Applications of stochastic semigroups to queueing models

Piotr Gwiżdż

Published: 2018-06-18

Vol. 33 (2019)

Applications of stochastic semigroups to queueing models

Piotr Gwiżdż

Abstract

Non-markovian queueing systems can be extended to piecewisedeterministic Markov processes by appending supplementary variables to the system. Then their analysis leads to an infinite system of partial differential equations with an infinite number of variables and non-local boundary conditions. We show how one can study such systems by using the theory of stochastic semigroups.

Keywords:

stochastic semigroup , invariant density , piecewise deterministic Markov process , queueing systems

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Gwiżdż, P. (2018). Applications of stochastic semigroups to queueing models. Annales Mathematicae Silesianae, 33, 121–142. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/13657

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References

1. Arendt W., Resolvent positive operators, Proc. London Math. Soc. (3) 54 (1987), no. 2, 321–349.
2. Asmussen S., Applied Probability and Queues, Applications of Mathematics, vol. 51, Second edition, Springer-Verlag, New York, 2003.
3. Banasiak J., On an extension of the Kato-Voigt perturbation theorem for substochastic semigroups and its application, Taiwanese J. Math. 5 (2001), no. 1, 169–191.
4. Banasiak J., Arlotti L., Perturbations of Positive Semigroups with Applications, Springer Monographs in Mathematics, Springer-Verlag London Ltd., London, 2006.
5. Biedrzycka W., Tyran-Kamińska M., Existence of invariant densities for semiflows with jumps, J. Math. Anal. Appl. 435 (2016), no. 1, 61–84.
6. Bobrowski A., Boundary conditions in evolutionary equations in biology, in: Banasiak J., Mokhtar-Kharroubi M. (eds.), Evolutionary Equations with Applications in Natural Sciences, Lecture Notes in Math., 2126, Springer, Cham, 2015, pp. 47–92.
7. Cohen J.W., The Single Server Queue, North-Holland Series in Applied Mathematics and Mechanics, vol. 8, North-Holland Publishing Co., Amsterdam–New York, 1982.
8. Cox D.R., The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables, Proc. Cambridge Philos. Soc. 51 (1955), 433–441.
9. Davis M.H.A., Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models, J. Roy. Statist. Soc. Ser. B 46 (1984), no. 3, 353–388.
10. Desch W., Perturbations of positive semigroups in AL-spaces (1988). Unpublished manuscript.
11. Greiner G., Perturbing the boundary conditions of a generator, Houston J. Math. 13 (1987), no. 2, 213–229.
12. Gupur G., Advances in queueing models’ research, Acta Anal. Funct. Appl. 13 (2011), no. 3, 225–245.
13. Gupur G., Ehmet R., Asymptotic behavior of the time-dependent solution of an M/G/1 queueing model, Bound. Value Probl. 2013, 2013:17, 21 pp.
14. Gwiżdż P., Tyran-Kamińska M., Positive semigroups and perturbations of boundary conditions. Preprint.
15. Haji A., Radl A., A semigroup approach to queueing systems, Semigroup Forum 75 (2007), no. 3, 610–624.
16. Kato T., On the semi-groups generated by Kolmogoroff’s differential equations, J. Math. Soc. Japan 6 (1954), 1–15.
17. Krishnamoorthy A., Pramod P.K., Chakravarthy S.R., Queues with interruptions: a survey, TOP 22 (2014), no. 1, 290–320.
18. Lasota A., Mackey M.C., Chaos, Fractals, and Noise, Applied Mathematical Sciences, vol. 97, Springer-Verlag, New York, 1994.
19. Mackey M.C., Tyran-Kamińska M., Dynamics and density evolution in piecewise deterministic growth processes, Ann. Polon. Math. 94 (2008), no. 2, 111–129.
20. Pichór K., Rudnicki R., Continuous Markov semigroups and stability of transport equations, J. Math. Anal. Appl. 249 (2000), no. 2, 668–685.
21. Rudnicki R., Tyran-Kamińska M., Piecewise Deterministic Processes in Biological Models, SpringerBriefs in Applied Sciences and Technology, SpringerBriefs in Mathematical Methods, Springer, Cham, 2017.
22. Takács L., Introduction to the Theory of Queues, University Texts in the Mathematical Sciences, Oxford University Press, New York, 1962.
23. Tyran-Kamińska M., Substochastic semigroups and densities of piecewise deterministic Markov processes, J. Math. Anal. Appl. 357 (2009), no. 2, 385–402.
24. Voigt J., On resolvent positive operators and positive C0-semigroups on AL-spaces, Semigroup Forum 38 (1989), no. 2, 263–266.
25. Zheng F., Guo B.Z., Quasi-compactness and irreducibility of queueing models, Semigroup Forum 91 (2015), no. 3, 560–572. Google Scholar

Vol. 33 (2019)
Published: 2019-07-18

ISSN: 0860-2107

eISSN: 2391-4238

10.1515/amsil

Publisher

University of Silesia Press

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