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On mathematical models of the service networks

  • New Lines of Development of the Queuing Theory
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Abstract

Queueing theory, as a part of probability theory, has classical problems and well-established mathematical apparatus. In this short review we follow queueing theory relations with other mathematical disciplines and different applied problems.

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References

  1. Erlang, A.K., The Theory of Probabilities and Telephone Conversations, Nyt Tidsskrift Mat., 1909, vol. 20, pp. 33–39.

    Google Scholar 

  2. Ganesh, A., O’Connell, N., and Wischik, D., Big Queues, in Lect. Notes in Math., 2004, vol. 1838, Berlin: Springer, 2004 (reprinted 2008).

    MATH  Google Scholar 

  3. Yashkov, S.F. and Yashkova, A.S., Processor Sharing: A Survey of the Mathematical Theory, Autom. Remote Control, 2007, no. 9, pp. 1662–1731.

  4. Yashkov, S.F., Processor-sharing Queues: Some Progress in Analysis, Queueing Syst., 1987, vol. 2, no. 1, pp. 1–17.

    Article  MATH  MathSciNet  Google Scholar 

  5. Kelly, F., Reversibility and Stochastic Networks, Chichester: Wiley, 1979 (reprinted 1987, 1994).

    MATH  Google Scholar 

  6. Nelson, R., The Mathematics of Product Form Queueing Networks, ACM Comput. Surveys, 1995, vol. 25, no. 3, pp. 339–369.

    Article  Google Scholar 

  7. Malyshev, V.A. and Pirogov, S.A., Invertibility and Noninvertibility in the Stochastic Chemical Kinetics, Rus. Math. Surv., 2008, vol. 63, no. 1, pp. pp. 1–34.

    Article  MATH  MathSciNet  Google Scholar 

  8. Malyshev, V.A., Wiener-Hopf Equations and Their Applications in the Probability Theory, J. Soviet Math., 1977, vol. 7, no. 2, pp. 129–148.

    Article  MathSciNet  Google Scholar 

  9. Malyshev, V.A., Sluchainye bluzhdaniya. Uravneniya Vinera-Khopfa v chetverti ploskosti. Avtomorfizmy Galua (Random Walks. Wiener-Hopf Equations in Quarter Plane. Galois Automorphisms), Moscow: Mosk. Gos. Univ., 1970.

    Google Scholar 

  10. Fayolle, G., Malyshev, V., and Iasnogorodski, R., Random Walks in the Quarter Plane (Algebraic Methods, Boundary Value Problems and Applications to Queuing Systems), Berlin: Springer, 1999 (reprinted 2008).

    Google Scholar 

  11. Kurkova, I.A. and Malyshev, V.A., Martin Boundary and Elliptic Curves, Markov Process. Related Fields, 1998, vol. 4, no. 2, pp. 203–272.

    MATH  MathSciNet  Google Scholar 

  12. Kurkova, I.A. and Suhov, Yu.M., Malyshev’s Theory and JS-queues. Asymptotics of Stationary Probabilities, Ann. Appl. Probab., 2003, vol. 13, no. 4, pp. 1313–1354.

    Article  MATH  MathSciNet  Google Scholar 

  13. Malyshev, V. and Spieksma, F., Intrinsic Convergence Rates of Countable Markov Chains, Markov Process. Related Fields, 1995, vol. 1, no. 2, pp. 203–266.

    MATH  MathSciNet  Google Scholar 

  14. Fayolle, G., Malyshev, V., and Menshikov, M., Topics in Constructive Theory of Countable Markov Chains, Cambridge: Cambridge Univ. Press, 1995.

    MATH  Google Scholar 

  15. Malyshev, V.A. and Men’shikov, M.V., Ergodicity, Continuity, and Analyticity of the Countable Markov Chains, Tr. Mosk. Mat. Obshch., 1979, vol. 39, pp. 3–48.

    MATH  MathSciNet  Google Scholar 

  16. Fayolle, G., Malyshev, V., Menshikov, M., and Sidorenko, A., Lyapounov Functions for Jackson Networks, Math. Oper. Res., 1993, vol. 18, no. 4, pp. 916–927.

    Article  MATH  MathSciNet  Google Scholar 

  17. Malyshev, V., Networks and Dynamical Systems, Adv. Appl. Prob., 1993, vol. 25, pp. 140–175.

    Article  MATH  MathSciNet  Google Scholar 

  18. Malyshev, V.A. and Ignatyuk, I.A., Classification of Random Walks in Z 4+ , Selecta Math. Sov., 1993, vol. 12, no. 2, pp. 129–194.

    MATH  MathSciNet  Google Scholar 

  19. Ignatyuk, I.A., Malyshev, V.A., and Shcherbakov, V.V., Boundary Effects in Large Deviation Problems, Rus. Math. Surv., 1994, vol. 49, no. 2, pp. 41–99.

    Article  MathSciNet  Google Scholar 

  20. Gajrat, A., Hordijk, A., Malyshev, V., and Spieksma, F., Fluid Approximation of Decision Markov Processes, Markov Process. Related Fields, 1997, vol. 3, no. 1, pp. 129–150.

    MATH  MathSciNet  Google Scholar 

  21. Malyshev, V.A., Evolution of a Random String: Stabilization Laws, Probl. Inform. Transm., 1994, vol. 30, no. 3, pp. 260–274.

    MATH  MathSciNet  Google Scholar 

  22. Gairat, A., Iasnogorodski, R., and Malyshev, V., Null Recurrent Random String, Markov Processes Related Fields, 1996, vol. 2, no. 3, pp. 427–460.

    Google Scholar 

  23. Malyshev, V., Gairat, A., and Zamyatin, A., Two-sided Evolution of a Random String, Markov Process. Related Fields, 1995, vol. 1, no. 2, pp. 281–316.

    MATH  MathSciNet  Google Scholar 

  24. Malyshev, V., Gairat, A., Men’shikov, M., and Pelikh, K., Classification of Markov Chains Describing the Evolution of Random Strings, Rus. Math. Surv., 1995, vol. 50, no. 2, pp. 237–255.

    Article  MATH  MathSciNet  Google Scholar 

  25. Malyshev, V.A., Interacting Strings of Symbols, Rus. Math. Surv., 1997, vol. 52, no. 2, pp. 299–326.

    Article  MATH  MathSciNet  Google Scholar 

  26. Malyshev, V.A., Random Grammars, Rus. Math. Surv., 1998, vol. 53, no. 2, pp. 345–370.

    Article  MATH  MathSciNet  Google Scholar 

  27. Karpelevich, F., Malyshev, V., Petrov, A., et al., Context Free Evolution of Words, in Analytic Methods in Applied Probability. In Memory of Fridrih Karpelevich, Suhov, Yu., Ed., Providence: Am. Math. Soc. Transl., Ser. 2, 2002, vol. 207, pp. 91–115.

    Google Scholar 

  28. Blythe, R.A. and Evans, M.R., Nonequilibrium Steady States of Matrix-Product Form: A Solver’s Guide, J. Phys. A: Math. Theor., 2007, vol. 40, pp. R333–R441 (arXiv: 0706.1678).

    Article  MATH  MathSciNet  Google Scholar 

  29. Malyshev, V. and Manita, A., Stochastic Micromodel of the Couette Flow, Theor. Prob. Appl., 2008, vol. 53, no. 4.

  30. Malyshev, V. and Tsaregradskii, I., Queuing System with Local Interaction, Theor. Prob. Appl., 1982, vol. 27, no. 3, pp. 619–624.

    Article  MATH  MathSciNet  Google Scholar 

  31. Lee, T., The Mathematical Parallels between Packet Switching and Parallel Transmission, Preprint, 2006, arXiv:cs/0610050v2.

  32. Berezner, S.A. and Malyshev, V.A., The Stability of Infinite Server Network with Random Routing, J. Appl. Prob., 1989, vol. 26, pp. 363–371.

    Article  MATH  MathSciNet  Google Scholar 

  33. Botvich, D., Fayolle, G., and Malyshev, V., Loss Networks in Thermodynamic Limit, in Lecture Notes in Control and Information Sciences, Berlin: Springer, 1994, vol. 199, pp. 465–489.

    Google Scholar 

  34. Malyshev, V. and Robert, Ph., Phase Transitions in a Load Sharing Model, Ann. Appl. Probab., 1995, vol. 5, no. 4, pp. 1161–1176.

    Article  MathSciNet  Google Scholar 

  35. Kelly, Loss Networks, Ann. Appl. Probab., 1991, vol. 1, pp. 319–378.

    Article  MATH  MathSciNet  Google Scholar 

  36. Vladimirov, A.A., Rybko, A.N., and Shlosman, S.B., Self-averaging Property of Queueing Systems, Probl. Inform. Transm., 2006, vol. 42, no. 4, pp. 344–355.

    Article  MathSciNet  Google Scholar 

  37. Rybko, A., Shlosman, S., and Vladimirov, A., Absence of Breakdown of the Poisson Hypothesis I. Closed Networks at Low Load, Preprint, 2008, arXiv:0811.3577.

  38. Rybko, A., Shlosman, S., and Vladimirov, A., Spontaneous Resonances and the Coherent States of the Queuing Networks, Preprint, 2007, arXiv:0708.3073.

    Google Scholar 

  39. Rybko, A., Shlosman, S., and Vladimirov, A., Spontaneous Resonances and the Coherent States of the Queuing Networks, J. Statist. Phys., 2009, vol. 134, no. 1, pp. 67–104 (arXiv:0708.3073).

    Article  MATH  MathSciNet  Google Scholar 

  40. Greenberg, A., Malyshev, V., and Popov, S., Stochastic Model of Massively Parallel Simulation, Markov Process. Related Fields, 1995, vol. 1, no. 4, pp. 473–490.

    MATH  MathSciNet  Google Scholar 

  41. Manita, A.D., Collective Behavior in Multidimensional Probabilistic Models of Synchronization, Obozr. Prikl. Prom. Mat., 2007, vol. 14, no. 6, pp. 1001–1021.

    MATH  Google Scholar 

  42. Malyshev, V. and Yakovlev, A., Condensation in Large Closed Jackson Networks, Ann. Appl. Probab., 1996, vol. 6, no. 1, pp. 92–115.

    Article  MATH  MathSciNet  Google Scholar 

  43. Malyshev, V., Pirogov, S., and Rybko, A., Random Walks and Chemical Networks, Moscow Math. J., 2004, no. 2, pp. 441–453.

  44. Malyshev, V.A., Microscopic Models For Chemical Thermodynamics, J. Statist. Phys., 2005, vol. 119, no. 5/6, pp. 997–1026.

    Article  MATH  MathSciNet  Google Scholar 

  45. Zamyatin, A.A., Malyshev, V.A., and Manita, A.D., Homeostasis in Chemical Reaction Pathways, Theor. Prob. Appl., 2006, vol. 51, no. 4, pp. 714–723.

    MathSciNet  Google Scholar 

  46. The Handbook of Brain Theory and Neural Networks, Arbib, M., Ed., Cambridge: MIT Press, 2003.

    MATH  Google Scholar 

  47. Haykin, S., Neironnye seti: polnyi kurs (Neural Networks: A Comprehensive Foundation), Moscow: Williams, 2006.

    Google Scholar 

  48. Dotsenko, V., Introduction to the Theory of Spin Glasses and Neural Networks, Singapore: World Scientific, 1994.

    MATH  Google Scholar 

  49. Malyshev, V.A., Random Graphs and Grammars on Graphs, Diskretn. Mat. Primen., 1998, vol. 10, no. 2, pp. 30–44.

    Google Scholar 

  50. Karpelevich, F., Malyshev, V., and Rybko, A., Stochastic Evolution of Neural Networks, Markov Process. Related Fields, 1995, vol. 1, no. 1, pp. 141–161.

    MATH  MathSciNet  Google Scholar 

  51. Malyshev, V. and Turova, T., Gibbs Measures on Attractors in Biological Neural Networks, Markov Process. Related Fields, 1997, vol. 3, no. 4, pp. 443–464.

    MATH  MathSciNet  Google Scholar 

  52. Shcherbakov, V. and Volkov, S., Queueing with Neighbours, Preprint of Glasgow Univ., 2008, arXiv: 0907.1826.

  53. Zamyatin, A.A. and Malyshev, V.A., Accumulation on the Boundary for a One-Dimensional Stochastic Particle System, Probl. Inform. Transm., 2007, vol. 43, no. 4, pp. 331–343.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Lomonosov State University, Moscow, Russia

    V. A. Malyshev

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  1. V. A. Malyshev
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Original Russian Text © V.A. Malyshev, 2009, published in Avtomatika i Telemekhanika, 2009, No. 12, pp. 9–15.

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Malyshev, V.A. On mathematical models of the service networks. Autom Remote Control 70, 1947–1953 (2009). https://doi.org/10.1134/S0005117909120029

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