Sample-path analysis of the proportional relation and its constant for discrete-time single-server queues (Q2498192)

Summary: In their previous work [Queueing Syst. 42, No.~2, 201--212 (2002; Zbl 1035.90024)], 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 \(K_1\) is completely expressed in terms of the stationary distribution for the system with capacity \(K_0\) \((>K_1)\). 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 \(K_1\) is completely expressed only in terms of the quantities concerning the corresponding system with capacity \(K_0\) \((>K_1)\). 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 authors' previous result, because the probabilistic assumptions are less restrictive.