Skip to main content
SpringerLink
Log in

An asymptotic property of output streams in queueing systems with unbounded number of servers and a Markov arrival process

  • Stochastic Systems, Queueing Systems
  • Published:
Automation and Remote Control Aims and scope Submit manuscript

Abstract

We study the outgoing stream of queueing systems with unbounded number of servers and a Markov arrival process. We consider models with exponential and arbitrary distributions of the service duration. For the case of arbitrary service duration, we propose the sieved stream method that reduces the study of the output stream to studying a sieved stream. We show that for growing service duration, the output stream is asymptotically a simple stream independent of the service duration distribution.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Gnedenko, B.V. and Kovalenko, I.N., Vvedenie v teoriyu massovogo obsluzhivaniya (Introduction to Queueing Theory), Moscow: KomKniga, 2005, 3rd ed.

    Google Scholar 

  2. Nazarov, A.A. and Terpugov, A.F., Teoriya massovogo obsluzhivaniya (Queueing Theory), Tomsk: NTL, 2004.

    Google Scholar 

  3. Saaty, T.L., Elements of Queueing Theory, with Applications, New York: McGraw-Hill, 1961. Translated under the title Elementy teorii massovogo obsluzhivaniya i ee primeneniya, Moscow: Sovetskoe Radio, 1971.

    MATH  Google Scholar 

  4. Lopukhova, S.V. and Nazarov, A.A., Studying a MAP Stream with Asymptotic Analysis of the Nth Order, Vestnik TGU, Ser.: Informat. Kibern. Mat., 2006, no. 293, pp. 110–115.

  5. Nazarov, A.A. and Kulikova, O.A., Studying Infinite-Dimensional Queueing Systems with the Sieved Stream Method, in Massovoe obsluzhivanie. Potoki, sistemy, seti. Materialy Mezhdunarodnoi nauchnoi konferentsii “Matematicheskie metody povysheniya effektivnosti funktsionirovaniya telekommunikatsionnykh setei” (Queueing Systems. Streams, Systems, Networks. Proc. Int. Conf. “Mathematical Methods for Increasing the Operation Efficiency of Telecommunication Networks”), Minsk: BGU, 2005, pp. 98–102.

    Google Scholar 

  6. Baum, D., The Infinite Server Queue with Markov Additive Arrivals in Space, Proc. Int. Conf. “Probabilistic Analysis of Rare Events: Theory and Problems of Safety, Insurance and Ruin,” Riga: RAU, 1999, pp. 136–142.

    Google Scholar 

  7. Jackson, J.R., Networks of Waiting Lines, Oper. Res., 1957, vol. 5, no. 4, pp. 518–521.

    Article  Google Scholar 

  8. Moor, F.R., Computational Model of a Closed Queueing Network with Exponential Servers, IBM J. Res. Dep., 1972, vol. 16, no. 6, pp. 567–573.

    Article  Google Scholar 

  9. Zhozhikashvili, V.A. and Vishnevskii, V.M., Seti massovogo obsluzhivaniya. Teoriya i primenenie k setyam EVM (Queueing Networks. Theory and Applications to Computer Networks), Moscow: Radio i Svyaz’, 1988.

    Google Scholar 

  10. Massey, W.A. and Whitt, W., Networks of Infinite-Server Queues with Nonstationary Poisson Input, Queuing Syst., 1993, no. 13, pp. 183–250.

  11. Burke, P.J., The Output of Queueing Systems, Oper. Res., 1956, vol. 4, pp. 699–704.

    Article  MathSciNet  Google Scholar 

  12. Reich, E., Waiting Times When Queues are in Tandem, Ann. Math. Stat., 1957, vol. 28, no. 3, p. 768.

    Article  MATH  Google Scholar 

  13. Finch, P.D., The Output Process of the Queueing System M|G|1, J. Roy. Stat. Soc., 1959, vol. 21, no. 2, pp. 375–380.

    MathSciNet  MATH  Google Scholar 

  14. Akulinichev, N.M. and Gorskii, L.K., On Asymptotic Distributions of Output Streams for Certain Queueing Systems, Cybernetics, 1973, no. 1, pp. 71–78.

  15. Disney, R.L., Farell, R.L., and de Morais, P.R., Characterization of M|G|1 Queues with Renewal Departure Processes, Manag. Sci., 1973, vol. 19, no. 11, pp. 1222–1228.

    Article  MATH  Google Scholar 

  16. Goswami, V., Distribution of the Number of Customs Served During a Busy Period in a Discrete Time Geom/Geom/1, Indian J. Pure Appl. Math., 2002, vol. 33, no. 9, pp. 1405–1508.

    MathSciNet  MATH  Google Scholar 

  17. Proidakova, E.V. and Fedotkin, M.A., Finding Existence Conditions for a Stationary Output Stream Distribution in a System with Cyclic Control, Vestn. Nizhn. Novgorod. Gos. Univ., Mat., 2006, no. 1 (4), pp. 92–102.

  18. Proidakova, E.V. and Fedotkin, M.A., Finding a Sufficient Existence Condition for a Stationary Output Stream Distribution in a System with Cyclic Control, Vestn. Nizhn. Novgorod. Gos. Univ., Mat. Modelir. Optimal. Upravlen., 2006, no. 3 (32), pp. 57–68.

  19. Proidakova, E.V. and Fedotkin, M.A., Control of Output Flows in the System with Cyclic Servicing and Readjustments, Autom. Remote Control, 2008, vol. 69, no. 6, pp. 993–1002.

    Article  MathSciNet  MATH  Google Scholar 

  20. Bean, N.G., Green, D.A., and Taylor, P.G., The Output Process of an MMPP |M| 1 Queue, J. Appl. Probab., 1998, no. 35(4), pp. 998–1002.

    Google Scholar 

  21. Mirasol, N.M., The Output of an M|G| ∞ Queueing System is Poisson, Oper. Res., 1963, no. 11, pp. 282–284.

  22. Neuts, M.F., A Versatile Markovian Arrival Process, J. Appl. Prob., 1979, vol. 16, pp. 764–779.

    Article  MathSciNet  MATH  Google Scholar 

  23. Lucantoni, D., New Results for the Single Server Queue with a Batch Markovian Arrival Process, Stoch. Models, 1991, vol. 7, pp. 1–46.

    Article  MathSciNet  MATH  Google Scholar 

  24. Dudin, A.N. and Klimenok, V.I., Sistemy massovogo obsluzhivaniya s korrelirovannymi potokami, (Queueing Systems with Correlated Streams), Minsk: BGU, 2000.

    Google Scholar 

  25. Nazarov, A.A. and Moiseeva, S.P., Metod asimptoticheskogo analiza v teorii massovogo obsluzhivaniya (The Asymptotic Analysis Method in Queueing Theory), Tomsk: NTL, 2006.

    Google Scholar 

  26. Koenig, D. and Stoyan, D., Methoden der Bedienungstheorie, Berlin: Akademie-Verlag, 1976. Translated under the title Metody teorii massovogo obsluzhivaniya, Klimov, G.P., Ed., Moscow: Radio i Svyaz’, 1981.

    Google Scholar 

  27. Kendall, D.G., Stochastic Processes Occurring in the Theory of Queues and Their Analysis by the Method of the Imbedded Markov Chain, Ann. Math. Stat., 1953, vol. 24, pp. 338–354.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Tomsk State University, Tomsk, Russia

    I. L. Lapatin & A. A. Nazarov

Authors
  1. I. L. Lapatin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. A. Nazarov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Original Russian Text © I.L. Lapatin, A.A. Nazarov, 2012, published in Avtomatika i Telemekhanika, 2012, No. 5, pp. 57–70.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lapatin, I.L., Nazarov, A.A. An asymptotic property of output streams in queueing systems with unbounded number of servers and a Markov arrival process. Autom Remote Control 73, 810–820 (2012). https://doi.org/10.1134/S0005117912050050

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117912050050

Keywords

Search

Navigation