Journal of Applied Mathematics and Stochastic Analysis
Volume 2006 (2006), Article ID 47125, 10 pages
doi:10.1155/JAMSA/2006/47125
Sample-path analysis of the proportional relation and its constant for discrete-time single-server queues
1Department of Mathematics and Telecommunication Mathematics Research Center, Korea University, Seoul 136-701, Korea
2Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo 152-8552, Japan
Received 22 December 2004; Revised 8 April 2005; Accepted 12 April 2005
Copyright © 2006 Fumio Ishizaki and Naoto Miyoshi. 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
In the previous work, the authors have considered a discrete-time
queueing system and they have established that, under some
assumptions, the stationary queue length distribution for the
system with capacity K1 is completely expressed in terms of the
stationary distribution for the system with capacity K0
(>K1). In this paper, we study a sample-path version of this
problem in more general setting, where neither stationarity nor
ergodicity is assumed. We establish that, under some assumptions,
the empirical queue length distribution (along through a sample
path) for the system with capacity K1 is completely expressed
only in terms of the quantities concerning the corresponding
system with capacity K0 (>K1). Further, we consider a
probabilistic setting where the assumptions are satisfied with
probability one, and under the probabilistic setting, we obtain a
stochastic version of our main result. The stochastic version is
considered as a generalization of the author's previous result,
because the probabilistic assumptions are less restrictive.