Little's Law

Little's Law

In the mathematical theory of queues, Little's result, theorem, lemma, law or formula is a theorem by John Little which states:

The long-term average number of customers in a stable system L is equal to the long-term average effective arrival rate, λ, multiplied by the (Palm-)average time a customer spends in the system, W; or expressed algebraically: L = λW.

Although it looks intuitively reasonable, it's a quite remarkable result, as the relationship is "not influenced by the arrival process distribution, the service distribution, the service order, or practically anything else."

The result applies to any system, and particularly, it applies to systems within systems. So in a bank, the customer line might be one subsystem, and each of the tellers another subsystem, and Little's result could be applied to each one, as well as the whole thing. The only requirements are that the system is stable and non-preemptive; this rules out transition states such as initial startup or shutdown.

In some cases it is possible to mathematically relate not only the average number in the system to the average wait but relate the entire probability distribution (and moments) of the number in the system to the wait.