Little's law

Little's law

In queueing theory, Little's result, theorem, or law are three names for the same law.

In plain English, it says:

The average number of customers in a stable system (over some time interval) is equal to their average arrival rate, multiplied by their average time in the system.

Although it looks intuitively reasonable, it's a quite remarkable result, as it implies that this behavior is entirely independent of any of the detailed probability distributions involved, and hence requires no assumptions about the schedule according to which customers arrive or are serviced, or whether they are served in the order in which they arrive.

It is also a comparatively recent result - it was first proved by John Little in 1961.

Handily his result applies to any system, and particularly, it applies to systems within systems. So in a bank, the queue 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 requirement is that the system is stable - it can't be in some transition state such as just starting up or just shutting down.

Small Example

Imagine a small shop with a single counter and an area for browsing, where only one person can be at the counter at a time, so the system is roughly:

Entrance → Browsing → Counter → Exit

The three important measures are the average time people take at the counter, the utilisation of the counter, and the rate at which people move through the system. The rate is what the shop wants to maximise.

So all we do is apply Little's result to the counter. This shows that the number of people on average at the counter is the rate at which they move through the system, multiplied by the time it takes to serve them. Since the number of people at the counter is just the utilisation, it can therefore be shown that the rate is given by the utilisation, divided by the time per customer.

Therefore, to make a really productive shop you should strive to take as little time as possible ringing up the bill, and you should try to keep your counter as busy as possible. In practice, the latter means walking up to people who seem to be taking their time browsing and saying 'Can I help you' in an annoying fashion. Other ways to increase the counter utilisation might be to have more people in the shop browsing, or to have a queue.