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Queueing Systems

, Volume 76, Issue 2, pp 125–147 | Cite as

Coupling method for asymptotic analysis of queues with regenerative input and unreliable server

  • L. G. Afanasyeva
  • E. E. BashtovaEmail author
Article

Abstract

First we define a regenerative flow and describe its properties. Then a single-server queueing system with regenerative input flow and an unreliable server are considered. By applying coupling we establish the ergodicity condition and prove the limit theorem in the heavy traffic situation (traffic coefficient \(\rho <1, \rho \uparrow 1\)). The asymptotic analysis of the super-heavy traffic situation (\(\rho \ge 1\)) is also realized.

Keywords

Regenerative flow Unreliable server Ergodicity  Limit theorems 

Mathematics Subject Classification (2000)

60K25 90B22 

Notes

Acknowledgments

This work is partially supported by RFBR-grant 13-01-00653.

References

  1. 1.
    Afanasyeva, L.G.: Queueing systems with cyclic control processes. Cybern. Syst. Anal. 41(1), 43–55 (2005)CrossRefGoogle Scholar
  2. 2.
    Afanasyeva, L.G., Bashtova, E.E.: Limit theorems for queuing systems with doubly stochastic poisson arrivals (heavy traffic conditions). Probl. Inf. Transm. 44(4), 352–369 (2008)CrossRefGoogle Scholar
  3. 3.
    Afanasyeva, L.G., Bashtova, E.E.: Limit theorems for queues in heavy traffic situation. Mod. Probl. Math. Mech. IV, 1 140–54 (in Russian) (2009)Google Scholar
  4. 4.
    Afanasyeva, L., Bashtova, E., Bulinskaya, E.: Limit theorems for semi-Markov queues and their applications. Commun. Stat. Simul. Comput. 41(6), 688–709 (2012)CrossRefGoogle Scholar
  5. 5.
    Afanasyeva, L.G., Bulinskaya, E.V.: Stochasic models of transport flows. Commun. Stat. Theory Methods 40(16), 2830–2846 (2011)CrossRefGoogle Scholar
  6. 6.
    Afanasyeva, L.G., Rudenko, I.V.: Queueing systems \(GI|G|\infty \) and there applications to transport models analysis. Theory Probab. Appl. 57(3), 427–452 (2012)Google Scholar
  7. 7.
    Asmussen, S.: Applied Probability and Queues. Wiley, Chichester (1987)Google Scholar
  8. 8.
    Bikjalis, A.: Estimates of the remainder term in the central limit theorem. (Russian). Litovsk Mat. Sb. 6, 323–346 (1966)Google Scholar
  9. 9.
    Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968)Google Scholar
  10. 10.
    Borovkov, A.A.: Some limit theorems in the theory of mass service, II. Theory Probab. Appl. 10, 375–400 (1965)CrossRefGoogle Scholar
  11. 11.
    Borovkov, A.A.: Stochastic Processes in Queuing Theory. Springer, Berlin (1976)CrossRefGoogle Scholar
  12. 12.
    Borovkov, A.A.: Asymptotic Methods in Queuing Theory. Wiley, Chichester (1984)Google Scholar
  13. 13.
    Cox, D.R.: Renewal Theory, London: Methuen and Co LTD. Wiley, New York (1962)Google Scholar
  14. 14.
    Djellab, N.V.: On the \(M|G|1\) retrial queue subjected to breakdowns RAIRO. Oper. Res. 36, 299–310 (2002)CrossRefGoogle Scholar
  15. 15.
    Gaver, D.P.: A waiting line with interrupted service, including priorities. J. R. Soc. B24, 73–90 (1962)Google Scholar
  16. 16.
    Foss, S., Kovalevskii, A.: A stability criterion via fluid limits and its application to a polling model. Queueing Syst. 32, 131–168 (1999)CrossRefGoogle Scholar
  17. 17.
    Foss, S.G., Kalashnikov, V.V.: Regeneration and renovation in queues. Queueing Syst. 8(3), 211–223 (1991)CrossRefGoogle Scholar
  18. 18.
    Foster, F.G.: On the stochastic matrices associated with queueing processes. Ann. Math. Stat. 24(3), 355–360 (1953)CrossRefGoogle Scholar
  19. 19.
    Gideon, R., Pyke, R.: Markov Renewal Modeling of Poisson Traffic at Intersection having Separate Turn Lanes. Semi-Markov Models and Applications. Kluwek Academic Publishers, Dordrecht (1999)Google Scholar
  20. 20.
    Grandell, J.: Double Stochastic Poisson Processes. Lect. Notes Math., vol. 529. Springer, Berlin (1976)Google Scholar
  21. 21.
    Haight, F.A.: Mathematical Theories of Traffic Flow, Mathematical in Science and Eng., vol. 7. Academic Press, New York (1963)Google Scholar
  22. 22.
    Iglehart, D.L., Witt, L.W.: Multipple cannel queues in heavy traffic, I and II. Adv. Appl. Prob. 2(150–177), 355–369 (1970)CrossRefGoogle Scholar
  23. 23.
    Lindvall, T.: The probabilistic proof of Blackwell’s renewal theorem. Ann. Probab. 5(3), 482–485 (1977)CrossRefGoogle Scholar
  24. 24.
    Lindvall, T.: Lectures on the Coupling Method. J. Wiley, New York (1992)Google Scholar
  25. 25.
    Loynes, R.: The stability of a queue with non-independent inter-arrival and service times. Proc. Camb. Phil. Soc. 58(3), 497–520 (1962)CrossRefGoogle Scholar
  26. 26.
    Malyshev, V.A., Men’shikov, M.V.: Ergodicity continuity and analyticity of countable Markov chains. Trans. Moscow Math. 1, 1–48 (1982)Google Scholar
  27. 27.
    Meyn, S.P., Tweedie, R.L.: Markov Chains and Stochastic Stability. Springer, New York (1994)Google Scholar
  28. 28.
    Morozov, E.: The stability of non-homogeneous queueing system with regenerative input. J. Math. Sci. 89, 407–421 (1997)CrossRefGoogle Scholar
  29. 29.
    Mousstafa, M.D.: Input-output Markov processes. Proc. Koninkijke Nederlands Akad. Wetenschappen 60, 112–118 (1957)Google Scholar
  30. 30.
    Serfozo, R.: Applications of the key renewal theorem: crudely regenerative process. J. Appl. Probab. 29, 384–395 (1992)CrossRefGoogle Scholar
  31. 31.
    Sherman, N., Kharoufen, J., Abramson, M.: An \(M|G|1\) retrial queue with unreliable server for streaming multimedia applications. Prob. Eng. Inf. Sci. 23, 281–304 (2009)CrossRefGoogle Scholar
  32. 32.
    Smith, W.L.: Renewal theory and its ramifications. J. R. Stat. Soc. B,20, N2. (1958)Google Scholar
  33. 33.
    Tanner, J.C.: The delay to pedestrians crossing a road. Biometrika 38, 383–392 (1951)Google Scholar
  34. 34.
    Thorisson, H.: Coupling, Stationary and Regeneration. Springer, New York (2000)CrossRefGoogle Scholar
  35. 35.
    Whitt, W.: Heavy Traffic Limit Theorems for Queues: A Survey. Lecture Notes in Economics and Math. Systems, vol. 98. Springer, Berlin (1974)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Mathematics and MechanicsMoscow State UniversityMoscowRussia

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