ANALYSIS OF THE LAW OF THE ITERATED LOGARITHM FOR THE IDLE TIME OF A CUSTOMER IN MULTIPHASE QUEUES

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1 International Journal of Pure and Applied Mathematics Volume 66 No , ANALYSIS OF THE LAW OF THE ITERATED LOGARITHM FOR THE IDLE TIME OF A CUSTOMER IN MULTIPHASE QUEUES Saulius Minkevičius 1, Vladimiras Dolgopolovas 2 1 VU Institute of Mathematics and Informatics 4, Akademijos, Vilnius, 08663, LITHUANIA 1,2 Vilnius University 24, Naugarduko, Vilnius, 03225, LITHUANIA 1 minkevicius.saulius@gmail.com Abstract: The modern queueing theory is one of the powerful tools for a quantitative and qualitative analysis of communication systems, computer networks, transportation systems, and many other technical systems. The paper is designated to the analysis of queueing systems, arising in the network and communications theory called multiphase queues. We have proved here the law of the iterated logarithm LIL for the idle time of a customer in a multiphase queueing system. AMS Subject Classification: 60K25, 60G70, 60F17 Key Words: mathematical models of technical systems, queueing theory, multiphase queueing systems, heavy traffic, law of iterated logarithm, idle time of a customer 1. Introduction The modern queueing theory is one of the powerful tools for a quantitative and qualitative analysis of communication systems, computer networks, transportation systems, and many other technical systems. The paper is designated to the analysis of queueing systems, arising in the network and communications theory called multiphase queues. We have proved here LIL for the idle time of a customer in a multiphase queueing system. Received: October 28, 2010 c 2011 Academic Publications Correspondence author

2 184 S. Minkevičius, V. Dolgopolovas Limit theorems diffusion approximations and the LIL for the queueing system under the conditions of heavy traffic are closely connected they belong to the same field of research, i.e. investigations on the theory of queueing systems in heavy traffic. Therefore, first we shall try to trace the development of research on the general theory of a queueing system in heavy traffic. One of the main directions of research in the theory of queues corresponds to the asymptotic analysis of formulas or equations that describe the distribution of this and other probabilistic characteristics of a queue. For the development of this analysis, formulas or equations should be found as well as, unlimited convergence of a queue to the own critical point. Thus, the first results on limited behavior of single channel queues in heavy traffic are achieved in [5, 6]. Single phase case, where intervals of times between the arrival of customers are independent identically distributed random variables and there is one single device, working independently of the output in heavy traffic, is completely investigated in [2, 3]. Functional limit theorems for a virtual waiting time of a customer in a single queue are proved under various conditions of heavy traffic see [7]. Functional limit theorems for a virtual waiting time of a customer and the idle time of a customer are closely connected. Also, functional limit theorems for the waiting time of a customer and the queue length of customers in a multiphase queue are proved in various conditions of heavy traffic see [8]-[10]. So, in the following work we will prove LIL in conditions of heavy traffic for other important probabilistic characteristic of multiphase queues idle time of a customer. Note that there are only some works designated for the investigation of idle time of a customer in a single-server queue see surveys [12, 14] and paper [13]. The natural setting for limit theorems in this paper is a weak convergence of probability measures on the function space D[0,1] D. Since an excellent treatment of this subject has been recently published in [1], we shall only make a few remarks here about our terminology and notation. Stochastic processes that characterize the queueing system give rise to sequences of random functions in D, the space of all right-continuous functions on [0,1] with left limits and endowed with a Skorohod metric, d. Let D be the class of Borel sets of D. Then, if P n and P are probability measures on D which satisfy lim fdp n = fdp n D for every bounded, continuous, real-valued function f on D, we say that P n weakly converges to P, as n and write P n P. A random function X is D

3 ANALYSIS OF THE LAW OF THE ITERATED LOGARITHM a measurable mapping from some probability space Ω, B, P into D with the distribution P = PX 1 on D, D. We say that a sequence of random functions {X n } weakly converges to the random function X, and write X n X if the distribution P n of X n converges to the distribution P of X. In this paper, we will constantly use the analog of the theorem on converging together see, for example, [1]: Theorem 1.1. Let ε > 0 and X n,y n,x D. If P lim dx n,x > ε = n 0 and P lim dx n,y n > ε = 0, then P lim dy n,x > ε = 0. n n So we prove here the LIL for the idle time of a customer in multiphase queueing systems. The service discipline is first come, first served FCFS. We consider multiphase queueing systems with the FCFS service discipline at each station and general distributions of interarrival and service times. We apply the methods of [9, 10]. The idle time of a customer in each phase of a queue is unlimited, all random variables are defined in a common probability space Ω,β,P. Let us consider a k-phase queue. When a customer is served in the j- th phase of a queue, he goes to the j + 1-st phase of the queue. When a customer is served in the k-th phase of the queue, he leaves the queue. Let us denote t n as the time of arrival of the n-th customer at a multiphase queue, as the service time of the n-th customer in the j-th phase of a queue, z n = t n+1 t n, x j t = max{l : l i=1 Sj i t} such a total number of customers can be served in the j-th phase of a queue if devices in the queue are working without time out, et = max{l : l i=1 z i t} as the total number of customers that arrive at the multiphase queue until time t; τ j t as the total number of customers after service at the j-th phase of the queue until time t, τ 0 t = et, j = 1,2,...,k. x j t,j = 1,2,...,k and et are counting processes see [3]. S j n Also, let us denote by I j t the idle time of a customer in the j-th phase of a queue at time t, by w j t a virtual waiting time in the j-th phase of a queue at time t, by S j t = τ j 1 t the time, obtained after summarizing i=1 S j i the times of service customers that arrive at the j-th phase of queue until time t; by Q j t the queue length of customers in the j-th phase at time t; y j t = S j t t, ŷ j t = x j 1 t i=1 S j i t, x 0 t = et, f t y = yt inf ys,ŵ jt = f t ŷ j, v j t = j i=1 Q it, j = 1,2,...,k and t > 0. In particular, the j-th phase can be considered as the queue < G,G,G,1 >

4 186 S. Minkevičius, V. Dolgopolovas see the definition in [3]. Hence, if S j 0 = w j 0 = 0 also see [3], w j t = f t y j, j = 1,2,...,k and t > 0. 1 Also, note that see, for example, [3] I j t = inf ys, j = 1,2,...,k and t > 0. 2 Let us define α j = MS j n, α 0 = Mz n, σ j = DS j n > 0, σ 0 = Dz n > 0, σ 2 j = σ j α j 1 + σ j 1 α j 1 3 α j 2 > 0, ˆβ j = α j α j 1 1,β j = ˆβ j, j = 1,2,...,k, n 1. Suppose that α k < α k 1 <... < α 1 < α 0. 3 We will prove such a theorem. 2. Main Result Theorem 2.1. If conditions 3 are fulfilled, then I j t β j t I j t β j t P lim = 1 = P lim = 1 = 1, t σ j at t σ j at where at = 2t ln ln t and j = 1,2,...,k. Proof. At first we write that I j t + ŷ j t = y j t inf y js + ŷ j t y j t w j t + ŷ j t y j t = f t y j + f t y j f t ŷ j + f t ŷ j + ŷ j t y j t 2 ŷ j t y j t + ŷ j t y j t + f t ŷ j 3 ŷ j t y j t + f t ŷ j, j = 1,2,...,k and t > 0. 4 Thus, we obtain for ε > 0 that P sup I j s + ŷ j s > ε at P3 sup P + P sup ŷ j s y j s + sup f s ŷ j > ε at ŷ j s y j s > ε at sup f s ŷ j > ε at

5 ANALYSIS OF THE LAW OF THE ITERATED LOGARITHM If conditions 3 are satisfied for ε > 0, we prove P sup ŷ j s y j s lim t at > ε = 0, j = 1,2,...,k. 6 Similarly as in [9], we obtain that for ε > 0 sup ŷ j s y j s > ε t at sup l=1 2 P lim t at x j 1 s τ j 1 s S j l α j > ε P sup x j 1 s τ j 1 s lim t at > ε, 4 j = 1,2,...,k. Hence, estimate [9] implies that x j 1 t τ j 1 t v k t k { } sup x i s x i 1 s x i t x i 1 t, i=1 8 j = 1,2,...,k, x 0 = e. If conditions 3 are fulfilled for ε > 0, by virtue of 8 we obtain that see, for example, [10] P sup x j 1 s τ j 1 s lim t at Thus, from 7-9 it follows for ε > 0 that see [4] P sup ŷ j s y j s lim t at > ε = 0, j = 1,2,...,k. 9 > ε = 0, j = 1,2,...,k. 10

6 188 S. Minkevičius, V. Dolgopolovas Now we will prove if conditions 3 are satisfied for ε > 0, sup f s ŷ j > ε = 0, j = 1,2,...,k. 11 t at Note that ŷ j nt n = xj 1 nt l=1 S j l α j nt α j 1 n + β j n t ẑ j t, 12 where ẑ j t = σ j z j t + b t b= and z j t are independent standard Wiener processes, j = 1,2,...,k and 0 t 1. The continuous mapping theorem see again [1] implies that f nt ŷ j n f t ẑ j, j = 1,2,...,k and 0 t Also note that f t ẑ j is distributed as sup t > 0. Consequently, ẑ j s, j = 1,2,...,k and f nt ŷ j n 0, j = 1,2,...,k and 0 t Again, applying 14 and the continuous mapping theorem to and the maximum function, we prove that sup f ns ŷ j 0, j = 1,2,...,k and 0 t n Hence, applying the Strassen invariance principle, we obtain that for ε > 0 sup f s ŷ j > ε = 0, j = 1,2,...,k. 16 t at Using 5, 10 and 16, we derive that sup I j s + ŷ j s > ε = 0, j = 1,2,...,k. 17 t at

7 ANALYSIS OF THE LAW OF THE ITERATED LOGARITHM Similarly as in 17, we find that sup ŷ j s + ˆβ j s t at = 1 = 1, j = 1,2,...,k. 18 However, using 17, 18 and Theorem 1.1, we obtain that I j t β j t P lim = 1 = 1, j = 1,2,...,k. 19 t at Therefore, the proof of Theorem 2.1 is complete. 3. Concluding Remarks The theorems of this paper are proved for a class of multiphase queues in heavy traffic with the service principle first come, first served, endless idle time of a customer in each phase of the queueing system, and times between the arrival of customers at the multiphase queue are independent identically distributed random variables. However, analogous theorems can be applied to a wider class of multiphase queues in heavy traffic: when the arrival and service of customers in a queue is by group, when interarrival times of customers at the multiphase queue are weakly dependent random variables, etc. Acknowledgments The authors thank Professor Br. Grigelionis and referees for helpful advice and remarks on this and other topics. References [1] P. Billingsley, Convergence of probability measures, Wiley, New York [2] A.A. Borovkov, Probability Processes in Theory of Queue, Nauka, Moscow 1972, In Russian. [3] A. Borovkov, Asymptotic Methods in Theory of Queues, Nauka, Moscow 1980, In Russian.

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