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.
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Araujo, A., Giné, E.: The Central Limit Theorem for Real and Banach Valued Random Variables. Wiley, Hoboken (1980)
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)
Bertsimas, D., Mourtzinou, G.: Transient laws of non-stationary queueing systems and their applications. Que. Syst. 25, 115–155 (1997)
Billingsley, P.: Probability and Measure, 3rd edn. Wiley, New York (1995)
Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley, New York (1999)
Bradley, R.C.: On the central limit question under absolute regularity. Ann. Probab. 13, 1314–1325 (1985)
El-Taha, M., Stidham, S.: Sample-Path Analysis of Queueing Systems. Kluwer, Boston (1999)
Fralix, B.H., Riano, G.: A new look at transient versions of Little’s law. J. Appl. Probab. 47, 459–473 (2010)
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.)
Glynn, P.W., Whitt, W.: Sufficient conditions for functional limit theorem versions of \(L = \lambda W\). Que. Syst. 1, 279–287 (1987)
Glynn, P.W., Whitt, W.: Ordinary CLT and WLLN versions of \(L = \lambda W\). Math. Oper. Res. 13, 674–692 (1988)
Glynn, P.W., Whitt, W.: Indirect estimation via \(L = \lambda W\). Oper. Res. 37(1), 82–103 (1989)
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
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 2017
Ibragimov, I.A., Linnik, Y.V.: Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen (1971)
Kim, S., Whitt, W.: Statistical analysis with Little’s law. Oper. Res. 61(4), 1030–1045 (2013)
Lauwers, L., Willekens, M.: Five hundred years of bookkeeping: a portrait of Luca Pacioli. Tijdschr. voor Econ. Manag. 39(3), 289–304 (1994)
Ledoux, M., Talagrand, M.: Probability in Banach Spaces: Isoperimetry and Processes. Springer, Berlin (1991)
Little, J.D.C.: A proof of the queueing formula: \(L = \lambda W\). Oper. Res. 9, 383–387 (1961)
Little, J.D.C.: Little’s law as viewed on its 50th anniversary. Oper. Res. 59, 536–539 (2011)
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)
Megginson, R.E.: An Introduction to Banach Space Theory, vol. 183. Springer, Berlin (2012)
Munkres, J.R.: Topology, 2nd edn. Prentice Hall, Upper Saddle River (2000)
Pacioli, L.: Summa de Arithmetica, Geometria, Proportioni et Proportionalita. Venice, (1494)
Stidham, S.: A last word on \(L = \lambda W\). Oper. Res. 22, 417–421 (1974)
Whitt, W.: A review of \(L = \lambda W\). Que. Syst. 9, 235–268 (1991)
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.)
Whitt, W.: Stochastic-Process Limits. Springer, New York (2002)
Whitt, W.: Extending the FCLT version of \(L = \lambda W\). Oper. Res. Lett. 40, 230–234 (2012)
Whitt, W., Zhang, X.: A data-driven model of an emergency department. Oper. Res. Health Care 12(1), 1–15 (2017)
Whitt, W., Zhang, X.: Periodic Little’s Law (2017). Submitted to Operations Research, available at Columbia University, http://www.columbia.edu/~ww2040/allpapers.html
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Support was received from NSF grants CMMI 1634133.
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Whitt, W., Zhang, X. A central-limit-theorem version of the periodic Little’s law. Queueing Syst 91, 15–47 (2019). https://doi.org/10.1007/s11134-018-9588-8
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DOI: https://doi.org/10.1007/s11134-018-9588-8
Keywords
- Little’s law
- \(L = \lambda W\)
- Periodic queues
- Central limit theorem
- Emergency departments
- Weak convergence in \((\ell _1)^d\)