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

In the previous work, the authors have considered a discrete-timequeueing system and they have established that, under someassumptions, the stationary queue length distribution for thesystem with capacity K 1 is completely expressed in terms of thestationary distribution for the system with capacity K 0 ( > K 1 ). In this paper, we study a sample-path version of thisproblem in more general setting, where neither stationarity norergodicity is assumed. We establish that, under some assumptions,the empirical queue length distribution (along through a samplepath) for the system with capacity K 1 is completely expressedonly in terms of the quantities concerning the correspondingsystem with capacity K 0 ( > K 1 ). Further, we consider aprobabilistic setting where the assumptions are satisfied withprobability one, and under the probabilistic setting, we obtain astochastic version of our main result. The stochastic version isconsidered as a generalization of the author's previous result,because the probabilistic assumptions are less restrictive.