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.

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.

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.

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.

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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