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PASTA for Remaining Service Time in Stable and Unstable Queues

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Information Technologies and Mathematical Modelling. Queueing Theory and Applications (ITMM 2021)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 1605))

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Abstract

In this paper, we establish the PASTA property for the limiting distribution of the remaining service time in stable and non-stable multiclass M/G/1 queueing systems. Our asymptotic analysis heavily exploits the regenerative property of the queueing system. The proof is first given for the stable system. A key observation for non-stable systems is that while the basic processes diverge to infinity, a proper time-average limit of the remaining service time exists. Some numerical results demonstrating the PASTA property are included as well.

This work is supported by the Russian Science Foundation, Project No. 21-71-10135, https://rscf.ru/en/project/21-71-10135/.

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References

  1. Asmussen, S.: Applied Probability and Queues, 2nd edn. Springer, New York (2003). https://doi.org/10.1007/b97236

    Book  MATH  Google Scholar 

  2. Boer, P.T.D., Nicola, V.F., van Ommeren, J.K.C.: The remaining service time upon reaching a high level in \(M/G/1\) queues. Questa 39, 55–7823 (2001). https://doi.org/10.1023/A:1017935616446

    Article  MathSciNet  MATH  Google Scholar 

  3. Serfozo, R.F.: Basics of Applied Stochastic Processes. Springer-Verlag, Heidelberg (2009). https://doi.org/10.1007/978-3-540-89332-5

    Book  MATH  Google Scholar 

  4. Borovkov, A.A.: Probability Theory. Springer Science and Business Media, London (2013). https://doi.org/10.1007/978-1-4471-5201-9

    Book  MATH  Google Scholar 

  5. Morozov, E., Steyaert, B.: Stability Analysis of Regenerative Queueing Models. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-82438-9

    Book  MATH  Google Scholar 

  6. Iglehart, D., Whitt, W.: Multiple channel queues in heavy traffic. Adv. Appl. Prob. 2, 150–170 (1970)

    Article  MathSciNet  Google Scholar 

  7. Kerner, Y.: The conditional distribution of the residual service time in the \(M_n/G/1\) queue. Stochast. Models 24(3), 364–375 (2008)

    Article  Google Scholar 

  8. Kerner, Y.: Equilibrium joining probabilities for an \(M/G/1\) queue. Games Econ. Behav. 71(2), 521–526 (2011)

    Article  MathSciNet  Google Scholar 

  9. Morozov, E.: The tightness in the ergodic analysis of regenerative queueing processes. Queueing Syst. 27, 179–203 (1997). https://doi.org/10.1023/A:1019114131583

    Article  MathSciNet  MATH  Google Scholar 

  10. Morozov, E., Delgado, R.: Stability analysis of regenerative queues. Autom. Remote Contr. 70, 1977–1991 (2009). https://doi.org/10.1134/S0005117909120066

    Article  MATH  Google Scholar 

  11. Morozov, E., Steyaert, B.: A stability analysis method of regenerative queueing systems. In: Anisimov, V., Limnios, N. (eds.) Queueing Theory 2. Advanced Trends, Wiley/ISTE, New York (2021)

    MATH  Google Scholar 

  12. Morozov, E., Morozova, T.: On the stationary remaining service time in the queuieng systems. CEUR Workshop Proc. 2792, 140–149 (2020)

    Google Scholar 

  13. Ross, S.M., Seshadri, S.: Hitting time in an \(M/G/1\) queue. J. Appl. Prob. 36, 934–940 (1999)

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Petrozavodsk State University, Petrozavodsk, Russia

    Evsey Morozov & Taisia Morozova

  2. Institute of Applied Mathematical Research of the Karelian research Centre of RAS, Petrozavodsk, Russia

    Evsey Morozov & Taisia Morozova

  3. Moscow Center for Fundamental and Applied Mathematics, Moscow State University, Moscow, Russia

    Evsey Morozov

  4. Uppsala University, Uppsala, Sweden

    Taisia Morozova

Authors
  1. Evsey Morozov
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  2. Taisia Morozova
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Corresponding author

Correspondence to Taisia Morozova .

Editor information

Editors and Affiliations

  1. Belarusian State University, Minsk, Belarus

    Alexander Dudin

  2. Tomsk State University, Tomsk, Russia

    Anatoly Nazarov

  3. Tomsk State University, Tomsk, Russia

    Alexander Moiseev

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Morozov, E., Morozova, T. (2022). PASTA for Remaining Service Time in Stable and Unstable Queues. In: Dudin, A., Nazarov, A., Moiseev, A. (eds) Information Technologies and Mathematical Modelling. Queueing Theory and Applications. ITMM 2021. Communications in Computer and Information Science, vol 1605. Springer, Cham. https://doi.org/10.1007/978-3-031-09331-9_23

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  • DOI: https://doi.org/10.1007/978-3-031-09331-9_23

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-09330-2

  • Online ISBN: 978-3-031-09331-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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