's equation) has not been found in the literature. In this paper we develop a non-linear integral equation for the idle period distribution of the GI/G/1 queue. We also show that this non-linear system defines a unique solution. This development makes possible the analysis of the GI/G/1 queue in a different perspective.