# Little's law

In queueing theory, Little's result, theorem, lemma, or law says:

The long-term average number of customers in a stable system L, is equal to the long-term average arrival rate, λ, multiplied by the long-term average time a customer spends in the system, W, or:
$\, L= \lambda W.$

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.

It is also a comparatively recent result; the first proof was published in 1961 by John Little, then at Case Western Reserve University. Handily his 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.

## 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, and no one leaves without buying something. So the system is roughly:

Entrance → Browsing → Counter → Exit

This is a stable system, so the rate at which people enter the store is the rate at which they arrive at the counter and the rate at which they exit as well. We call this the arrival rate.

Little's Law tells us that the average number of customers in the store, L, is the arrival rate, λ, times the average time that a customer spends in the store, W, or simply:

$\, L= \lambda W.$

Assume customers arrive at the rate of 10 per hour and stay an average of 0.5 hour. This means we should find the average number of customers in the store at any time to be 5.

Now suppose the store is considering doing more advertising to raise the arrival rate to 20 per hour. The store must either be prepared to host an average of 10 occupants or must reduce the time each customer spends in the store to 0.25 hour. The store might achieve the latter by ringing up the bill faster or by walking up to customers who seem to be taking their time browsing and saying, "Can I help you?".

We can apply Little's Law to systems within the shop. For example, the counter and its queue. Assume we notice that there are on average 2 customers in the queue and at the counter. We know the arrival rate is 10 per hour, so customers must be spending 0.2 hour on average checking out.

We can even apply Little's Law to the counter itself. The average number of people at the counter would be in the range (0,1) since no more than one person can be at the counter at a time. In that case, the average number of people at the counter is also known as the counter's utilisation.

## Use in performance testing of computer systems

Little's law can be used in software performance testing to ensure that the observed performance results are not due to bottlenecks imposed by the testing apparatus. See: