Skip to main content
SpringerLink
Log in

Extremal GI/GI/1 queues given two moments: exploiting Tchebycheff systems

  • Published:
Queueing Systems Aims and scope Submit manuscript

A Correction to this article was published on 16 April 2022

This article has been updated

Abstract

This paper studies tight upper bounds for the mean and higher moments of the steady-state waiting time in the GI/GI/1 queue given the first two moments of the interarrival-time and service-time distributions. We apply the theory of Tchebycheff systems to obtain sufficient conditions for classical two-point distributions to yield the extreme values. These distributions are determined by having one mass at 0 or at the upper limit of support.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Change history

References

  1. Asmussen, S.: Phase-type representations in random walk and queueing problems. Ann. Probab. 20(2), 772–789 (1992)

    Article  Google Scholar 

  2. Asmussen, S.: Applied Probability and Queues, 2nd edn. Springer, New York (2003)

    Google Scholar 

  3. Bergmann, R., Daley, D.J., Rolski, T., Stoyan, D.: Bounds for cumulants of waiting times in \(GI/GI/1\) queues. Math. Op. Schung Stat. Ser. Optim. 10(2), 257–263 (1979)

    Google Scholar 

  4. Billingsley, P.: Convergence of Probability Measures, 1st edn. Wiley, New York (1968)

    Google Scholar 

  5. Chen, Y., Whitt, W.: Algorithms for the upper bound mean waiting time in the \(GI/GI/1\) queue. Queue. Syst. 94, 327–356 (2020a)

    Article  Google Scholar 

  6. Chen, Y., Whitt, W.: Extremal \(GI/GI/1\) queues given two moments: numerical results exploiting three-point distrdibutions. Columbia University, http://www.columbia.edu/~ww2040/allpapers.html(2020b)

  7. Chen, Y., Whitt.W.: Extremal models for the \(GI/GI/K\) waiting-time tail-probability decay rate. Oper. Res. Letters 48, 770–776 (2020c)

  8. Cohen, J.W.: The Single Server Queue, 2nd edn. North-Holland, Amsterdam (1982)

    Google Scholar 

  9. Daley, D.J.: Inequalities for moments of tails of random variables with queueing applications. Zeitschrift fur Wahrscheinlichkeitsetheorie Verw. Gebiete 41, 139–143 (1977)

  10. Daley, D.J., Kreinin, A.Ya., Trengove, C.D.: Inequalities concerning the waiting-time in single-server queues: a survey. In: Bhat, U.N., Basawa, I.V. (eds.) Queueing and Related Models. pp. 177–223. Clarendon Press, Oxford (1992)

  11. Denuit, M., Lefevre, C., Shaked, M.: The s-convex orders among real random variables with applications. Math. Inequal. Appl. 1(4), 585–613 (1998)

    Google Scholar 

  12. Eckberg, A.E.: Sharp bounds on Laplace-Stieltjes transforms, with applications to various queueing problems. Math. Op. Res. 2(2), 135–142 (1977)

    Article  Google Scholar 

  13. Gupta, V., Osogami, T.: On Markov-Krein characterization of the mean waiting time in \(M/G/K\) and other queueing systems. Queue. Syst. 68, 339–352 (2011)

    Article  Google Scholar 

  14. Holtzman, J.M.: The accuracy of the equivalent random method with renewal in outs. Bell Syst. Tech. Ical J. 52(9), 1673–1679 (1973)

    Article  Google Scholar 

  15. Johnson, M.A., Taaffe, M.R.: An investigation of phase-distribution moment-matching algorithms for use in queueing models. Queie. Syst. 8(1–2), 129–148 (1991)

    Article  Google Scholar 

  16. Johnson, M.A., Taaffe, M.R.: Tchebycheff systems for probability analysis. Am. J. Math. Manag. Sci. 13(1–2), 83–111 (1993)

    Google Scholar 

  17. Karlin, S., Studden, W.J.: Tchebycheff Systems; With Applications in Analysis and Statistics, vol. 137. Wiley, New York (1966)

    Google Scholar 

  18. Kingman, J.F.C.: The single server queue in heavy traffic. Proc. Camb. Phil. Soc. 77, 902–904 (1961)

    Article  Google Scholar 

  19. Kingman, J.F.C.: Inequalities for the queue \(GI/G/1\). Biometrika 49(3/4), 315–324 (1962)

    Article  Google Scholar 

  20. Muller, A., Stoyan, D.: Comparison Methods for Stochastic Models and Risks. John Wiley and Sons, New York (2002)

    Google Scholar 

  21. Neuts, M.F.: Matrix-Geometric Solutions of Stochastic Models. The Johns Hopkins University Press, Baltimore (1981)

    Google Scholar 

  22. Ott, T.J.: Simple inequalties for the \(D/G/1\) queue. Op. Res. 35(4), 589–597 (1987)

    Article  Google Scholar 

  23. Rogozin, B.A.: Some extremal problems in the theory of mass service. Theor. Probab. Appl. 11(1), 144–151 (1966)

    Article  Google Scholar 

  24. Rolski, T.: Some inequalities for \(GI/M/n\) queues. Zast. Math. 13(1), 43–47 (1972)

    Google Scholar 

  25. Rolski, T.: Order relations in the set of probability distribution functions and their applications in queueing theory. Diss. Math. Pol. Acad. Sci. 132(1), 3–47 (1976)

    Google Scholar 

  26. Stoyan, D.: Comparison Methods for Queues and Other Stochastic Models. John Wiley and Sons, New York (1983). Translated and edited from 1977 German Edition by D. J. Daley

  27. Szekli, R.: On the complete monotonicity of the waiting time density in some \(GI/G/k\) systems. Queue Syst. 1, 401–406 (1987)

    Article  Google Scholar 

  28. Szekli, R.: Stochastic Ordering and Dependence in Applied Probability. Springer, Berlin (1995)

    Book  Google Scholar 

  29. Whitt, W.: On approximations for queues, I: extremal distributions. AT&T Bell Lab. Tech. J. 63(1), 115–137 (1984a)

    Article  Google Scholar 

  30. Whitt, W.: On approximations for queues, III: mixtures of exponential distributions. AT&T Bell Lab. Tech. J. 63(1), 163–175 (1984b)

    Article  Google Scholar 

  31. Wolff, R.W., Wang, C.: Idle period approximations and bounds for the \(GI/G/1\) queue. Adv. Appl. Probab. 35(3), 773–792 (2003)

    Article  Google Scholar 

Download references

Acknowledgements

This research was supported by NSF CMMI 1634133. We thank Tomasz Rolski and an anonymous referee for exposing a gap in our first proof of Theorem 1.

Author information

Authors and Affiliations

  1. Industrial Engineering and Operations Research, Columbia University, New York, USA

    Yan Chen & Ward Whitt

Authors
  1. Yan Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ward Whitt
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ward Whitt.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, Y., Whitt, W. Extremal GI/GI/1 queues given two moments: exploiting Tchebycheff systems. Queueing Syst 97, 101–124 (2021). https://doi.org/10.1007/s11134-020-09675-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11134-020-09675-7

Keywords

Mathematics Subject Classification

Search

Navigation