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CNAA / Theses / 2009 / July /

The Analysis of Priority Systems and Heavy Traffic


Author: Benderschi Olga
Degree:doctor of physics and mathematics
Speciality: 01.01.09 - Mathematical cybernetics and operation research
Year:2009
Scientific adviser: Gheorghe Mişcoi
doctor habilitat, professor, Institute of Mathematics and Computer Science
Institution:
Scientific council:

Status

The thesis was presented on the 3 July, 2009
Approved by NCAA on the 5 November, 2009

Abstract

Adobe PDF document – 0.20 Mb / in romanian

Keywords

priority queueing systems, priorities, generalized queueing systems, switchover times, busy period, traffic coefficient, heavy traffic, Kendall equation, Laplace-Stieltjes transform, numerical algorithms

Summary

This thesis is concerned with the analysis of priority queueing systems with r > 1 classes of priority with switchover times. It is also devoted to the study of traffic problems, including the heavy traffic regime, for these systems.

The priority queueing systems with switchover times are those systems where the incoming requests are distinguished by their importance and served according to the assigned priority levels. It is also assumed that the server needs some random time to switch from one priority queue of requests to another. These models have a number of new features which are more flexible and more advanced than those of classical models. The thesis contains a review of analytical results obtained by different researchers for classical queueing systems, in particular for the systems operating under the heavy traffic regime. Analytical results available for the priority queueing systems are also analyzed. Under the assumption that the incoming flows of requests are Poisson the main system characteristics can be described in terms of the Laplace-Stieltjes transform and generally can be evaluated only numerically.

The traffic coefficient is an important measure of the performance of a queueing system and it characterizes the workload of the system. Analysis of queueing systems delivers formulae for system performance characteristics - many of such analytical expressions involve the traffic coefficient. In the case of priority queueing systems with random switchover times one should be able to evaluate the Laplace-Stieltjes transforms of the system busy period in order to estimate the value of the traffic coefficient. Generally this can only be done numerically.

Numerical algorithms for evaluating the traffic coefficient and the Laplace-Stieltjes transforms of the auxiliary periods of system are described. The algorithms are implemented in the programming language C++ and were tested for a range of distribution function. The developed algorithms allow one to analyze the changes in behaviour of the various generalized queueing systems for different types of service time distributions and different values of the incoming flow rates.

Methods and techniques discussed in the thesis allow one to extend the presented algorithms in order to use them in the analysis of queueing systems that appear in various applications (manufacturing, transport, medicine, economics), where phenomena of interest can be described by the models analyzed in the paper.