Daniel Vacanti

# One Law. Two Equations.

Updated: May 27

This is post 4 of 9 in our Little's Law series.

In the __previous post__, we demonstrated how the two different forms of Little's Law (LL) can lead to two very different answers even when using the same dataset. How can one law lead to two answers?

As was suggested, the applicability of any theory depends completely on one's understanding of the assumptions that need to be in place in order for that given theory to be valid. However, in the case of LL, we have two different equations that purport to express one single theory. Does having two equations require having two sets of assumptions (and potentially two types of applicability)? In a word, yes.

Recall that the L = λW (this is the version based on **arrival **rate) came first, and in his 1961 proof, Little stated his assumptions for the formula to be correct: "if the three means are finite and the corresponding stochastic process strictly stationary, and, if the arrival process is metrically transitive with nonzero mean, then L = λW." There's a lot of mathematical gibberish in there that you don't need to know anyway because it turns out Little's initial assumptions were overly restrictive, as was demonstrated by subsequent authors (__reference #1__). All you really need to know is that--very generally speaking--LL is applicable to any process that is relatively stable over time [__see note below__].

For our earlier thought experiment, I took this notion of stability to an extreme in order to (hopefully) prove a point. In the example data I provided, you'll see that arrivals are infinitely stable in that they never change. In this ultra-stable world, you'll note that the arrivals form of LL works--quite literally--exactly the way that it should. That is to say, when you plug two numbers into the equation, you get the exact answer for the third.

Things change dramatically, however, when we start talking about the WIP = TH * CT version of the law. Most people assume--quite erroneously--that this latter form of LL only requires the same assumptions as the arrivals version. However, Dr. Little is very clear that changing the perspective of the equation from arrivals to departures has a very specific impact on the assumptions that are required for the law to be valid. Let's use Little's own words for this discussion: "At a minimum, we must have conservation of flow. Thus, the average output or departure rate (TH) equals the average input or arrival rate (λ). Furthermore, we need to assume that all jobs that enter the shop will eventually be completed and will exit the shop; there are no jobs that get lost or never depart from the shop...we need the size of the WIP to be roughly the same at the beginning and end of the time interval so that there is neither significant growth nor decline in the size of the WIP, [and] we need some assurance that the average age or latency of the WIP is neither growing nor declining." (__reference #2__)

"At a minimum, we must have conservation of flow."

Allow me to put these in a bulleted list that will be easier for your reference later. For a system being observed for an arbitrarily long amount of time:

Average arrival rate equals average departure rate

All items that enter a workflow must exit

WIP should neither be increasing nor decreasing

Average age of WIP is neither increasing nor decreasing

Consistent units must be used for all measures

I added that last bullet point for clarity. It should make sense that if Cycle Time is measured in days, then Throughput cannot be measured in weeks. And don't even talk to me about story points. If you have a system that obeys all of these assumptions, then you have a system in which the TH form of Little's Law will apply.

If you have a system that obeys all of these assumptions, then you have a system in which the TH form of Little's Law will apply.

Wait, what's that you say? Your system doesn't follow these assumptions? I'm glad you pointed that out because that will be the topic of our next post.

#### A note on stability

Most people have an incorrect notion of what stability means. "Stable" does not necessarily mean "not changing." For example, Little explicitly states aspects of a system that L = λW is **NOT** dependent on and, therefore, may reasonably change over time: size of items, order of items worked on, number of servers, etc. That means situations like adding or removing team members over time may not be enough to consider to a process "unstable." However, to take an extreme example, it would be easy to see that all of the restrictions/changes imposed during the 2020 COVID pandemic would cause a system to be unstable. From a LL perspective, only when all 5 assumptions are met can a system reasonably be considered stable (assuming we are talking about the TH form of LL).

#### References

Whitt, W. 1991. A review of L = λW and extensions. Queueing Systems 9(3) 235–268.

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.

**Explore all entries in this series**

__One Law. Two Equations__(this article)__The Most Important Metric of Little's Law Isn't In the Equation__

**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. *