Little’s Law as Viewed on its 50th Anniversary

In the May-June, 2011 issue of Operations Research, the journal revisits one of its most influential publications: “A Proof for the Queuing Formula: L = λW” by John Little. The formula, now known widely as Little’s Law, has been critical in many following results and applications. As David Simchi-Levi (the Editor) and Michael Trick (the Area Editor for the OR Forum) write in their introduction:

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This paper has become one of the five most cited papers ever published in the journal. And its influence goes far beyond the professional literature.

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The importance of the paper is due to its simplicity:

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Little proves that under very general conditions, the average length of a queue, in steady state, will be equal to the arrival rate into the queue times the average wait in the queue. Remarkably, this relationship is not influenced by the arrival process distribution, the service distribution, the service order, or practically anything else. Nor does it depend on the structure of the queueing system: “Little’s Law” holds not just at the individual queue level but also at the system level.

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In the OR Forum paper pdf “Little’s Law as Viewed on its 50th Anniversary” , John Little explores the effect Little’s Law has had over the past half-century. Much of the emphasis is on how Little’s Law is used in practice. As Little points out in his article:

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Practice is a different story. By practice we mean problem solving in the “real world,” aimed at improving an operation or system for an organization, preferably with measured results. That is what operations research originally tried to do and, of course, many operations researchers continue to do. There is no ready source of written material of this sort for a specialized topic like Little’s Law. Nevertheless, the author has collected a few interesting cases to report.

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Little also discusses the effect Little’s Law has had on theory, and talks about the finite time period case (a case of strong practical importance).

Little’s Law has been extremely influential over the past 50 years, and we are sure its impact will continue to be felt.

As part of the OR Forum, the editors of Operations Research have invited four eminent scholars to provide a commentary on Little’s paper, and Little’s Law in general.

Ed Kaplan is the William N. and Marie A. Beach Professor of Management Sciences, Professor of Public Health, and Professor of Engineering at Yale University. In pdf his commentary , Kaplan discusses a number of other application areas for Little’s Law, including epidemiology, physical systems, and counter-terrorism. In his closing, he extolls the simplicity of Little’s Law:

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…one of the most exciting things about operations research is that with a little (!) effort, one can find operations everywhere, enabling the application of our modeling mindset to create relevant operations models for whatever problem is of interest. Little’s Law is a key piece of the “sophisticated common sense” that comprises our operations research toolkit, and so it shall remain.

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John Little provided pdf a response to Ed’s comments , thanking him for his insights.

Timothy Lowe is the Chester Phillips Professor of Operations Management at the Tippie College of Business at the University of Iowa. In pdf his commentary , Lowe talks about how he uses Little’s Law in class to develop insight in students.

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Since the University of Iowa has a large teaching hospital, we tend to get two or three physicians in each EMBA [Executive MBA] class. I have found that the MDs are particularly interested in learning how to apply the concepts we discuss to their work environments. Many of these students have been forced into becoming “business managers” in their work environment in the hospital, but of course have not been formally trained to take on that task. One of the key metrics that they need to keep track of (and reduce as much as possible) is patient flow time. Clearly Little’s Law can be of assistance to them as they grapple with the flow time problem.

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Sridhar Tayur is Ford Distinguished Research Chair at the Tepper School of Business, Carnegie Mellon. In pdf his commentary , Tayur points out the value of Little’s Law in teaching basic concepts, while suggesting that real operational problems need to go beyond Little’s Law for their solution.

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From a viewpoint of teaching some basic concepts to undergraduates, MBA students, and executives, I have found Littles Law to be useful in conjunction with something else. For example, in a situation where an arrival may choose not to enter a line if he has to wait, then by combining Pollaczek-Khinchin formula for M/G/1 queues (or with M/M/1 queue with a base stock policy for inventories with level B), we can study the interaction between customer patience, inventory investment, service rate, service variability and profitability. In terms of actual practice, either in manufacturing or service, this relationship apart from perhaps verifying that the data being used are “consistent” has not been very useful to me. That is likely because I have found that queuing models are rarely useful to actually solve operational problems, but instead are helpful in guiding the selection of their design parameters.

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Ronald Wolff is Professor Emeritus at the University of California, Berkeley. John Little was Wolff’s dissertation advisor, and pdf his commentary provides some historical perspective:

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In those days, methods of analysis were tailored to individual queuing models. There were no theorems of any consequence that held across the board from model to model. Nevertheless, as particular models were analyzed, Morse [the textbook used] would go out of his way to point out that LL held for them. Anyone reading Morse was likely to believe that LL was true in general without understanding why. In this sense, LL was a folk theorem back then.

Or maybe not. On p. 75, Morse points this out again, but also warns of the intractability of “really general theorems” in this area and suggests instead that readers try to find a counterexample. The alternative to this of course is to come up with a really general proof.

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And that is what Little’s Law is all about.

As part of this discussion, John Little and Ron Wolff have provided a further discussion of issues related to Little’s Law entitled “The ‘Flaw’ in Little (1961), its identification, and its fixes”. In this commentary, Little and Wolff discuss the history and resolution of issues in Little’s original proof of L = λW.

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Little (1961) has a well known “flaw,” first pointed out by Brumelle (1971). Our Commentary provides a brief history of the flaw and its various fixes. The latter include Brumelle (1971), Franken (1976), Franken et al., (1982), Whitt (1991), Stidham (2002), and Wolff (2011).

Brumelle (1971) set out to generalize the formula L=λW , which Little (1961) had apparently proved under quite general stationary conditions. In the course of his work, Brumelle found that not all the conditions required by Little could be made simultaneously.

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You can read the full commentary pdf here .

How has Little’s Law affected your life? You are welcome to add your own thoughts and experiences in the comments.

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