Controlled queues in heavy traffic

1975 ◽  
Vol 7 (3) ◽  
pp. 656-671 ◽  
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.

Controlled queues in heavy traffic

1975 ◽  
Vol 7 (03) ◽  
pp. 656-671 ◽  
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.

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

2020 ◽  
Vol 30 (4) ◽  
Author(s):  
Saulius Minkevičius
Keyword(s):  
Queue Length ◽  
Limit Theorems ◽  
Queueing System ◽  
Heavy Traffic ◽  
Queueing Systems ◽  
Probability Limit ◽  
Iterated Logarithm ◽  
Multiphase Queueing Systems ◽  
Multi Phase ◽  
Complex Computer

The model of a Hybrid Multi-phase Queueing System (HMQS) under conditions of heavy traffic is developed in this paper. This is a mathematical model to measure the performance of complex computer networks working under conditions of heavy traffic. Two probability limit theorems (Laws of the iterated logarithm, LIL) are presented for a queue length of jobs in HMQS.

scholarly journals Asymptotic analysis of a queueing system by a two-dimensional state space

1984 ◽  
Vol 21 (4) ◽  
pp. 870-886 ◽  
Author(s):  
J. P. C. Blanc
Keyword(s):  
Asymptotic Analysis ◽  
Queue Length ◽  
Queueing System ◽  
Queueing Systems ◽  
Length Distribution ◽  
Time Dependent ◽  
Queue Length Distribution ◽  
Long Run ◽  
Dimensional State Space ◽  
Hilbert Type

A technique is developed for the analysis of the asymptotic behaviour in the long run of queueing systems with two waiting lines. The generating function of the time-dependent joint queue-length distribution is obtained with the aid of the theory of boundary value problems of the Riemann–Hilbert type and by introducing a conformal mapping of the unit disk onto a given domain. In the asymptotic analysis an extensive use is made of theorems on the boundary behaviour of such conformal mappings.

scholarly journals Asymptotic analysis of a queueing system by a two-dimensional state space

1984 ◽  
Vol 21 (04) ◽  
pp. 870-886
Author(s):  
J. P. C. Blanc
Keyword(s):  
Asymptotic Analysis ◽  
Queue Length ◽  
Queueing System ◽  
Queueing Systems ◽  
Length Distribution ◽  
Time Dependent ◽  
Queue Length Distribution ◽  
Long Run ◽  
Dimensional State Space ◽  
Hilbert Type

A technique is developed for the analysis of the asymptotic behaviour in the long run of queueing systems with two waiting lines. The generating function of the time-dependent joint queue-length distribution is obtained with the aid of the theory of boundary value problems of the Riemann–Hilbert type and by introducing a conformal mapping of the unit disk onto a given domain. In the asymptotic analysis an extensive use is made of theorems on the boundary behaviour of such conformal mappings.

On berry–esseen rate for queue length of the GI/G/K system in heavy traffic

1988 ◽  
Vol 25 (03) ◽  
pp. 596-611
Author(s):  
Xing Jin
Keyword(s):  
Limit Theorem ◽  
Waiting Time ◽  
Queue Length ◽  
Queueing System ◽  
Heavy Traffic ◽  
Traffic Intensity ◽  
Main Method ◽  
Number Of Customers

This paper provides Berry–Esseen rate of limit theorem concerning the number of customers in a GI/G/K queueing system observed at arrival epochs for traffic intensity ρ > 1. The main method employed involves establishing several equalities about waiting time and queue length.

On berry–esseen rate for queue length of the GI/G/K system in heavy traffic

1988 ◽  
Vol 25 (3) ◽  
pp. 596-611 ◽  
Author(s):  
Xing Jin
Keyword(s):  
Limit Theorem ◽  
Waiting Time ◽  
Queue Length ◽  
Queueing System ◽  
Heavy Traffic ◽  
Traffic Intensity ◽  
Main Method ◽  
Number Of Customers

This paper provides Berry–Esseen rate of limit theorem concerning the number of customers in a GI/G/K queueing system observed at arrival epochs for traffic intensity ρ > 1. The main method employed involves establishing several equalities about waiting time and queue length.

Stochastic Bounds for Queueing Systems with Multiple On–Off Sources

1998 ◽  
Vol 12 (1) ◽  
pp. 25-48 ◽  
Author(s):  
Ger Koole ◽  
Zhen Liu
Keyword(s):  
Queue Length ◽  
Queueing System ◽  
Queueing Systems ◽  
Stochastic Order ◽  
Arrival Process ◽  
Buffer System ◽  
Finite Capacity ◽  
Convex Order ◽  
Increasing Convex Order ◽  
Background Traffic

Consider a queueing system where the input traffic consists of background traffic, modeled by a Markov Arrival Process, and foreground traffic modeled by N ≥ 1 homogeneous on–off sources. The queueing system has an increasing and concave service rate, which includes as a particular case multiserver queueing systems. Both the infinite-capacity and the finite-capacity buffer cases are analyzed. We show that the queue length in the infinite-capacity buffer system (respectively, the number of losses in the finite-capacity buffer system) is larger in the increasing convex order sense (respectively, the strong stochastic order sense) than the queue length (respectively, the number of losses) of the queueing system with the same background traffic and M N homogeneous on–off sources of the same total intensity as the foreground traffic, where M is an arbitrary integer. As a consequence, the queue length and the loss with a foreground traffic of multiple homogeneous on–off sources is upper bounded by that with a single on–off source and lower bounded by a Poisson source, where the bounds are obtained in the increasing convex order (respectively, the strong stochastic order). We also compare N ≥ 1 homogeneous arbitrary two-state Markov Modulated Poisson Process sources. We prove the monotonicity of the queue length in the transition rates and its convexity in the arrival rates. Standard techniques could not be used due to the different state spaces that we compare. We propose a new approach for the stochastic comparison of queues using dynamic programming which involves initially stationary arrival processes.

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

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.

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

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.

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

1972 ◽  
Vol 4 (2) ◽  
pp. 382-391 ◽  
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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