Applications of Stochastic Semigroups to Queueing Models

[1] Arendt W., Resolvent positive operators, Proc. London Math. Soc. (3) 54 (1987), no. 2, 321–349.10.1112/plms/s3-54.2.321Open DOISearch in Google Scholar

[2] Asmussen S., Applied Probability and Queues, Applications of Mathematics, vol. 51, Second edition, Springer-Verlag, New York, 2003.Search in Google Scholar

[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, 169–191.10.11650/twjm/1500574893Search in Google Scholar

[4] Banasiak J., Arlotti L., Perturbations of Positive Semigroups with Applications, Springer Monographs in Mathematics, Springer-Verlag London Ltd., London, 2006.Search in Google Scholar

[5] Biedrzycka W., Tyran-Kamińska M., Existence of invariant densities for semiflows with jumps, J. Math. Anal. Appl. 435 (2016), no. 1, 61–84.10.1016/j.jmaa.2015.10.019Search in Google Scholar

[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. 47–92.10.1007/978-3-319-11322-7_2Search in Google Scholar

[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.Search in Google Scholar

[8] Cox D.R., The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables, Proc. Cambridge Philos. Soc. 51 (1955), 433–441.10.1017/S0305004100030437Search in Google Scholar

[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, 353–388.10.1111/j.2517-6161.1984.tb01308.xSearch in Google Scholar

[10] Desch W., Perturbations of positive semigroups in AL-spaces (1988). Unpublished manuscript.Search in Google Scholar

[11] Greiner G., Perturbing the boundary conditions of a generator, Houston J. Math. 13 (1987), no. 2, 213–229.Search in Google Scholar

[12] Gupur G., Advances in queueing models’ research, Acta Anal. Funct. Appl. 13 (2011), no. 3, 225–245.Search in Google Scholar

[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.10.1186/1687-2770-2013-17Search in Google Scholar

[14] Gwiżdż P., Tyran-Kamińska M., Positive semigroups and perturbations of boundary conditions. Preprint.Search in Google Scholar

[15] Haji A., Radl A., A semigroup approach to queueing systems, Semigroup Forum 75 (2007), no. 3, 610–624.10.1007/s00233-007-0727-5Search in Google Scholar

[16] Kato T., On the semi-groups generated by Kolmogoroff’s differential equations, J. Math. Soc. Japan 6 (1954), 1–15.10.2969/jmsj/00610001Search in Google Scholar

[17] Krishnamoorthy A., Pramod P.K., Chakravarthy S.R., Queues with interruptions: a survey, TOP 22 (2014), no. 1, 290–320.10.1007/s11750-012-0256-6Search in Google Scholar

[18] Lasota A., Mackey M.C., Chaos, Fractals, and Noise, Applied Mathematical Sciences, vol. 97, Springer-Verlag, New York, 1994.10.1007/978-1-4612-4286-4Search in Google Scholar

[19] Mackey M.C., Tyran-Kamińska M., Dynamics and density evolution in piecewise deterministic growth processes, Ann. Polon. Math. 94 (2008), no. 2, 111–129.10.4064/ap94-2-2Open DOISearch in Google Scholar

[20] Pichór K., Rudnicki R., Continuous Markov semigroups and stability of transport equations, J. Math. Anal. Appl. 249 (2000), no. 2, 668–685.10.1006/jmaa.2000.6968Search in Google Scholar

[21] Rudnicki R., Tyran-Kamińska M., Piecewise Deterministic Processes in Biological Models, SpringerBriefs in Applied Sciences and Technology, SpringerBriefs in Mathematical Methods, Springer, Cham, 2017.10.1007/978-3-319-61295-9Search in Google Scholar

[22] Takács L., Introduction to the Theory of Queues, University Texts in the Mathematical Sciences, Oxford University Press, New York, 1962.Search in Google Scholar

[23] Tyran-Kamińska M., Substochastic semigroups and densities of piecewise deterministic Markov processes, J. Math. Anal. Appl. 357 (2009), no. 2, 385–402.10.1016/j.jmaa.2009.04.033Search in Google Scholar

[24] Voigt J., On resolvent positive operators and positive C₀-semigroups on AL-spaces, Semigroup Forum 38 (1989), no. 2, 263–266.10.1007/BF02573236Search in Google Scholar

[25] Zheng F., Guo B.Z., Quasi-compactness and irreducibility of queueing models, Semi-group Forum 91 (2015), no. 3, 560–572.10.1007/s00233-014-9663-3Search in Google Scholar