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Switchover time modelling in priority queueing systems

Author: Andrei Bejan
Degree:doctor of physics and mathematics
Speciality: 01.01.09 - Mathematical cybernetics and operation research
Scientific adviser: Gheorghe Mişcoi
doctor habilitat, professor, Institute of Mathematics and Computer Science of the ASM
Scientific council:


The thesis was presented on the 16 October, 2007
Approved by NCAA on the 20 December, 2007


Adobe PDF document0.23 Mb / in romanian


CZU 519.872

Adobe PDF document 1.19 Mb / in romanian
117 pages


queueing systems, priorities, prioritisation, device's switching, switchover times, busy period, Kendall equation, Quality of Service, imitational modelling


The thesis is devoted to the study of priority queueing systems with random switchover times. It is developed the classification for the reach class of queueing systems with one server which switches between waiting lines of requests distinguished by their importance, and thus prioritised.

We recapitulate the analytic results for such systems under the assumption of Poisson incoming flows and special structure of switchover times. These results are formulated by means of systems of functional recurrent equations stated in terms of Laplace transforms of the characteristics of interest. These results are thoroughly analysed. The functional Kendall equation makes an important part of them.

It is mentioned that the case of switchover times of general structure was not studied in the literature. Yet, even for the systems Mr |Gr| 1 , generally, it is impossible to obtain the solutions to the theoretical results in analytic form.

Therefore, it is suggested to treat such kind of results numerically. For this purpose the Kendall equation is studied by exploring the introduced Kendall functional operator. The classical iterative algorithm of numerical solution of the Kendall equation is improved in such a way that it can be efficiently used in solving mentioned systems of functional recurrent equations. The accelerations schemes (Salzer summation and Wynn's Rho algorithm) for the Laplace transform’s inversion method based on Gaver functionals are used to invert the solutions and obtain complete information on the system performance characteristics. The methodology is developed for the busy periods of the systems under study. This allows one to study many other characteristics of the system, in particular, the workload coefficient ρ.

The method of imitational modelling is applied to the study of priority systems with switchover times of general structure. The Java package PQSST was developed and it allows one to imitate such systems and to obtain full empirical information on the system performance characteristics, particularly on the busy periods, idle periods, mean waiting times, loss probabilities. The detailed chronology of the processes which take place in priority system is provided by the package PQSST.

The comparative analysis of the solutions obtained by using these two different approaches – numerical and imitational modelling methods – is carried out. The paper is accompanied with illustrative examples. There are also discussed applications of priority queueing systems with switchover times in traffic network QoS analysis.


1 Priority queueing systems with switchover times and their characteristics
  • 1.1 Classification of the priority queueing systems with switchover times and one server
  • 1.2 Performance characteristics of priority queueing systems

2 Analytical methods in the theory of priority queueing systems
  • 2.1 Introductive note
  • 2.2 Preliminary requisites
  • 2.2.1 Laplace and Laplace-Stieltjes transform technique
  • 2.2.2 Complete monotonicity and Bernstein theorem
  • 2.2.3 The z-transform technique
  • 2.3 Priority queueing systems with nonzero switchover times and Poisson incoming flows
  • 2.3.1 General case: decomposed switchover times
  • 2.3.2 Special case: no termination works
  • 2.3.3 The problem of switchover times decomposition
  • 2.4 Priority queueing systems with zero switchover times
  • 2.5 Busy period and its relationship to other characteristics

3 Numerical methods for priority queueing systems
  • 3.1 Introduction
  • 3.2 Ingredients for numerical algorithms
  • 3.2.1 Laplace transform inversion
  • 3.2.2 Kendall equation treatment
  • 3.3 Evaluation of the busy period: algorithms
  • 3.3.1 Description of the algorithms

4 Imitational modelling and statistical inference for the performance character-istics
  • 4.1 Imitational modelling
  • 4.1.1 Introduction
  • 4.1.2 Modelling
  • 4.2 Realisation: package PQSST
  • 4.3 Comparative analysis of the busy period's evaluation in the modelling of the
  • switchover times and analytical and numerical results
  • 4.3.1 Statistical analysis of simulation output: busy periods
  • 4.3.2 Examples

5 Conclusions, applications and open problems