Little’s Law

Synonyms

Little’s lemma; Little’s principle; Little’s result; Little’s theorem

Definition

Little’s Law says that in the long-term, steady state of a production system, the average number of items L in the system is the product of the average arrival rate λ and the average time W that an item spends in the system, that is,

$$L = \lambda W.$$

Discussion

At first glance, Little’s Law looks like common sense. If items arrive faster than the system can process them, the system will overflow. Perhaps the earliest mention of the relation is by A. Cobham in 1954, and he states it as a fact without proof [2]. However, the law is insightful in two ways. First, it does not depend on the probability distributions of any of the variables. Second, since arrival rates are generally less than maximum processing rates, it says what the capacity of the system must be to handle the queue in a system design.

Figure 1 shows a visualization of a queue in steady state that explains Little’s Law:

Little’s...