Geodesic
Lord Kelvin and Andrey Andreyevich Markov in a queue with single server
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 11 (2018) no. 3, pp. 29-43
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We use Lord Kelvin's method of images to show that a certain infinite system of equations with interesting boundary conditions leads to a Markovian dynamics in an $L^1$-type space. This system originates from the queuing theory.
Keywords: queue; method of images; generation theorem; boundary conditions; Markovian dynamics. 
@article{VYURU_2018_11_3_a2,
     author = {A. Bobrowski},
     title = {Lord {Kelvin} and {Andrey} {Andreyevich} {Markov} in a queue with single server},
     journal = {Vestnik \^U\v{z}no-Uralʹskogo gosudarstvennogo universiteta. Seri\^a, Matemati\v{c}eskoe modelirovanie i programmirovanie},
     pages = {29--43},
     year = {2018},
     volume = {11},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/VYURU_2018_11_3_a2/}
}
TY  - JOUR
AU  - A. Bobrowski
TI  - Lord Kelvin and Andrey Andreyevich Markov in a queue with single server
JO  - Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie
PY  - 2018
SP  - 29
EP  - 43
VL  - 11
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/VYURU_2018_11_3_a2/
LA  - en
ID  - VYURU_2018_11_3_a2
ER  - 
%0 Journal Article
%A A. Bobrowski
%T Lord Kelvin and Andrey Andreyevich Markov in a queue with single server
%J Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie
%D 2018
%P 29-43
%V 11
%N 3
%U http://geodesic.mathdoc.fr/item/VYURU_2018_11_3_a2/
%G en
%F VYURU_2018_11_3_a2
A. Bobrowski. Lord Kelvin and Andrey Andreyevich Markov in a queue with single server. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 11 (2018) no. 3, pp. 29-43. http://geodesic.mathdoc.fr/item/VYURU_2018_11_3_a2/

[1] Engel K.-J., Nagel R., One-Parameter Semigroups for Linear Evolution Equations, Springer, N.Y., 2000 | MR | Zbl

[2] Bátkai A., Piazzera S., Semigroups for Delay Equations, CRC Press, Wellesley, 2005 | MR

[3] Siegle P., Goychuk I., Talkner P., Hänggi P., “Markovian Embedding of Non-Markovian Superdiffusion”, Physical Review, 81 (2010), 011136, 10 pp.

[4] Durbin R., Eddy S., Krogh A., Mitchison G., Biological Sequence Analysis. Probabilistic Models of Proteins and Nucleic Acids, Cambridge University Press, Cambridge, 1998 | Zbl

[5] Hoek J., Elliott R. J., Introduction to Hidden Semi-Markov Models, Cambridge University Press, Cambridge, 2018 | MR | Zbl

[6] Goldstein S., “On Diffusion by Discontinuous Movements, and on the Telegraph Equation”, The Quarterly Journal of Mechanics and Applied Mathematics, 4:2 (1986), 129–156 | DOI | MR

[7] Kac M., Some Stochastic Problems in Physics and Mechanics, Literary Licensing, N.Y., 1956

[8] Kisyński J., “On M. Kac's Probabilistic Formula for the Solutions of the Telegraphist's Equation”, Annales Polonici Mathematici, 29 (1974), 259–272 | DOI | MR | Zbl

[9] Ethier S. N., Kurtz T. G., Markov Processes. Characterization and Convergence, Wiley, N.Y., 1986 | MR | Zbl

[10] Pinsky M. A., Lectures on Random Evolutions, World Scientific, Singapore, 1991 | MR

[11] Asmussen S., Applied Probability and Queues, Springer, N.Y., 2003 | MR | Zbl

[12] Cox D. R., “The Analysis of Non-Markovian Stochastic Processes by the Inclusion of Supplementary Variables”, Mathematical Proceedings of the Cambridge Philosophical Society, 51:3 (1955), 433–441 | DOI | MR | Zbl

[13] Gwiżdż P., “Application of Stochastic Semigroups to Queueing Models”, Annales Mathematicae Silesianae, 2018 (to appear) | DOI

[14] Davis M. H. A., Lectures on Stochastic Control and Nonlinear Filtering, Springer, N.Y., 1984 | MR | Zbl

[15] Davis M. H. A., “Piece-Wise Deterministic Markov Processes”, Journal of the Royal Statistical Society, 46:3 (1984), 353–388 | MR | Zbl

[16] Davis M. H. A., Markov Processes and Optimization, Chapman and Hall, London, 1993 | MR

[17] Rudnicki R., Tyran-Kamińska M., Piecewise Deterministic Processes in Biological Models, Springer, N.Y., 2017 | MR | Zbl

[18] Greiner G., “Perturbing the Boundary Conditions of a Generator”, Houston Journal of Mathematics, 13:2 (1987), 213–229 | MR | Zbl

[19] Bobrowski A., “Generation of Cosine Families via Lord Kelvin's Method of Images”, Journal of Evolution Equations, 10:3 (2010), 663–675 | DOI | MR | Zbl

[20] Bobrowski A., “Lord Kelvin's Method of Images in the Semigroup Theory”, Semigroup Forum, 81:3 (2010), 435–445 | DOI | MR | Zbl

[21] Bobrowski A., “Families of Operators Describing Diffusion Through Permeable Membranes”, Operator Semigroups Meet Complex Analysis, Harmonic Analysis and Mathematical Physics, 250 (2015), 87–105 | MR | Zbl

[22] Bobrowski A., Gregosiewicz A., “A General Theorem on Generation of Moments-Preserving Cosine Families by Laplace Operators in $\mathrm{C}[0,1]$”, Semigroup Forum, 88:3 (2014), 689–701 | DOI | MR | Zbl

[23] Bobrowski A., Gregosiewicz A., Murat M., “Functionals-Preserving Cosine Families Generated by Laplace Operators in $\mathrm{C}[0, 1]$”, Discrete and Continuous Dynamical System, 20:7 (2015), 1877–1895 | DOI | MR | Zbl

[24] Bobrowski A., Mugnolo D., “On Moments-Preserving Cosine Families and Semigroups in $\mathrm{C}[0, 1]$”, Journal of Evolution Equations, 13:4 (2013), 715–735 | DOI | MR | Zbl

[25] Bobrowski A., “From Diffusions on Graphs to Markov Chains via Asymptotic State Lumping”, Annales Henri Poincare, 13:6 (2012), 1501–1510 | DOI | MR | Zbl

[26] Bobrowski A., Kaźmierczak B., Kunze M., “An Averaging Principle for Fast Diffusions in Domains Separated by Semi-Permeable Membranes”, Mathematical Models and Methods in Applied Sciences, 27:4 (2017), 663–706 | DOI | MR | Zbl