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Phase Transitions in Multiserver Queuing Systems

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Information Technologies and Mathematical Modelling - Queueing Theory and Applications (ITMM 2016)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 638))

Abstract

In this paper synergetic effects in queuing systems and networks are investigated. In the systems M|M|n|0 with failures phase transition is established in cases when competition between servers for customers is present and when it is absent. In the systems \(M|M|n|\infty \) and \(M|G|n|\infty \) with heavy traffic, where distribution of service times are hyperexponential, for some stationary characteristics phase transitions are established. The critical parameters of these phase transitions are defined by load coefficients. The obtained results are spread onto queuing networks with nodes which have the type \(M|G|n|\infty \) and the effect of a queue disappearence is investigated. The main approach of these systems and networks analysis is their transformation into Jackson networks.

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References

  1. Basharin, G.P., Tolmachev, A.L.: Queuing network theory and its applications to the analysis of information-computing networks. In: Itogi Nauki Tech. Ser. Teor. Veroiatn. Mat. Stat. Teor. Kibern, vol. 21, pp. 3–119. VINITI, Moscow (1983) (In Russian)

    Google Scholar 

  2. Mokrov, E.V., Samuilov, K.E.: Model of systems of claud computing as queuing systems with few queues and group arrival of customers. Telecommun. Transp. 7(11), 139–141 (2013). (In Russian)

    Google Scholar 

  3. Vishnevskiy, V.M.: Theoretical Principles of Computer Network Design. Technosila, Moscow (2003)

    Google Scholar 

  4. Grachev, V.V., Moiseev, A.N., Nazarov, A.A., Iampolskiy, V.Z.: Multiphase model of queuing system with distributed data mining. Lect. TUSUR 2(26), 248–251 (2012). (In Russian)

    Google Scholar 

  5. Ivnitskiy, V.A.: Theory of Queuing Networks. Edition of Phys. - Mat. Lit., Moscow (2004). (In Russian)

    Google Scholar 

  6. Ivnitskiy, V.A.: Theory of Arbitrary Input Flow. Palmarium Academic Publishing, Saarbrucken (2012). (In Russian)

    Google Scholar 

  7. Nazarov, A.A.: Asymptotic Analysis of Markovian Systems. Edition of Tomsk State University, Tomsk (1995). (In Russian)

    Google Scholar 

  8. Nazarov, A.A., Moiseeva, S.P.: Method of Asymptotic Analysis in Queuing Theory. Edition of Tomsk State University, Tomsk (2006). (In Russian)

    Google Scholar 

  9. Nazarov, A.A., Moiseev, A.N.: Investigation of opened NonMarkovian queuing network \(GI-(GI|\infty )^K\) with high intensive recurrent input flow. Probl. Inf. Trans. 49(2), 78–91 (2013). (In Russian)

    Article  MathSciNet  Google Scholar 

  10. Moiseev, A.N., Nazarov, A.A.: Infinite Server Queuing Systems and Networks. Edition of Tomsk State University, Tomsk (2015). (In Russian)

    Google Scholar 

  11. Moiseev, A.N., Nazarov, A.A.: Asymptotic analysis of high intensive half Markovian flow of events. Lect. TUSUR 3(29), 109–115 (2013). (In Russian)

    Google Scholar 

  12. Moiseev, A.N., Nazarov, A.A.: Asymptotic analysis of multiphase queuing system with high intensive recurrent input flow. Autometrics 50(2), 67–76 (2014). (In Russian)

    Google Scholar 

  13. Matveev, S.A., Moiseev, A.N., Nazarov, A.A.: Application of method of initial moments for investigation of multiphase queuing systems \(GI/(M|\infty )^K\). Lect. TUSUR 3(33), 129–134 (2014)

    Google Scholar 

  14. Lopukhova, S.V.: Investigation of MMP flow by asymptotic method of m-th order. Herald of Tomsk State Univ. Control Comput. Sci. Inf. 3(4), 71–76 (2008). (In Russian)

    Google Scholar 

  15. Jackson, J.R.: Networks of waiting lines. Oper. Res. 5(4), 518–521 (1957)

    Article  MathSciNet  Google Scholar 

  16. Serfozo, R.: Introduction to Stochastic Networks, p. 301. Springer, New York (1999)

    Google Scholar 

  17. Ivchenko, G.I., Kashtanov, V.A., Kovalenko, I.N.: Queuing Theory. High school, Moscow (1982). (In Russian)

    MATH  Google Scholar 

  18. Shiriaev, A.N.: Probability. Moscow, Science (1989). (In Russian)

    Google Scholar 

  19. Feldmann, A., Whitt, W.: Fitting mixtures of exponentials to long tailed distributions to analyze network perfomance models. Perfom. Eval. 31(3–4), 245–279 (1998)

    Article  Google Scholar 

  20. Vatamidou, E., et al.: On the accuracy of phase-type approximations of heavy-tailed risk models. Scand. Actuarial J. 2014(6), 510–534 (2014)

    Article  MathSciNet  Google Scholar 

  21. Embrechts, P., Cluppelberg, C., Mikosch, T.: Modelling extremal events: for insurance and finance. Applications of Mathematics, vol. 33, p. 648. Springer, Heidelberg (1997)

    Google Scholar 

  22. Asmussen, S.: Ruin probabilities. In: Advanced Series on Statistical Science and Applied Probability, vol. 2. World Scientific Publishing Co. Inc., Singapore (2000)

    Google Scholar 

  23. Borovkov, A.A.: Limit theorems for queuing networks. Probab. Theor. Appl. 31(3), 474–490 (1986). (In Russian)

    MathSciNet  Google Scholar 

  24. Borovkov, A.A.: Probability Theory. Science, Moscow (1986)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Institute for Applied Mathematics Far Eastern Branch of RAS, Radio street 7, 690041, Vladivostok, Russia

    Gurami Tsitsiashvili

  2. Far Eastern Federal University, Sukhanova street 8, 690091, Vladivostok, Russia

    Marina Osipova

Authors
  1. Gurami Tsitsiashvili
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  2. Marina Osipova
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Corresponding author

Correspondence to Gurami Tsitsiashvili .

Editor information

Editors and Affiliations

  1. Belarusian State University , Minsk, Belarus

    Alexander Dudin

  2. Tomsk State University , Tomsk, Russia

    Alexander Gortsev

  3. Tomsk State University , Tomsk, Russia

    Anatoly Nazarov

  4. Kemerovo State University , Anzhero-Sudzhensk, Russia

    Rafael Yakupov

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Tsitsiashvili, G., Osipova, M. (2016). Phase Transitions in Multiserver Queuing Systems. In: Dudin, A., Gortsev, A., Nazarov, A., Yakupov, R. (eds) Information Technologies and Mathematical Modelling - Queueing Theory and Applications. ITMM 2016. Communications in Computer and Information Science, vol 638. Springer, Cham. https://doi.org/10.1007/978-3-319-44615-8_30

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  • DOI: https://doi.org/10.1007/978-3-319-44615-8_30

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-44614-1

  • Online ISBN: 978-3-319-44615-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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