Analysis of the GI/G/1 queue by the supplementary variables approach

Abstract

Abstract This paper analyzes the GI/G/ 1 queue on FIFO discipline by the supplementary variables approach. Using the residual interar‐rival time and the remaining service time of a customer being served as supplementary variables, we derive the basic equations describing the behavior of the system. From the basic equations, characteristic quantities of the system in the steady state are discussed. On the basis of Laplace transform and generating function expression, Laplace transform of actual waiting time is obtained by performing a Wiener‐Hopf decomposition. From this result, Laplace transforms of virtual waiting time, length of an idle period and inter‐departure time and the distributions of the number of customers at the moment just before an arrival and at an arbitrary moment, are derived. Furthermore, it is revealed that the analysis of busy period is reduced to a generalized Wiener‐Hopf decomposition problem. Finally, an application of the general results to the GI/Kn/1 queue is considered and by indicating that the results agree with the known results, the validity of this approach is confirmed.

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