Applications of stochastic semigroups to queueing models
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
References
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 Unpublished manuscript.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.
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
The Copyright Holders of the submitted text are the Author and the Journal. The Reader is granted the right to use the pdf documents under the provisions of the Creative Commons 4.0 International License: Attribution (CC BY). The user can copy and redistribute the material in any medium or format and remix, transform, and build upon the material for any purpose.
- License
This journal provides immediate open access to its content under the Creative Commons BY 4.0 license (http://creativecommons.org/licenses/by/4.0/). Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license. - Author’s Warranties
The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s. - User Rights
Under the Creative Commons Attribution license, the users are free to share (copy, distribute and transmit the contribution) and adapt (remix, transform, and build upon the material) the article for any purpose, provided they attribute the contribution in the manner specified by the author or licensor. - Co-Authorship
If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.
