Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-22T11:06:43.488Z Has data issue: false hasContentIssue false
  • English
  • Français

ESTIMATING WAITING TIMES WITH THE TIME-VARYING LITTLE'S LAW

Published online by Cambridge University Press:  13 August 2013

Song-Hee Kim  and
Ward Whitt
Song-Hee Kim
Affiliation:
Industrial Engineering and Operations Research, Columbia University, New York, NY 10027 E-mails: sk3116@columbia.edu; ww2040@columbia.edu
Ward Whitt
Affiliation:
Industrial Engineering and Operations Research, Columbia University, New York, NY 10027 E-mails: sk3116@columbia.edu; ww2040@columbia.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

When waiting times cannot be observed directly, Little's law can be applied to estimate the average waiting time by the average number in system divided by the average arrival rate, but that simple indirect estimator tends to be biased significantly when the arrival rates are time-varying and the service times are relatively long. Here it is shown that the bias in that indirect estimator can be estimated and reduced by applying the time-varying Little's law (TVLL). If there is appropriate time-varying staffing, then the waiting time distribution may not be time-varying even though the arrival rate is time varying. Given a fixed waiting time distribution with unknown mean, there is a unique mean consistent with the TVLL for each time t. Thus, under that condition, the TVLL provides an estimator for the unknown mean wait, given estimates of the average number in system over a subinterval and the arrival rate function. Useful variants of the TVLL estimator are obtained by fitting a linear or quadratic function to arrival data. When the arrival rate function is approximately linear (quadratic), the mean waiting time satisfies a quadratic (cubic) equation. The new estimator based on the TVLL is a positive real root of that equation. The new methods are shown to be effective in estimating the bias in the indirect estimator and reducing it, using simulations of multi-server queues and data from a call center.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 27 , Issue 4 , October 2013 , pp. 471 - 506
Copyright
Copyright © Cambridge University Press 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Bertsimas, D. & Mourtzinou, G. (1997). Transient laws of nonstationary queueing systems and their applications. Queueing Systems 25: 315–359.CrossRefGoogle Scholar
2.Bharucha-Reid, A.T. & Sambandham, M. (1986). Random polynomials. New York: Academic Press.Google Scholar
3.Brown, L., Gans, N., Mandelbaum, A., Sakov, A., Shen, H., Zeltyn, S., & Zhao, L. (2005). Statistical analysis of a telephone call center: a queueing-science perspective. Journal of the American Statistical Association 100: 36–50.CrossRefGoogle Scholar
4.Eick, S.G., Massey, W.A., & Whitt, W. (1993). M t/G/∞ queues with sinusoidal arrival rates. Management Science 39: 241–252.CrossRefGoogle Scholar
5.Eick, S.G., Massey, W.A., & Whitt, W. (1993). The physics of the M t/G/∞ queue. Operations Research 41: 731–742.CrossRefGoogle Scholar
6.El-Taha, M. & Stidham, S Jr. (1999). Sample-path analysis of queueing systems. Boston: Kluwer.CrossRefGoogle Scholar
7.Feldman, Z., Mandelbaum, A., Massey, W.A., & Whitt, W. (2008). Staffing of time-varying queues to achieve time-stable performance. Management Science 54(2): 324–338.CrossRefGoogle Scholar
8.Fralix, B.H. & Riano, G. (2010). A new look at transient versions of Little's Law. Journal of Applied Probability 47: 459–473.CrossRefGoogle Scholar
9.Glynn, P.W. & Whitt, W. (1989). Indirect estimation via L = λW. Operations Research 37: 82–103.CrossRefGoogle Scholar
10.Hamblen, J.W. (1956). Distribution of roots of quadratic equations with random coefficients. The Annals of Mathematical Statistics 27: 1136–1143.CrossRefGoogle Scholar
11.Jennings, O.B., Mandelbaum, A., Massey, W.A., & Whitt, W. (1996). Server staffing to meet time-varying demand. Management Science 42: 1383–1394.CrossRefGoogle Scholar
12.Kim, S.-H. & Whitt, W. (2013). Statistical analysis with Little's Law. Operations Research, forthcoming.CrossRefGoogle Scholar
13.Kim, S.-H. & Whitt, W. (2012). Statistical analysis with Little's Law, supplementary material: Technical report. Columbia University, http://www.columbia.edu/~ww2040/allpapers.html.Google Scholar
14.Larson, R.C. (1990). The queue inference engine: Deducing queue statistics from transactional data. Management Science 36: 586–601.CrossRefGoogle Scholar
15.Little, J.D.C. (1961). A proof of the queueing formula: L = λW. Operations Research 9: 383–387.CrossRefGoogle Scholar
16.Little, J.D.C. (2011). Little's Law as viewed on its 50th anniversary. Operations Research 59: 536–539.CrossRefGoogle Scholar
17.Little, J.D.C. & Graves, S.C. (2008). Building intuition: insights from basic operations management models and principles, chapter 5, Little's Law. New York: Springer, pp. 81–100.CrossRefGoogle Scholar
18.Lovejoy, W.S. & Desmond, J.S. (2011). Little's Law flow analysis of observation unit impact and sizing. Academic Emergency Medicine 18: 183–189.CrossRefGoogle ScholarPubMed
19.Mandelbaum, A. (2010). Lecture notes on Little's Law, course on service engineering. The Technion, Israel. http://iew3.technion.ac.il/serveng/Lectures/lectures.html.Google Scholar
20.Mandelbaum, A. (2012). Service Engineering of Stochastic Networks web page: http://iew3.technion.ac.il/serveng/.Google Scholar
21.Massey, W.A., Parker, G.A., & Whitt, W. (1996). Estimating the parameters of a nonhomogeneous Poisson process with linear rate. Telecommunication Systems 5: 361–388.CrossRefGoogle Scholar
22.Massey, W.A. & Whitt, W. (1993). Networks of infinite-server queues with nonstationary Poisson input. Queueing Systems 13(1): 183–250.CrossRefGoogle Scholar
23.Mazumdar, R., Kannurpatti, R., & Rosenberg, C. (1991). On rate conservation for non-stationary processes. Journal of Applied Probability 28: 762–770.CrossRefGoogle Scholar
24.Stidham, S. Jr. (1974). A last word on L = λW. Operations Research 22: 417–421.CrossRefGoogle Scholar
25.Whitt, W. (1982). Approximating a point process by a renewal process: two basic methods. Operations Research 30: 125–147.CrossRefGoogle Scholar
26.Whitt, W. (1991). A review of L = λW. Queueing Systems 9: 235–268.CrossRefGoogle Scholar
27.Whitt, W. (2005). Engineering solution of a basic call-center model. Management Science 51: 221–235.CrossRefGoogle Scholar
28.Wolfe, R.W. (1989). Stochastic modeling and the theory of queues. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
29.Wolfe, R.W. (2011). Wiley encyclopedia of operations research and management science: Little's law and related results. New York: Wiley.Google Scholar