Journal of Applied Mathematics and Stochastic Analysis
Volume 2004 (2004), Issue 1, Pages 19-71
Asymptotic analysis by the saddle point method of the
Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, M/C 249, 851 South Morgan Street, Chicago 60607-7045, IL, USA
Received 6 June 2003; Revised 12 December 2003
Copyright © 2004 Diego Dominici and Charles Knessl. 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 a fluid queue where the input process consists of identical
sources that turn on and off
at exponential waiting times. The server works at the constant
rate and an on source generates fluid at unit rate.
This model was first formulated and analyzed by Anick et
al. (1982). We obtain an alternate representation of the joint
steady-state distribution of
the buffer content and the number of on sources. This is
given as a contour integral that we then analyze in the limit
. We give detailed asymptotic results for the
joint distribution as well as the associated marginal and
conditional distributions. In particular, simple conditional
limits laws are obtained. These show how the buffer content
behaves conditioned on the number of active sources and vice
versa. Numerical comparisons show that our asymptotic results are
very accurate even for .