Alternative Numerical Solutions of Stationary Queueing-Time Distributions in Discrete-Time Queues: GI/G/1

Abstract

This paper gives closed-form expressions, in terms of the roots of certain equations, for the distribution of the waiting time in queue, W_q, in the steady-state for the discrete-time queue GI/G/1. Essentially, this is done by finding roots of the denominator of the probability generating function of W_q and then resolving the generating function into partial fractions. Numerical examples are given showing the use of the required roots, even when there is a large number of them. The method discussed in this paper avoids spectrum factorization and uses both closed- and non-closed forms of interarrival- and service-time distributions. Approximations for the tail probabilities in terms of one or three roots taken in ascending order of magnitude are also discussed. The exact computational results that can be obtained from the methods of this study should prove useful to both practitioners and queueing theorists dealing with bounds, inequalities, approximations, and simulation results.

Keywords:

• Characteristic Equations
• Discrete-Time Queues
• Probability Generating Functions