On the law of the iterated logarithm in hybrid multiphase queueing systems

This paper studies a controlled queueing system in which the decisionmaker may change servers according to rules which depend only on the queue length. It is proved that for a given control policy a properly normalised sequence of these controlled queue length processes converges weakly to a controlled diffusion process as the queueing systems approach a state of heavy traffic.

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Controlled queues in heavy traffic

Advances in Applied Probability ◽

10.1017/s0001867800040854 ◽

1975 ◽

Vol 7 (03) ◽

pp. 656-671 ◽

Cited By ~ 1

Author(s):

John H. Rath

Keyword(s):

Diffusion Process ◽

Queue Length ◽

Queueing System ◽

Systems Approach ◽

Heavy Traffic ◽

Queueing Systems ◽

Control Policy ◽

Controlled Diffusion ◽

Controlled Queues

This paper studies a controlled queueing system in which the decisionmaker may change servers according to rules which depend only on the queue length. It is proved that for a given control policy a properly normalised sequence of these controlled queue length processes converges weakly to a controlled diffusion process as the queueing systems approach a state of heavy traffic.

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Conditioned Limit Theorems for the Difference of Waiting Time and Queue Length

Advances in Applied Probability ◽

10.1017/s0001867800026100 ◽

1994 ◽

Vol 26 (01) ◽

pp. 242-257

Author(s):

Władysław Szczotka ◽

Krzysztof Topolski

Keyword(s):

Waiting Time ◽

Queue Length ◽

Limit Theorems ◽

Queueing System ◽

Queueing Systems ◽

Traffic Intensity ◽

Virtual Waiting Time ◽

The Difference ◽

Scheme Of Series ◽

Brownian Meander

Consider the GI/G/1 queueing system with traffic intensity 1 and let wk and lk denote the actual waiting time of the kth unit and the number of units present in the system at the kth arrival including the kth unit, respectively. Furthermore let τ denote the number of units served during the first busy period and μ the intensity of the service. It is shown that as k →∞, where a is some known constant, , , and are independent, is a Brownian meander and is a Wiener process. A similar result is also given for the difference of virtual waiting time and queue length processes. These results are also extended to a wider class of queueing systems than GI/G/1 queues and a scheme of series of queues.

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Conditioned Limit Theorems for the Difference of Waiting Time and Queue Length

Advances in Applied Probability ◽

10.2307/1427589 ◽

1994 ◽

Vol 26 (1) ◽

pp. 242-257

Author(s):

Władysław Szczotka ◽

Krzysztof Topolski

Keyword(s):

Waiting Time ◽

Queue Length ◽

Limit Theorems ◽

Queueing System ◽

Queueing Systems ◽

Traffic Intensity ◽

Virtual Waiting Time ◽

The Difference ◽

Scheme Of Series ◽

Brownian Meander

Consider the GI/G/1 queueing system with traffic intensity 1 and let wk and lk denote the actual waiting time of the kth unit and the number of units present in the system at the kth arrival including the kth unit, respectively. Furthermore let τ denote the number of units served during the first busy period and μ the intensity of the service. It is shown that as k →∞, where a is some known constant, , , and are independent, is a Brownian meander and is a Wiener process. A similar result is also given for the difference of virtual waiting time and queue length processes. These results are also extended to a wider class of queueing systems than GI/G/1 queues and a scheme of series of queues.

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Rates of convergence for queues in heavy traffic. II: Sequences of queueing systems

Advances in Applied Probability ◽

10.2307/1426005 ◽

1972 ◽

Vol 4 (2) ◽

pp. 382-391 ◽

Cited By ~ 3

Author(s):

Douglas P. Kennedy

Keyword(s):

Central Limit ◽

Queue Length ◽

Limit Theorems ◽

Heavy Traffic ◽

Queueing Systems ◽

Rates Of Convergence ◽

Traffic Intensity ◽

Multiple Channel ◽

Functional Central Limit Theorems ◽

Functional Central Limit

Estimates are given for the rates of convergence in functional central limit theorems for the queue length process in a sequence of general multiple channel queues. The situation is considered where the traffic intensity in the nth. queue, ρn, tends to ρ ≧ 1 as n approaches infinity. This extends previous work by the author, [6], in which the traffic intensity was fixed ≧ 1.

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Rates of convergence for queues in heavy traffic. II: Sequences of queueing systems

Advances in Applied Probability ◽

10.1017/s0001867800038416 ◽

1972 ◽

Vol 4 (02) ◽

pp. 382-391 ◽

Cited By ~ 2

Author(s):

Douglas P. Kennedy

Keyword(s):

Central Limit ◽

Queue Length ◽

Limit Theorems ◽

Heavy Traffic ◽

Queueing Systems ◽

Rates Of Convergence ◽

Traffic Intensity ◽

Multiple Channel ◽

Functional Central Limit Theorems ◽

Functional Central Limit

Estimates are given for the rates of convergence in functional central limit theorems for the queue length process in a sequence of general multiple channel queues. The situation is considered where the traffic intensity in thenth. queue, ρn, tends to ρ ≧ 1 asnapproaches infinity. This extends previous work by the author, [6], in which the traffic intensity was fixed ≧ 1.

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Heavy traffic limit theorems for a queueing system in which customers join the shortest line

Advances in Applied Probability ◽

10.1017/s0001867800018632 ◽

1989 ◽

Vol 21 (02) ◽

pp. 451-469 ◽

Cited By ~ 1

Author(s):

Zhang Hanqin ◽

Wang Rongxin

Keyword(s):

Stochastic Processes ◽

Central Limit ◽

Limit Theorems ◽

Queueing System ◽

Heavy Traffic ◽

Traffic Intensity ◽

Heavy Traffic Limit ◽

Functional Central Limit Theorems ◽

Service Channels ◽

Functional Central Limit

The queueing system considered in this paper consists of r independent arrival channels and s independent service channels, where, as usual, the arrival and service channels are independent. In the queueing system, each server of the system has his own queue and arriving customers join the shortest line in the system. We give functional central limit theorems for the stochastic processes characterizing this system after appropriately scaling and translating the processes in traffic intensity ρ > 1.

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On the waiting-time process in a single-line queue with repeated calls

Journal of Applied Probability ◽

10.2307/3214127 ◽

1986 ◽

Vol 23 (1) ◽

pp. 185-192 ◽

Cited By ~ 18

Author(s):

G. I. Falin

Keyword(s):

Waiting Time ◽

Limit Theorems ◽

Queueing System ◽

Queueing Systems ◽

Retrial Queue ◽

Time Process ◽

Light Traffic ◽

Repeated Calls ◽

Waiting Time Process ◽

Mean Variance

Waiting time in a queueing system is usually measured by a period from the epoch when a subscriber enters the system until the service starting epoch. For repeated orders queueing systems it is natural to measure the waiting time by the number of repeated attempts, R, which have to be made by a blocked primary call customer before the call enters service. We study this problem for the M/M/1/1 retrial queue and derive expressions for mean, variance and generating function of R. Limit theorems are stated for heavy- and light-traffic cases.

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The virtual waiting time of the GI/G/1 queue in heavy traffic

Advances in Applied Probability ◽

10.2307/1426170 ◽

1971 ◽

Vol 3 (2) ◽

pp. 249-268 ◽

Cited By ~ 16

Author(s):

E. Kyprianou

Keyword(s):

Markov Chains ◽

Waiting Time ◽

Limit Theorems ◽

Queueing System ◽

Heavy Traffic ◽

Waiting Times ◽

Double Sequence ◽

Traffic Intensity ◽

Virtual Waiting Time

Investigations in the theory of heavy traffic were initiated by Kingman ([5], [6] and [7]) in an effort to obtain approximations for stable queues. He considered the Markov chains {Wni} of a sequence {Qi} of stable GI/G/1 queues, where Wni is the waiting time of the nth customer in the ith queueing system, and by making use of Spitzer's identity obtained limit theorems as first n → ∞ and then ρi ↑ 1 as i → ∞. Here &rHi is the traffic intensity of the ith queueing system. After Kingman the theory of heavy traffic was developed by a number of Russians mainly. Prohorov [10] considered the double sequence of waiting times {Wni} and obtained limit theorems in the three cases when n1/2(ρi-1) approaches (i) - ∞, (ii) -δ and (iii) 0 as n → ∞ and i → ∞ simultaneously. The case (i) includes the result of Kingman. Viskov [12] also studied the double sequence {Wni} and obtained limits in the two cases when n1/2(ρi − 1) approaches + δ and + ∞ as n → ∞ and i → ∞ simultaneously.

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