The distribution of the line length in a discrete time [formula omitted] queue

Abstract

We consider methods to find the distribution of the line length in a discrete-time GI∕G∕1 queue with bounded inter-arrival and service times. Here, the term line is meant to include all elements in the system. The methods in question can be divided into two groups: direct and indirect methods. The direct methods formulate the problem as a three-dimensional Markov chain, with one dimension indicating the line length, one the arrival process, and the third one the departure process. The indirect methods find first the waiting time distribution or the roots defining this distribution, and then use this information to determine the distribution of the line length. As it turns out, the indirect methods are faster, often by many orders of magnitude. All indirect methods discussed in literature use characteristic roots. However, we present a new indirect method that does not use roots. We also present a new indirect method based on roots. This method leads to much simpler formulas. Numerical tests show that the indirect methods we developed are indeed much faster than any direct method described in literature.