Little's Law

In Chapter 5 of Building Intuition: Insights from Basic Operations Management Models and Principles, edited by D. Chhajed and T. J. Lowe, Springer Science+Business Media, LLC, New York, 2008, Steven Graves and John Little discuss Little's Law:

A “queuing system” consists of discrete objects we shall call “items” that “arrive” at some rate to the “system.” Within the system the items may form one or more queues and eventually receive “service” and exit.

Little's Law says that, under steady state conditions, the average number of items in a queuing system equals the average rate at which items arrive multiplied by the average time that an item spends in the system. Letting

L = average number of items in the queuing system,
W = average waiting time in the system for an item, and
λ = average number of items arriving per unit time, the law is

L=λW

Graves and Little give several examples, and I'll paraphrase one here.

Caroline is a wine buff. Her wine rack in the cellar holds 240 bottles. She seldom fills the rack to the top but sometimes after a good party the rack is empty. On average it seems to be about 2/3rds full. She buys, on average, about eight bottles per month. How long, on average, does a bottle languish in the cellar before being consumed?

Using Little's Law, we let L be the average number of bottles in the wine rack, 2/3×240=160. λ is the average number of items arriving per unit time, 12×8=96 bottles/year. Divide L by λ and we get W≅1.67 years per bottle.

(It seems to me that Little's Law is related to Stoke's Theorem. The size of the queue is the integral of the “flux” across the queue boundaries.)

more to come...