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On Estimates of the Mean Queue Length for Single-Channel Queuing Systems in Terms of Statistical Unconditional Second-Order Moments of the Modified Arrival Flow

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Abstract

We consider a mathematical model of the simplest single-channel queuing system (QS) with a deterministic service time in the case of an arbitrarily correlated arrival flow. Various generalizations of the Pollaczek–Khinchine formula for the mean queue length are obtained for this QS. An interval model of the arrival flow is proposed. Within the framework of this model, an expression is obtained for the mean queue length in terms of statistical unconditional moments of the second order. All results are obtained under very general assumptions of ergodicity and stationarity. The results of numerical experiments confirming the theoretical conclusions are presented.

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REFERENCES

  1. Leland, W.E., Taqqu Murad, S., Willinger, W., and Wilson, D.V., On the self-similar nature of Ethernet traffic, J. IEEE/ACM Trans. Networking, 1994, vol. 2, no. 1, pp. 1–15.

    Article  Google Scholar 

  2. Neuts, M.F., Versatile Markovian point process, J. Appl. Probab., 1979, vol. 16, no. 4, pp. 764–779.

    Article  MathSciNet  Google Scholar 

  3. Ramaswami, V., The N/G/1 queue and its detailed analysis, Adv. Appl. Probab., 1980, vol. 12, no. 1, pp. 222–261.

    Article  MathSciNet  Google Scholar 

  4. Jagerman, D.L., Balcioglu, B., Altiok, T., and Melamed, B., Mean waiting time approximations in the G/G/1 queue, Queueing Syst., 2004, vol. 46, pp. 481–506.

    Article  MathSciNet  Google Scholar 

  5. Balcioglu, B., Jagerman, D.L., and Altiok, T., Approximate mean waiting time in a GI/D/1 queue with autocorrelated time to failures, IEEE Trans., 2007, vol. 39, no. 10, pp. 985–996.

    Article  Google Scholar 

  6. Kartashevskii, I.V. and Saprykin, A.V., Processing of correlated traffic at a G/G/1 type node, Radiotekhnika, 2017, no. 10, pp. 119–125.

  7. Kartashevskii, I.V., Traffic model for SDNs, Radiotekhnika, 2016, no. 6, pp. 124–129.

  8. Tsybakov, B.S., Teletraffic model based on a self-similar random process, Radiotekhnika, 1999, no. 5, pp. 24–31.

  9. Shelukhin, O.I., Tenyakshev, A.M., and Osin, A.V., Fraktal’nye protsessy v telekommunikatsiyakh (Fractal Processes in Telecommunications), Moscow: Radiotekhnika, 2003.

    Google Scholar 

  10. Likhttsinder, B.Ya., Correlation links in pack flows of queuing systems, Telekommunikatsii (Telecommunications), Moscow: Nauka Tekhnol., 2015, no. 9, pp. 8–12.

  11. Likhttsinder, B.Ya., Correlation properties of queue lengths in queuing systems with general flows, Infokommun. Tekhnol., 2015, vol. 13, no. 3, pp. 276–280.

    Article  Google Scholar 

  12. Blatov, I.A. and Likhttsinder, B.Ya., On the estimation of queue lengths in QS with arbitrary correlation, in Sb. tr. Informatsionnye tekhnologii i nanotekhnologii (ITNT–2018) (Collect. Works Information Technology and Nanotechnology (ITNT–2018)), Samara: Samarskii Nats. Issled. Univ. im. akad. S.P. Koroleva, 2018, pp. 1607–1616.

    Google Scholar 

  13. Likhttsinder, B.Ya., On some generalizations of the Pollaczek–Khinchine formula, Infokommun. Tekhnol., 2007, vol. 5, no. 4, pp. 253–258.

    Google Scholar 

  14. Vishnevskii, V.M. and Dudin, A.N., Queueing systems with correlated arrival flows and their applications to modeling telecommunication networks, Autom. Remote Control, 2017, vol. 78, no. 8, pp. 1361–1403.

    Article  MathSciNet  Google Scholar 

  15. Shul’ga, Yu.N., Extension of the Pollaczek–Khinchine formula to 3D stochastic networks, Autom. Remote Control, 1989, vol. 50, no. 3, pp. 355–365.

    MathSciNet  MATH  Google Scholar 

  16. Jain, G. and Sigman, K., A Pollaczek–Khinchine formula for M/G/1 queues with disasters, J. Appl. Probab., 1996, vol. 33, pp. 1191–1200.

    Article  MathSciNet  Google Scholar 

  17. Likhttsinder, B.Ya., Interval method for analyzing multiservice traffic of access networks, Elektrosvyaz’, 2015, no. 12, pp. 52–54.

  18. Likhttsinder, B.Ya., Trafik mul’tiservisnykh setei dostupa (interval’nyi analiz i proektirovanie) (Traffic of Multiservice Access Networks (Interval Analysis and Design)), Moscow: Goryachaya Liniya–Telekom, 2018.

    Google Scholar 

  19. Venttsel’, E.S., Issledovanie operatsii. Zadachi, printsipy, metodologiya (Operations Research. Objectives, Principles, Methodology), Moscow: Nauka, 1988.

    MATH  Google Scholar 

  20. Likhttsinder, B.Ya., Interval’nyi metod analiza trafika mul’tiservisnykh setei dostupa (Interval Method for Traffic Analysis of Multiservice Access Networks), Samara: PGUTI, 2015.

    Google Scholar 

  21. Kleinrock, L., Queueing Systems. Vol. 1: Theory, New York–London–Sydney–Toronto: Wiley Interscience, 1975. Translated under the title: Teoriya massovogo obsluzhivaniya, М.: Mashinostroenie, 1979.

    MATH  Google Scholar 

  22. Gortsev, A.M. and Nezhel’skaya, L.A., On the relationship between MS-flows and MAP-flows of events, Vestn. TGU. Upr. Vychisl. Tekh. Inf., 2010, vol. 13, no. 4, pp. 50–60.

    Google Scholar 

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Authors and Affiliations

  1. Povolzhskiy State University of Telecommunications and Informatics, Samara, 443010, Russia

    B. Ya. Likhttsinder & I. A. Blatov

  2. Samara University, Samara, 443086, Russia

    E. V. Kitaeva

Authors
  1. B. Ya. Likhttsinder
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  2. I. A. Blatov
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  3. E. V. Kitaeva
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Correspondence to B. Ya. Likhttsinder, I. A. Blatov or E. V. Kitaeva.

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Translated by V. Potapchouck

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Likhttsinder, B.Y., Blatov, I.A. & Kitaeva, E.V. On Estimates of the Mean Queue Length for Single-Channel Queuing Systems in Terms of Statistical Unconditional Second-Order Moments of the Modified Arrival Flow. Autom Remote Control 83, 92–105 (2022). https://doi.org/10.1134/S0005117922010076

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