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 queue
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 queue described by either the workload (unfinished work) or the number of customers in the system. We compute the
mean time until reaches excess of the level , and also the mean time until
reaches . For the and models, we obtain exact contour
integral representations for these mean first passage times. We then compute the
mean times asymptotically, as and , by evaluating these contour integrals. For the general 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 and .