posted on any other website, including the author’s site. Please send any questions regarding this policy to permissions@informs.org. Statistical Analysis with Little’s Law

The theory supporting Little's Law (L = λW) is now well developed, applying to both limits of averages and expected values of stationary distributions, but applications of Little's Law with actual system data involve measurements over a finite-time interval, which are neither of these. We advocate taking a statistical approach with such measurements. We investigate how estimates of L and λ can be used to estimate W when the waiting times are not observed. We advocate estimating confidence intervals. Given a single sample-path segment, we suggest estimating confidence intervals using the method of batch means, as is often done in stochastic simulation output analysis. We show how to estimate and remove bias due to interval edge effects when the system does not begin and end empty. We illustrate the methods with data from a call center and simulation experiments.

[1]  Ward Whitt,et al.  Server Staffing to Meet Time-Varying Demand , 1996 .

[2]  W. A. Massey,et al.  M t /G/∞ queues with sinusoidal arrival rates , 1993 .

[3]  R. Willink,et al.  A Confidence Interval and Test for the Mean of an Asymmetric Distribution , 2005 .

[4]  Ward Whitt,et al.  The Physics of the Mt/G/∞ Queue , 1993, Oper. Res..

[5]  R. Srikant,et al.  Simulation run lengths to estimate blocking probabilities , 1996, TOMC.

[6]  Shang Zhi,et al.  A proof of the queueing formula: L=λW , 2001 .

[7]  Benjamin Weiss,et al.  A SIMPLE PROOF OF , 1982 .

[8]  Ward Whitt,et al.  Coping with Time‐Varying Demand When Setting Staffing Requirements for a Service System , 2007 .

[9]  Søren Asmussen,et al.  Applied probability and queues, Second Edition , 2003, Applications of mathematics.

[10]  John D. C. Little,et al.  OR FORUM - Little's Law as Viewed on Its 50th Anniversary , 2011, Oper. Res..

[11]  P. Glynn,et al.  Central-limit-theorem version of L = λW , 1987 .

[12]  Richard C. Larson.  The queue inference engine: deducing queue statistics from transactional data , 1990 .

[13]  N. J. Johnson,et al.  Modified t Tests and Confidence Intervals for Asymmetrical Populations , 1978 .

[14]  Karl Sigman.  Stationary marked point processes , 1995 .

[15]  R. Serfozo.  Introduction to Stochastic Networks , 1999 .

[16]  Richard A. Davis,et al.  Time Series: Theory and Methods , 2013 .

[17]  Richard A. Davis,et al.  Time Series: Theory and Methods (2nd ed.). , 1992 .

[18]  James R. Wilson,et al.  Efficient Computation of Overlapping Variance Estimators for Simulation , 2007, INFORMS J. Comput..

[19]  R. R. P. Jackson,et al.  Introduction to the Theory of Queues , 1963 .

[20]  K. Sigman.  Stationary marked point processes : an intuitive approach , 1995 .

[21]  Martin B. Hansen,et al.  Nonparametric inference from the M/G/1 workload , 2006 .

[22]  Ward Whitt,et al.  A review ofL=λW and extensions , 1991, Queueing Syst. Theory Appl..

[23]  Stephen C. Graves,et al.  Little's Law , 2008 .

[24]  Ward Whitt,et al.  Indirect Estimation Via L = λW , 1989, Oper. Res..

[25]  J. G. Matthews,et al.  To Batch or Not to Batch? , 2007 .

[26]  W. Lovejoy,et al.  Little's law flow analysis of observation unit impact and sizing. , 2011, Academic emergency medicine : official journal of the Society for Academic Emergency Medicine.

[27]  W. Whitt,et al.  The correlation functions of rbm and m/m/1 , 1988 .

[28]  Ward Whitt,et al.  The last departure time from an Mt/G/∞ queue with a terminating arrival process , 2008, Queueing Syst. Theory Appl..

[29]  Ward Whitt,et al.  ESTIMATING WAITING TIMES WITH THE TIME-VARYING LITTLE'S LAW , 2012 .

[30]  James R. Wilson,et al.  Skart: A skewness- and autoregression-adjusted batch-means procedure for simulation analysis , 2009 .

[31]  Ward Whitt,et al.  Sufficient conditions for functional-limit-theorem versions ofL = λW , 1987, Queueing Syst. Theory Appl..

[32]  David Simchi-Levi,et al.  Introduction to "Little's Law as Viewed on Its 50th Anniversary" , 2011, Oper. Res..

[33]  F. Baccelli,et al.  Elements of Queueing Theory: Palm Martingale Calculus and Stochastic Recurrences , 2010 .

[34]  Ward Whitt,et al.  A central-limit-theorem version ofL=λw , 1986, Queueing Syst. Theory Appl..

[35]  Paul Bratley,et al.  A guide to simulation , 1983 .

[36]  Richard W. Conway,et al.  Some Tactical Problems in Digital Simulation , 1963 .

[37]  Peter W. Glynn,et al.  Stochastic Simulation: Algorithms and Analysis , 2007 .

[38]  Timothy J. Lowe,et al.  Building intuition insights from basic operations management models and principles , 2008 .

[39]  Avishai Mandelbaum,et al.  Statistical Analysis of a Telephone Call Center , 2005 .

[40]  Peter J. Denning,et al.  The Operational Analysis of Queueing Network Models , 1978, CSUR.

[41]  Ward Whitt,et al.  Extending the FCLT version of L=λW , 2012, Oper. Res. Lett..

[42]  William S. Jewell,et al.  A Simple Proof of: L = λW , 1967, Oper. Res..

[43]  J. Little.  A Proof for the Queuing Formula: L = λW , 1961 .

[44]  Alexandre Boucher,et al.  Applied Geostatistics with SGeMS: Stochastic simulation algorithms , 2009 .

[45]  Shaler Stidham,et al.  Technical Note - A Last Word on L = λW , 1974, Oper. Res..

[46]  Upendra Dave,et al.  Applied Probability and Queues , 1987 .

[47]  David Goldsman,et al.  To batch or not to batch? , 2004, TOMC.

[48]  W. Whitt.  Planning queueing simulations , 1989 .

[49]  Jeffrey P. Buzen,et al.  Fundamental operational laws of computer system performance , 1976, Acta Informatica.

[50]  Emily K. Lada,et al.  Skart: A skewness- and autoregression-adjusted batch-means procedure for simulation analysis , 2008, 2008 Winter Simulation Conference.

[51]  Avishai Mandelbaum,et al.  Workload forecasting for a call center: Methodology and a case study , 2009, 1009.5741.

[52]  S. Stidham,et al.  Sample-Path Analysis of Queueing Systems , 1998 .