International Journal of Stochastic Analysis

International Journal of Stochastic Analysis / 1996 / Article

Open Access

Volume 9 |Article ID 515492 | https://doi.org/10.1155/S1048953396000147

Charles Knessl, Charles Tier, "Mean time for the development of large workloads and large queue lengths in the GI/G/1 queue", International Journal of Stochastic Analysis, vol. 9, Article ID 515492, 36 pages, 1996. https://doi.org/10.1155/S1048953396000147

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

Received01 Jul 1995
Revised01 Oct 1995

Abstract

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.

Copyright © 1996 Hindawi Publishing Corporation. 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.


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