Journal of Applied Mathematics and Stochastic Analysis
Volume 11 (1998), Issue 3, Pages 247-254
doi:10.1155/S1048953398000215
A heavy-traffic theorem for the GI/G/1 queue with a Pareto-type service time distribution
CWI, P.O. Box 94079, Amsterdam 1090 GB, The Netherlands
Received 1 July 1997; Revised 1 November 1997
Copyright © 1998 J. W. Cohen. 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.
Abstract
For the GI/G/1 queueing model with traffic load a<1, service time
distribution B(t) and interarrival time distribution A(t), whenever for
t→∞1−B(t)∼c(t/β)ν+O(e−δt),c>0,1<ν<2,δ>0,
and
∫0∞tμdA(t)<∞ for μ>ν,
(1−a)1ν−1w converges in distribution for a↑1. Here w is distributed as the
stationary waiting time distribution. The L.-S. transform of the limiting
distribution is derived and an asymptotic series for its tail probabilities is
obtained. The theorem actually proved in the text concerns a slightly
more general asymptotic behavior of 1−B(t), t→∞, than mentioned
above.