Multi-channel queues in heavy traffic

1973 ◽  
Vol 10 (4) ◽  
pp. 769-777 ◽  
Author(s):  
Richard Loulou
Keyword(s):  
Weak Convergence ◽  
Queue Length ◽  
Heavy Traffic ◽  
Convergence Theory ◽  
Intermediate Result ◽  
Convergence Theorems ◽  
Multi Server ◽  
Channel Systems

In this paper, convergence theorems for heavy traffic queues are extended to multi-channel systems under general assumptions. Whitt (1968), and Iglehart and Whitt (1970) have proved weak convergence of the queue length process and the wait process for one-channel queues. Extensions to multi-server queues were established for the queue length process but only partly for the wait process (ρ = 1 only). We give here convergence theorems for the wait process when ρ > 1. Our approach uses weak convergence theory, but is different from previous ones in that we use the virtual delay as an intermediate result. The class of queues considered is more general than GI/G/m.

Multi-channel queues in heavy traffic

1973 ◽  
Vol 10 (04) ◽  
pp. 769-777 ◽  
Author(s):  
Richard Loulou
Keyword(s):  
Weak Convergence ◽  
Queue Length ◽  
Heavy Traffic ◽  
Convergence Theory ◽  
Intermediate Result ◽  
Convergence Theorems ◽  
Multi Server ◽  
Channel Systems

In this paper, convergence theorems for heavy traffic queues are extended to multi-channel systems under general assumptions. Whitt (1968), and Iglehart and Whitt (1970) have proved weak convergence of the queue length process and the wait process for one-channel queues. Extensions to multi-server queues were established for the queue length process but only partly for the wait process (ρ = 1 only). We give here convergence theorems for the wait process when ρ > 1. Our approach uses weak convergence theory, but is different from previous ones in that we use the virtual delay as an intermediate result. The class of queues considered is more general than GI/G/m.

Stochastic bounds for heterogeneous-server queues with Erlang service times

1974 ◽  
Vol 11 (04) ◽  
pp. 785-796 ◽  
Author(s):  
Oliver S. Yu
Keyword(s):  
Steady State ◽  
Queue Length ◽  
Heavy Traffic ◽  
Waiting Times ◽  
Single Server ◽  
Shape Parameters ◽  
Server Queue ◽  
Service Times ◽  
Heavy Traffic Approximations ◽  
Multi Server

This paper establishes stochastic bounds for the phasal departure times of a heterogeneous-server queue with a recurrent input and Erlang service times. The multi-server queue is bounded by a simple GI/E/1 queue. When the shape parameters of the Erlang service-time distributions of different servers are the same, these relations yield two-sided bounds for customer waiting times and the queue length, which can in turn be used with known results for single-server queues to obtain characterizations of steady-state distributions and heavy-traffic approximations.

Fluctuation theory in continuous time

1975 ◽  
Vol 7 (04) ◽  
pp. 705-766 ◽  
Author(s):  
N. H. Bingham
Keyword(s):  
Weak Convergence ◽  
Local Time ◽  
Continuous Time ◽  
Regular Variation ◽  
Heavy Traffic ◽  
Fluctuation Theory ◽  
Convergence Theorems ◽  
Time Fluctuation ◽  
Exit Problems ◽  
Time Theory

Our aim here is to give a survey of that part of continuous-time fluctuation theory which can be approached in terms of functionals of Lévy processes, our principal tools being Wiener-Hopf factorisation and local-time theory. Particular emphasis is given to one- and two-sided exit problems for spectrally negative and spectrally positive processes, and their applications to queues and dams. In addition, we give some weak-convergence theorems of heavy-traffic type, and some tail-estimates involving regular variation.

Asymptotic analysis of queueing systems with identical service

1996 ◽  
Vol 33 (1) ◽  
pp. 267-281 ◽  
Author(s):  
F. I. Karpelevitch ◽  
A. Ya. Kreinin
Keyword(s):  
Weak Convergence ◽  
Asymptotic Analysis ◽  
Heavy Traffic ◽  
Queueing Systems ◽  
Stationary Regime ◽  
Convergence Theory ◽  
Transient Behaviour ◽  
Limit Behaviour ◽  
Phase Systems ◽  
Multi Phase

We consider a heavy traffic regime in queueing systems with identical service. These systems belong to the class of multi-phase systems with dependent service times in different service nodes. We study the limit behaviour of the waiting time vector in heavy traffic. Both transient behaviour and the stationary regime are considered. Our analysis is based on the conception of ‘approximated functionals', which appeared to be fruitful in weak convergence theory of stochastic processes.

Asymptotic analysis of queueing systems with identical service

1996 ◽  
Vol 33 (01) ◽  
pp. 267-281 ◽  
Author(s):  
F. I. Karpelevitch ◽  
A. Ya. Kreinin
Keyword(s):  
Weak Convergence ◽  
Asymptotic Analysis ◽  
Heavy Traffic ◽  
Queueing Systems ◽  
Stationary Regime ◽  
Convergence Theory ◽  
Transient Behaviour ◽  
Limit Behaviour ◽  
Phase Systems ◽  
Multi Phase

We consider a heavy traffic regime in queueing systems with identical service. These systems belong to the class of multi-phase systems with dependent service times in different service nodes. We study the limit behaviour of the waiting time vector in heavy traffic. Both transient behaviour and the stationary regime are considered. Our analysis is based on the conception of ‘approximated functionals', which appeared to be fruitful in weak convergence theory of stochastic processes.

Stochastic bounds for heterogeneous-server queues with Erlang service times

1974 ◽  
Vol 11 (4) ◽  
pp. 785-796 ◽  
Author(s):  
Oliver S. Yu
Keyword(s):  
Steady State ◽  
Queue Length ◽  
Heavy Traffic ◽  
Waiting Times ◽  
Single Server ◽  
Shape Parameters ◽  
Server Queue ◽  
Service Times ◽  
Heavy Traffic Approximations ◽  
Multi Server

This paper establishes stochastic bounds for the phasal departure times of a heterogeneous-server queue with a recurrent input and Erlang service times. The multi-server queue is bounded by a simple GI/E/1 queue. When the shape parameters of the Erlang service-time distributions of different servers are the same, these relations yield two-sided bounds for customer waiting times and the queue length, which can in turn be used with known results for single-server queues to obtain characterizations of steady-state distributions and heavy-traffic approximations.

Fluctuation theory in continuous time

1975 ◽  
Vol 7 (4) ◽  
pp. 705-766 ◽  
Author(s):  
N. H. Bingham
Keyword(s):  
Weak Convergence ◽  
Local Time ◽  
Continuous Time ◽  
Regular Variation ◽  
Heavy Traffic ◽  
Fluctuation Theory ◽  
Convergence Theorems ◽  
Time Fluctuation ◽  
Exit Problems ◽  
Time Theory

Our aim here is to give a survey of that part of continuous-time fluctuation theory which can be approached in terms of functionals of Lévy processes, our principal tools being Wiener-Hopf factorisation and local-time theory. Particular emphasis is given to one- and two-sided exit problems for spectrally negative and spectrally positive processes, and their applications to queues and dams. In addition, we give some weak-convergence theorems of heavy-traffic type, and some tail-estimates involving regular variation.

Heavy traffic queue length scaling in switches with reconfiguration delay

2021 ◽  
Author(s):  
Chang-Heng Wang ◽  
Siva Theja Maguluri ◽  
Tara Javidi
Keyword(s):  
Queue Length ◽  
Heavy Traffic ◽  
Length Scaling

Beyond the heuristic approach to Kolmogorov-Smirnov theorems

1982 ◽  
Vol 19 (A) ◽  
pp. 359-365 ◽  
Author(s):  
David Pollard
Keyword(s):  
Weak Convergence ◽  
Goodness Of Fit ◽  
Empirical Distribution ◽  
Distribution Functions ◽  
Heuristic Approach ◽  
Convergence Theory ◽  
Simple Method ◽  
Starting Point ◽  
Kolmogorov Smirnov ◽  
The Subject

The theory of weak convergence has developed into an extensive and useful, but technical, subject. One of its most important applications is in the study of empirical distribution functions: the explication of the asymptotic behavior of the Kolmogorov goodness-of-fit statistic is one of its greatest successes. In this article a simple method for understanding this aspect of the subject is sketched. The starting point is Doob's heuristic approach to the Kolmogorov-Smirnov theorems, and the rigorous justification of that approach offered by Donsker. The ideas can be carried over to other applications of weak convergence theory.

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