Note on the strong limiting behaviour of busy periods in GI/G/1 queues under heavy traffic

1993 ◽  
Vol 30 (2) ◽  
pp. 483-488
Author(s):  
Josef Steinebach ◽  
Hanqin Zhang
Keyword(s):  
Heavy Traffic ◽  
Traffic Intensity ◽  
Limiting Behaviour

In this note, the strong limiting behaviour of busy periods in GI/G/1 queues is studied under the condition that the traffic intensity equals unity.

GI/G/1 processor sharing queue in heavy traffic

1994 ◽  
Vol 26 (2) ◽  
pp. 539-555 ◽  
Author(s):  
Sergei Grishechkin
Keyword(s):  
Queue Length ◽  
Branching Processes ◽  
Heavy Traffic ◽  
Processor Sharing ◽  
Traffic Intensity ◽  
Random Measures ◽  
Processing Times ◽  
Processor Sharing Queues ◽  
Limiting Behaviour ◽  
Residual Processing

Consider GI/G/1 processor sharing queues with traffic intensity tending to 1. Using the theory of random measures and the theory of branching processes we investigate the limiting behaviour of the queue length, sojourn time and random measures describing attained and residual processing times of customers present.

GI/G/1 processor sharing queue in heavy traffic

1994 ◽  
Vol 26 (02) ◽  
pp. 539-555 ◽  
Author(s):  
Sergei Grishechkin
Keyword(s):  
Queue Length ◽  
Branching Processes ◽  
Heavy Traffic ◽  
Processor Sharing ◽  
Traffic Intensity ◽  
Random Measures ◽  
Processing Times ◽  
Processor Sharing Queues ◽  
Limiting Behaviour ◽  
Residual Processing

Consider GI/G/1 processor sharing queues with traffic intensity tending to 1. Using the theory of random measures and the theory of branching processes we investigate the limiting behaviour of the queue length, sojourn time and random measures describing attained and residual processing times of customers present.

Note on the strong limiting behaviour of busy periods in GI/G/1 queues under heavy traffic

1993 ◽  
Vol 30 (02) ◽  
pp. 483-488
Author(s):  
Josef Steinebach ◽  
Hanqin Zhang
Keyword(s):  
Heavy Traffic ◽  
Traffic Intensity ◽  
Limiting Behaviour

In this note, the strong limiting behaviour of busy periods in GI/G/1 queues is studied under the condition that the traffic intensity equals unity.

The depletion time for M/G/1 systems and a related limit theorem

1983 ◽  
Vol 15 (02) ◽  
pp. 420-443 ◽  
Author(s):  
Julian Keilson ◽  
Ushio Sumita
Keyword(s):  
Limit Theorem ◽  
Time Distribution ◽  
Numerical Evaluation ◽  
Heavy Traffic ◽  
Single Server ◽  
Service Time Distribution ◽  
Traffic Intensity ◽  
Traffic Distribution ◽  
Delay Times ◽  
The Relationship

Waiting-time distributions for M/G/1 systems with priority dependent on class, order of arrival, service length, etc., are difficult to obtain. For single-server multipurpose processors the difficulties are compounded. A certain ergodic post-arrival depletion time is shown to be a true maximum for all delay times of interest. Explicit numerical evaluation of the distribution of this time is available. A heavy-traffic distribution for this time is shown to provide a simple and useful engineering tool with good results and insensitivity to service-time distribution even at modest traffic intensity levels. The relationship to the diffusion approximation for heavy traffic is described.

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

1989 ◽  
Vol 21 (02) ◽  
pp. 451-469 ◽  
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.

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

1971 ◽  
Vol 3 (2) ◽  
pp. 249-268 ◽  
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.

Rates of convergence for queues in heavy traffic. I

1972 ◽  
Vol 4 (2) ◽  
pp. 357-381 ◽  
Author(s):  
Douglas P. Kennedy
Keyword(s):  
Limit Theorems ◽  
Heavy Traffic ◽  
Rates Of Convergence ◽  
Traffic Intensity ◽  
Channel System ◽  
Multiple Channel ◽  
Skorokhod Representation Theorem ◽  
Functional Central Limit Theorems ◽  
Functional Central Limit ◽  
Quantities Of Interest

Estimates are given for the rates of convergence in functional central limit theorems for quantities of interest in the GI/G/1 queue and a general multiple channel system. The traffic intensity is fixed ≧ 1. The method employed involves expressing the underlying stochastic processes in terms of Brownian motion using the Skorokhod representation theorem.

Complements to heavy traffic limit theorems for the GI/G/1 queue

1972 ◽  
Vol 9 (1) ◽  
pp. 185-191 ◽  
Author(s):  
Ward Whitt
Keyword(s):  
Rate Of Convergence ◽  
Limit Theorems ◽  
Heavy Traffic ◽  
Waiting Times ◽  
Sufficient Conditions ◽  
Critical Value ◽  
Traffic Intensity ◽  
Heavy Traffic Limit ◽  
Convergence Of Moments

A bound on the rate of convergence and sufficient conditions for the convergence of moments are obtained for the sequence of waiting times in the GI/G/1 queue when the traffic intensity is at the critical value ρ = 1.

Complements to heavy traffic limit theorems for the GI/G/1 queue

1972 ◽  
Vol 9 (01) ◽  
pp. 185-191 ◽  
Author(s):  
Ward Whitt
Keyword(s):  
Rate Of Convergence ◽  
Limit Theorems ◽  
Heavy Traffic ◽  
Waiting Times ◽  
Sufficient Conditions ◽  
Critical Value ◽  
Traffic Intensity ◽  
Heavy Traffic Limit ◽  
Convergence Of Moments

A bound on the rate of convergence and sufficient conditions for the convergence of moments are obtained for the sequence of waiting times in the GI/G/1 queue when the traffic intensity is at the critical value ρ = 1.

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.

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