Numerical Issues in Computing Steady-State Queueing-Time Distributions of Single-Server Bulk-Service Queues: M/Gb/1 and M/Gd/1

This paper gives closed-form expressions in terms of the roots of certain equations, for the distributions of the queueing time, W_q, in the steady state for the queues M/G^b/1 and M/G^d/1. Whereas in the former system the server can serve customers up to its capacity b, in the latter system be serves several customers in batches of exactly size d. Essentially, the solution is obtained by first finding the roots of the denominators of the Laplace transforms of W_q and then resolving these transforms into partial fractions. Numerical examples are given that show that the algorithm discussed in the paper can be used effectively to find the required roots, even when there is a large number of such roots. With one exception, all the examples use service times with rational Laplace transforms. The remaining example demonstrates the fact that the quantities for the queues M/D^b/1 and M/D^d/1 can be approximated by those from the queues M/E_k^b/1 and M/E_k^d/1 respectively, when k is large. Out of numerous computations, only representative results in the form of tables and graphs have been appended. It is hoped that the results presented here might be of use to practitioners or queueing theorists dealing with inequalities, bounds, et cetera.

INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.