Little’s Law

The Presence Calculus Project

What is Little’s Law?

Little’s Law follows from a simple conservation principle:

The total time accumulated in the system by a set of items over a window, when averaged per item, is proportional to the average number of items present over that window per unit of time.
The proportionality factor is the rate at which items enter during the window.

The classical presentation expresses this as a steady state law:

\[L = λW\]

In general, we imagine discrete items flowing through an input–output system over a chosen observation window:

• λ is the arrival rate over that window,
• W is the average time each item is present,
• L is the average number of items present per time unit.

The common steady-state version assumes these quantities are stable, long run values. But the conservation principle itself holds on any finite window.

This gap between a finite-window identity and the long-run formula is where the much richer structure of the dynamics of flow processes appears—one we develop in The Presence Calculus.

Little’s Law and The Presence Calculus

The techniques used to prove Little’s Law and the concept of sample path analysis of flow processes form the mathematical and computational foundation of The Presence Calculus.

Our technical paper A Deep Dive into Little’s Law is a detailed examination of the concepts here. It is the basis of the samplepath toolkit — the computational foundation of The Presence Calculus Project.

We have a series of more informal articles at The Polaris Flow Dispatch that might be an easier entry point if you dont want to jump into the deep end right away.

The Little’s Law Series

• The Many Faces of Little’s Law:

  An overview of what modern stochastic process theory says about Little’s Law. Little’s Law is not a law about queueing processes, it’s a very general law that lets us reason about the economic impacts of operational decisions. See why.

• A Brief History of Little’s Law:

  The version of Little’s Law that everyone knows is the one that Dr. Little proved in 1961. But operationally the sample path proof from Dr. Stidham is much more valuable.

  In particular, his proof showed that Little’s Law required no probabilistic assumptions and could be applied to non-stationary stochastic processes. This the key result we need to apply it to complex adaptive systems.

  This post talks about the history and evolution of the proofs of Little’s Law.

• The Causal Arrow in Little’s Law:

  The key thing that Little’s Law gives us is a deterministic causal rule that allows us to understand cause and effect in the dynamics of flow processes.

• Little’s Law in a Complex Adaptive System:

  All this is exactly the machinery we need to reason rigorously about flow processes in complex adaptive systems. This post is a detailed overview of how to apply sample path analysis with a case study of real-world product development team.

Little’s Law on the Internet

If you look up Little’s Law on the internet, say for example on Wikipedia or in most classical operations management or queueing theory texts, you will see the version that assumes it applies only to stable, stationary processes.

This view is not accurate anymore, and has not been accurate for over 50 years, but it is still the most popular version. This leads people to assume that it cannot apply in complex domains like software.

This is untrue.

So don’t go by the Wikipedia definition of Little’s Law. Read the posts above instead.

Alternatively, if you prefer more technical peer-reviewed sources, here are more accurate sources that give an up-to-date picture of what Little’s Law is and what it means. All these discuss the version of Little’s Law that we use in The Presence Calculus.

• Little’s Law on it’s 50th Anniversary by Dr. John Little.
• Little’s Law (Chapter 5) by Dr. John Little and Stephen Graves in “Building Intuition.”
• The Sample Path Proof of Little’s Law by Dr. Shaler Stidham.
• The Sample Path Analysis Textbook by El-Taha and Stidham.