Journal of Applied Mathematics and Stochastic Analysis
Volume 2006 (2006), Article ID 47125, 10 pages

Sample-path analysis of the proportional relation and its constant for discrete-time single-server queues

Fumio Ishizaki1 and Naoto Miyoshi2

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.


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.