Journal of Applied Mathematics and Stochastic Analysis
Volume 9 (1996), Issue 2, Pages 107-142

Mean time for the development of large workloads and large queue lengths in the GI/G/1 queue

Charles Knessl and Charles Tier

University of Illinois at Chicago, Department of Mathematics, Statistics, and Computer Science, 851 South Morgan Street, Chicago 60607-7045, IL, USA

Received 1 July 1995; Revised 1 October 1995

Copyright © 1996 Charles Knessl and Charles Tier. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We consider the GI/G/1 queue described by either the workload U(t) (unfinished work) or the number of customers N(t) in the system. We compute the mean time until U(t) reaches excess of the level K, and also the mean time until N(t) reaches N0. For the M/G/1 and GI/M/1 models, we obtain exact contour integral representations for these mean first passage times. We then compute the mean times asymptotically, as K and N0, by evaluating these contour integrals. For the general GI/G/1 model, we obtain asymptotic results by a singular perturbation analysis of the appropriate backward Kolmogorov equation(s). Numerical comparisons show that the asymptotic formulas are very accurate even for moderate values of K and N0.