This post is part 2 in our Little's Law series.
You might think that the history of the relationship
L = λ * W
(Eq. 1)
would start with the publication of Dr. Little's seminal paper in 1961 [reference #1]. The reality is that we must begin by going back a bit further. What the symbols in the above figure (Eq. 1) mean will be discussed a little later.
Evidence points to queuing theorists applying (Eq. 1) in their work well before 1961--seemingly without ever providing a rigorous mathematical proof as to its validity. The earliest pre-1961 example that I could find (in a semi-exhaustive search) was a paper written in 1953 called "Priority Assignment in Waiting Line Problems" by Alan Cobham [reference #2]. Somewhat coincidentally (for those who know me), this paper applies (Eq. 1) to prove the dangers of prioritization schemes to the overall predictability of queuing systems. (As an interesting aside, a quote from that paper is, "any increase in the relative frequency of priority 1 units increases not only the expected delay for units of that priority level but for units of all other levels as well."--in other words, we knew about the dangers of classes of service at least as early as the 1950s!) It would seem that (Eq. 1) was not only acknowledged well in advance of 1953, but it was also widely accepted as true even then.
We knew about the dangers of classes of service as early as the 1950s!
For the purposes of our story, however, the most important person before 1961 to recognize the need for a more rigorous proof of (Eq. 1) was Philip M. Morse. In 1958, Morse had published an Operations Research (OR) textbook called "Queues, Inventories, and Maintenance." [reference #3] In that book, Morse provided heuristic proofs that (Eq. 1) holds for very specific queuing models but commented that it would be useful to have the relationship proved for the general case (i.e., for all queues, not just for specific, individual models). In Morse's words, "we have now shown that...the relation between the mean number [L] and mean delay [W] is via the factor λ, the arrival rate: L = λW, and we will find, in all the examples encountered in this chapter and the next, for a wide variety of service and arrival distributions, for one or for several channels, that this same relationship holds. Those readers who would like to experience for themselves the slipperiness of fundamental concepts in this field and the intractability of really general theorems might try their hand at showing under what circumstances this simple relationship between L and W does not hold."
Somewhat serendipitously, circa 1960, Dr. John Little was teaching an OR course at Case Institute of Technology in Cleveland (now Case Western Reserve University) and was using Morse's textbook as part of the curriculum. During one class, Little had introduced (Eq. 1) and commented (as Morse had) that it seemed to be a very general relationship. According to Little himself, "After class, I was talking to a number of students, and one of them (Sid Hess) asked, 'How hard would it be to prove it in general?' On the spur of the moment, I obligingly said, 'I guess it shouldn't be too hard.' Famous last words. Sid replied, 'Then you should do it!'" [reference #4]
Little took up the challenge, went away for the summer in 1961 to come up with a general proof for (Eq. 1), wrote up his findings in a paper, submitted the proof to the periodical Operations Review, and had his submission accepted on the first round. His paper has since become one of the most frequently referenced articles in Operations Review's history. [reference #5] As such, the relationship
L = λ * W
quickly became more commonly known as Little's Law (LL).
The real beauty of Little's general proof--apart from not relying on any specific queuing model--was all of the other things you didn't need to know in order to apply the law. For instance, you didn't need to have any detailed knowledge about inter-arrival times, service times, number of servers, order of service, etc., that you normally needed for queuing theory. [This point will become of monumental importance when we talk about applying LL to Agile.]
In the years after its first publication, LL found applications far beyond OR. One such application was in the area of Operations Management (OM). OM is a bit different than OR because OM is generally more focused on output rather than input. Consider the perspective of an operations manager in a factory. A factory manager's primary focus is output because the whole reason a factory exists is to produce "things" (factories don't exist to take in "things"). Because of this potentially differing perspective, in the OM world, LL is usually stated in terms of throughput (TH or departures) rather than arrivals; work in progress (WIP) rather than queue length; and cycle time (CT) rather than wait time [reference #6]:
WIP = TH * CT
(Eq. 2)
It's fairly easy to see that (Eq. 1) and (Eq. 2) are equivalent; however, the change in focus from arrivals to departures will require a nontrivial amount of care that we will get into in a later post. The reason I mention (Eq. 2) is because this is the form of LL that the Agile community seems to have preferred, and so it is here that our brief history ends and the real story begins.
So why should you be concerned about any of this? There are a couple of reasons, really. First, practitioners should acknowledge that any doubts about the legitimacy of the theory have been settled for 70 years or more. There is simply no question about the validity of LL or its place in the management of flow. Second, because most agile practitioners have only seen LL in the form of (Eq. 2) and not (Eq. 1), it is important for them to understand where (Eq. 2) really comes from. It's not just a matter of simply substituting variable names, and Robert is your father's brother.
There is simply no question about the validity of Little's Law or its place in the management of flow.
This brings us to the fact that we actually have two forms of Little's Law to consider:
L = λ * W
and
WIP = TH * CT
But which one do we use and when? I'm glad you asked because that will be the topic of the next post in this series...
Explore all entries in this series
About Daniel Vacanti, Guest Writer
Daniel Vacanti is the author of the highly-praised books "When will it be done?" and "Actionable Agile Metrics for Predictability" and the original mind behind the ActionableAgile™️ Analytics Tool. Recently, he co-founded ProKanban.org, an inclusive community where everyone can learn about Professional Kanban, and he co-authored their Kanban Guide.
When he is not playing tennis in the Florida sunshine or whisky tasting in Scotland, Daniel can be found speaking on the international conference circuit, teaching classes, and creating amazing content for people like us.
References
Little, J. D. C. A proof for the queuing formula: L = λ W. Operations Research. 9(3) 383–387, 1961.
Alan Cobham, Journal of the Operations Research Society of America, Vol. 2, No. 1 (Feb. 1954), pp. 70-76
Morse, P. M. (1958) Queues, Inventories and Maintenance, Publications in Operations Research, No.1, John Wiley, New York.
Little, J. D. C., S. C. Graves. 2008. Little's Law. D. Chhajed, T. J. Lowe, eds. Building Intuition: Insights from Basic Operations Management Models and Principles. Springer Science + Business Media LLC, New York.
Whitt, W. 1991. A review of L = λW and extensions. Queueing Systems 9(3) 235–268.
Hopp, W. J., M. L. Spearman. 2000. Factory Physics: Foundations of Manufacturing Management, 2nd ed. Irwin/McGraw-Hill, New York.