Exact and approximate numerical solutions of steady-state distributions arising in the queueGI/G/1

Abstract

In this paper we first obtain, in a unified way, closed-form analytic expressions in terms of roots of the so-called characteristic equation (c.e.), and then discuss the exact numerical solutions of steady-state distributions of (i) actual queueing time, (ii) virtual queueing time, (iii) actual idle time, and (iv) interdeparture time for the queueGI/R/1, whereR denotes the class of distributions whose Laplace-Stieltjes transforms (LSTs) are rational functions (ratios of a polynomial of degree at mostn to a polynomial of degreen). For the purpose of numerical discussions of idle- and interdeparture-time distributions, the interarrival-time distribution is also taken to belong to the classR. It is also shown that numerical computations of the idle-time distribution ofR/G/1 queues can be done even ifG is not taken asR. Throughout the discussions it is assumed that the queue discipline is first-come-first-served (FCFS). For the tail of the actual queueing-time distribution ofGI/R/1, approximations in terms of one or more roots of the c.e. are also discussed. If more than one root is used, they are taken in ascending order of magnitude. Numerical aspects have been tested for a variety of complex interarrival- and service-time distributions. The analysis is not restricted to generalized distributions with phases such as Coxian-n (C n ), but also covers nonphase type distributions such as uniform (U) and deterministic (D). Some numerical results are also presented in the form of tables and figures. It is expected that the results obtained from the present study should prove to be useful not only to practitioners, but also to queueing theorists who would like to test the accuracies of inequalities, bounds or approximations.