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COPYRIGHT 2006 Hindawi Publishing Corp.
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 [K.sub.1] is completely expressed in terms of the stationary distribution for the system with capacity [K.sub.0] (>[K.sub.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.sub.1] is completely expressed only in terms of the quantities concerning the corresponding system with capacity [K.sub.0] (> [K.sub.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 author's previous result, because the probabilistic assumptions are less restrictive.
1. Introduction
It is well known that a simple relation holds between the stationary queue length distribution of an M/GI/1/[K.sub.0] queue and that of the corresponding M/GI/1/[K.sub.1] queue (the same system except for the capacity [K.sub.1] <[K.sub.0]), that is,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)
where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], i = 0, 1, denotes a random variable representing the steady-state queue length in the M/GI/1/[K.sub.i] queue and c is a constant which is called proportional constant. We call such a relation the proportional relation. It has been shown that similar proportional relations also hold for some other queueing systems (see, e.g., [1, 2, 6, 7]).
In [7], the authors have considered a discrete-time stationary single-server queueing system described by the recursion;
[X.sub.n+1] = min [[([X.sub.n] - [[delta].sub.n]).sup.+] + [A.sub.n+1] + [B.sub.n+1], K], n [member of] Z, (1.2)
where [x.sup.+] = max(x,0). Here, [X.sub.n] ([member of] {0, ... ,K}) denotes the queue length at time n and an ([member of] {0,1}) represents the virtual departure, that is, a customer leaves the system at time n only when [[delta].sub.n] = 1 and [X.sub.n] > 0, while no customer leaves at time n when [[delta].sub.n] = 0. Both [A.sub.n] and [B.sub.n] denote the numbers of customers arriving at time n; the former is from a stationary arrival process with some regenerative structure and the latter is from an arrival process controlled according to the queue length. Under some assumptions for [{[A.sub.n]}.sub.n[member of]Z] and [{[B.sub.n]}.sub.n[member of]Z], the authors have not only established the proportional relations like (1.1) but also have shown that the proportional constant can be expressed in terms of the distribution for the system with capacity K = [K.sub.0] (>[K.sub.1]). This implies that the stationary queue length distribution of a system with capacity [K.sub.1] can be completely expressed in terms of the distribution of the other system with capacity [K.sub.0] (see [7] for detail).
The proportional relation has rich applications when the proportional constant is completely expressed in terms of the distribution for one system. For instance, the proportional relation is useful for numerical method to compute performance measures in a finite-buffer queue (see, e.g., [5, 6]) and for online concurrent performance estimation for queueing systems with different buffer capacities (see, e.g., [3, 4]). In numerical computation, we can use the proportional relation to evaluate some performance measures of a finite-buffer queue in terms of the queue-length distribution of the corresponding infinite-buffer queue, where we exploit the fact that the distributions of infinite-buffer queues are, in general, computed in efficient and stable ways. On...
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