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-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 (gK1). 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 (gK1). 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.


Introduction
It is well known that a simple relation holds between the stationary queue length distribution of an M/GI/1/K 0 queue and that of the corresponding M/GI/1/K 1 queue (the same system except for the capacity K 1 < K 0 ), that is, P X (K1) 0 = j = cP X (K0) 0 = j for j = 0,...,K 1 − 1, ( where X (Ki) 0 , i = 0,1, denotes a random variable representing the steady-state queue length in the M/GI/1/K 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]).
2 Sample-path analysis of discrete-time queues In [7], the authors have considered a discrete-time stationary single-server queueing system described by the recursion; where x + = max(x,0).Here, X n (∈ {0, ...,K}) denotes the queue length at time n and δ n (∈ {0, 1}) represents the virtual departure, that is, a customer leaves the system at time n only when δ n = 1 and X n > 0, while no customer leaves at time n when δ n = 0.Both A n and B 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 n } n∈Z and {B n } n∈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 0 (> K 1 ).This implies that the stationary queue length distribution of a system with capacity K 1 can be completely expressed in terms of the distribution of the other system with capacity K 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 the other hand, in concurrent estimation, we can use the proportional relation to estimate the performance of multiple queueing systems with different buffer capacities simultaneously by simulating only one system.
In this paper, we study a sample-path (deterministic) version of the proportional relation in more general setting, where any probabilistic assumption (such as stationarity or ergodicity) is not needed.This is quite useful in applying our result to some online control of queueing systems such as dynamic buffer allocation.Our main result (Theorem 3.7) shows that under some assumptions (Assumptions 3.1 and 3.5), 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 ).In addition to the sample-path (deterministic) version, we consider a probabilistic setting where Assumptions 3.1 and 3.5 are satisfied with probability one.Under the probabilistic setting, we obtain a stochastic version (Corollary 4.2) of our main result.The stochastic version is considered as a generalization of [7, Theorem 1], because the probabilistic assumptions are less restrictive than those in [7].
The remainder of this paper is organized as follows.Section 2 describes a discretetime single-server queueing model considered in this paper.In Section 3, we first make a few assumptions and then show that under the assumptions, the empirical queue length distribution 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 ).In Section 4, we consider a probabilistic setting which satisfies the assumptions made in Section 3 and generalize [7, Theorem 1].

Model
Consider a discrete-time queueing system consisting of a single server and a buffer, where the buffer capacity may be finite or infinite and K denotes the total system capacity (the buffer capacity plus one for the server).Time is divided into unit-time intervals called slots.Let {γ n } n≥1 denote an arrival process on Z + , where γ n represents the batch size (the number of customers) arriving at the beginning of the nth slot.An arriving customer that finds the buffer being full is discarded and lost.Let {δ n } n∈Z+ denote a sequence on {0, 1} closely related to the departure process, that is, δ n represents the number of customers departing at the end of the nth slot when the system is not empty in the slot.Let X n (∈ {0, ...,K}), n ∈ Z + , denote the queue length (including one in service if any) immediately after arrival in the nth slot.The dynamics of {X n } n∈Z+ is then represented by the following recursion with an initial state X 0 ; We can interpret {(X n ,δ n ,γ n+1 )} n∈Z+ as a sample path extracted from the corresponding stochastic process, where neither stationarity nor ergodicity is assumed.Hereafter, we consider two queueing systems both of which are identical to the above but their system capacities are K 0 and K 1 , respectively, where 0 < K 1 < K 0 ≤ +∞.We refer to the system with capacity K i as K i -system for i = 0,1, and to emphasize the system capacity, we put the superscript (K i ) on the quantities associated with the K i -system.

Main result
For any event sequence {E n } n∈Z+ , let ᏼ(E n ) denote the long-run average of the indicator 1 En , that is, provided that the limit exists.Note that the subscript n has no rigorous meaning in the left-hand side of (3.1), which is just the abbreviation of ᏼ({E n } n∈Z+ ).In this section, we show under some mild assumptions that ᏼ(X (K1) n = j), j = 0,...,K 1 , is (if it exists) completely expressed in terms of the long-run averages concerning the K 0 -system.The proof consists of two steps: first, we show in Section 3.1 that ᏼ(X (K0) n = j) and ᏼ(X (K1) n = j), j = 0,...,K 1 − 1, are proportionally related.This part may be straightforward from the classical truncation property of censored processes.However, this is not yet enough to say that ᏼ(X (K1) n = j), j = 0,...,K 1 , is completely expressed in terms of the long-run averages concerning the K 0 -system.Then in Section 3.2, we verify that the proportional constant is also expressed in terms of the long-run averages concerning the K 0 -system.4 Sample-path analysis of discrete-time queues

Proportional relation. First we put a basic assumption.
Assumption 3.1.(i) For j = 0,1,...,K 1 and i = 0,1, ᏼ(X (Ki) n = j) exists.In addition, for at least one j = 0,1,...,K 1 − 1, both ᏼ(X (K0) n = j) and ᏼ(X (K1) n = j) are positive, (ii) there exist N 0 and The latter part of Assumption 3.1(i) excludes the cases where ᏼ(X (K1) In other words, φ(m,K i ) represents the first epoch at which the cumulative time of {X (Ki)   n ≤ K 1 − 1} reaches m after (and including) N i .Note that under Assumption 3.1, φ(m,K i ) exists for every m = 1,2,... and i = 0,1, and lim m→∞ φ(m,K i ) = +∞.The next assumption, which imposes a restriction on the arrival and the departure processes, is a key to establish the following lemma.
Lemma 3.3 shows that the censored (truncated) process of {X (K0) n } n∈Z+ such that the state space is truncated at K 1 − 1 coincides with that of {X (K1)  n } n∈Z+ under the stated assumptions.Lemma 3.3 readily leads to the goal of this section.
6 Sample-path analysis of discrete-time queues

Proportional constant.
In general, to determine the proportional constant C 0 /C 1 appearing in (3.6), we need some quantities for both K 0 -and K 1 -systems, for example, both ᏼ(X (K0) n ≤ K 1 − 1) and ᏼ(X (K1) n ≤ K 1 − 1).However, with some additional assumptions, we can express the proportional constant C 0 /C 1 only in terms of the quantities concerning K 0 -system.For such a purpose, we replace Assumption 3.2 with the following.
The following lemma shows that the proportional constant C 0 /C 1 can be expressed only in terms of the quantities concerning K 0 -system under Assumptions 3.1 and 3.5.Lemma 3.6.Under Assumptions 3.1 and 3.5, we have (3.15) , and the result is trivial, where j * is such that j * = X (K0) N0 = X (K1) N1 in Assumption 3.1.
F. Ishizaki and N. Miyoshi 7 Suppose that ᏼ(δ (K0) n = 1, A (K0) n+1 = 0) > 0.Then, we show below that together with the existence of the limits in (3.16) (note that the existence of the limits in (3.17) is assumed in (3.12)).Once (3.16) and (3.17) are provided, summing each side of them and noting that Therefore, (3.15) is derived by applying (3.13).We first derive (3.16).From Lemma 3.3 with Assumptions 3.1 and 3.5, Thus, dividing both the sides by m and taking m to infinity, we have (3.16)under Assumptions 3.1 and 3.5, where the existence of the limits in (3.16) is ensured since and the two terms in the right-hand side above exist under Assumption 3.5.Now, we show (3.17).From Lemma 3.3 with Assumptions 3.1 and 3.5, we have Thus, dividing both the sides by m and taking m to infinity, we have under Assumption 3.1, provided that the limits exist.When the limits above exist, due to the level-crossing argument under Assumption 3.1, we have for i = 0,1, 8 Sample-path analysis of discrete-time queues where the second equality follows from (2.1) through the property of "skip free to the left" for {X (Ki)  n } n∈Z+ , i = 0,1.We here see that all the limits in (3.23) (and then those in (3.22)) exist because ᏼ(X (Ki) , is assumed to exist in Assumption 3.5.From (3.22) and (3.23), we have Hereafter, we separately treat the three cases: (a) ᏼ(X (K0) In the case (a), we have from (3.12) that, for i = 0,1, (3.25) Hence, substituting (3.25) into (3.24) and then using (3.14), we derive (3.17).
The following corollary can be considered as a more general result than [7, Theorem 1] and it reduces to that if {X (Ki)  n }, i = 0,1, is stationary and ergodic and {(A (Ki) n , δ (Ki) n )} is stationary.
10 Sample-path analysis of discrete-time queues hold with probability one, provided that the limits in (4.1) exist with probability one.