Advertisement

Queueing Systems

, Volume 91, Issue 1–2, pp 15–47 | Cite as

A central-limit-theorem version of the periodic Little’s law

  • Ward Whitt
  • Xiaopei ZhangEmail author
Article
  • 23 Downloads

Abstract

We establish a central-limit-theorem (CLT) version of the periodic Little’s law (PLL) in discrete time, which complements the sample-path and stationary versions of the PLL we recently established, motivated by data analysis of a hospital emergency department. Our new CLT version of the PLL extends previous CLT versions of LL. As with the LL, the CLT version of the PLL is useful for statistical applications.

Keywords

Little’s law \(L = \lambda W\) Periodic queues Central limit theorem Emergency departments Weak convergence in \((\ell _1)^d\) 

Mathematics Subject Classification

60F05 60F25 60K25 90B22 

Notes

Acknowledgements

Support was received from NSF grants CMMI 1634133.

References

  1. 1.
    Araujo, A., Giné, E.: The Central Limit Theorem for Real and Banach Valued Random Variables. Wiley, Hoboken (1980)Google Scholar
  2. 2.
    Armony, M., Israelit, S., Mandelbaum, A., Marmor, Y., Tseytlin, Y., Yom-Tov, G.: On patient flow in hospitals: a data-based queueing-science perspective. Stochast. Syst. 5(1), 146–194 (2015)CrossRefGoogle Scholar
  3. 3.
    Bertsimas, D., Mourtzinou, G.: Transient laws of non-stationary queueing systems and their applications. Que. Syst. 25, 115–155 (1997)CrossRefGoogle Scholar
  4. 4.
    Billingsley, P.: Probability and Measure, 3rd edn. Wiley, New York (1995)Google Scholar
  5. 5.
    Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley, New York (1999)CrossRefGoogle Scholar
  6. 6.
    Bradley, R.C.: On the central limit question under absolute regularity. Ann. Probab. 13, 1314–1325 (1985)CrossRefGoogle Scholar
  7. 7.
    El-Taha, M., Stidham, S.: Sample-Path Analysis of Queueing Systems. Kluwer, Boston (1999)CrossRefGoogle Scholar
  8. 8.
    Fralix, B.H., Riano, G.: A new look at transient versions of Little’s law. J. Appl. Probab. 47, 459–473 (2010)CrossRefGoogle Scholar
  9. 9.
    Glynn, P.W., Whitt, W.: A central-limit-theorem version of \(L = \lambda W\). Que. Syst. 2, 191–215 (1986). (See Correction Note on \(L = \lambda W\), Queueing Systems, 12 (4), 1992, 431-432. The results are correct; minor but important change needed in proofs.)CrossRefGoogle Scholar
  10. 10.
    Glynn, P.W., Whitt, W.: Sufficient conditions for functional limit theorem versions of \(L = \lambda W\). Que. Syst. 1, 279–287 (1987)CrossRefGoogle Scholar
  11. 11.
    Glynn, P.W., Whitt, W.: Ordinary CLT and WLLN versions of \(L = \lambda W\). Math. Oper. Res. 13, 674–692 (1988)CrossRefGoogle Scholar
  12. 12.
    Glynn, P.W., Whitt, W.: Indirect estimation via \(L = \lambda W\). Oper. Res. 37(1), 82–103 (1989)CrossRefGoogle Scholar
  13. 13.
    Gut, A.: Stopped Random Walks: Limit Theorems and Applications. Springer Series in Operations Research and Financial Engineering, 2nd edn. Springer, Berlin (2009).  https://doi.org/10.1007/978-0-387-87835-5 CrossRefGoogle Scholar
  14. 14.
    Harford, T.: Is this the most influential work in the history of capitalism? (2017). 50 Things that Made the Modern Economy, BBC World Service, 23 October 2017Google Scholar
  15. 15.
    Ibragimov, I.A., Linnik, Y.V.: Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen (1971)Google Scholar
  16. 16.
    Kim, S., Whitt, W.: Statistical analysis with Little’s law. Oper. Res. 61(4), 1030–1045 (2013)CrossRefGoogle Scholar
  17. 17.
    Lauwers, L., Willekens, M.: Five hundred years of bookkeeping: a portrait of Luca Pacioli. Tijdschr. voor Econ. Manag. 39(3), 289–304 (1994)Google Scholar
  18. 18.
    Ledoux, M., Talagrand, M.: Probability in Banach Spaces: Isoperimetry and Processes. Springer, Berlin (1991)CrossRefGoogle Scholar
  19. 19.
    Little, J.D.C.: A proof of the queueing formula: \(L = \lambda W\). Oper. Res. 9, 383–387 (1961)CrossRefGoogle Scholar
  20. 20.
    Little, J.D.C.: Little’s law as viewed on its 50th anniversary. Oper. Res. 59, 536–539 (2011)CrossRefGoogle Scholar
  21. 21.
    Little, J.D.C., Graves, S.C.: Little’s law. In: Chhajed, D., Lowe, T.J. (eds.) Building Intuition: Insights from Basic Operations Management Models and Principles, chap. 5, pp. 81–100. Springer, New York (2008)CrossRefGoogle Scholar
  22. 22.
    Megginson, R.E.: An Introduction to Banach Space Theory, vol. 183. Springer, Berlin (2012)Google Scholar
  23. 23.
    Munkres, J.R.: Topology, 2nd edn. Prentice Hall, Upper Saddle River (2000)Google Scholar
  24. 24.
    Pacioli, L.: Summa de Arithmetica, Geometria, Proportioni et Proportionalita. Venice, (1494)Google Scholar
  25. 25.
    Stidham, S.: A last word on \(L = \lambda W\). Oper. Res. 22, 417–421 (1974)CrossRefGoogle Scholar
  26. 26.
    Whitt, W.: A review of \(L = \lambda W\). Que. Syst. 9, 235–268 (1991)CrossRefGoogle Scholar
  27. 27.
    Whitt, W.: Correction Note on \(L = \lambda W\). Que. Syst. 12, 431–432 (1992). (The results in the previous papers are correct, but minor important changes are needed in some proofs.)CrossRefGoogle Scholar
  28. 28.
    Whitt, W.: Stochastic-Process Limits. Springer, New York (2002)Google Scholar
  29. 29.
    Whitt, W.: Extending the FCLT version of \(L = \lambda W\). Oper. Res. Lett. 40, 230–234 (2012)CrossRefGoogle Scholar
  30. 30.
    Whitt, W., Zhang, X.: A data-driven model of an emergency department. Oper. Res. Health Care 12(1), 1–15 (2017)Google Scholar
  31. 31.
    Whitt, W., Zhang, X.: Periodic Little’s Law (2017). Submitted to Operations Research, available at Columbia University, http://www.columbia.edu/~ww2040/allpapers.html

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Industrial Engineering and Operations Research DepartmentColumbia UniversityNew YorkUSA

Personalised recommendations