Little's Law - Example

Example

Imagine a small store 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 store, and the rate at which they exit as well. We call this the arrival rate. By contrast, an arrival rate exceeding an exit rate would represent an unstable system, where the number of waiting customers in the store will gradually increase towards infinity.

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

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 adding more counters.

We can apply Little's Law to systems within the store. 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 hours 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 utilisation of the counter.

However, because a store in reality generally has a limited amount of space, it cannot become unstable. Even if the arrival rate is much greater than the exit rate, the store will eventually start to overflow, and thus any new arriving customers will simply be rejected (and forced to go somewhere else or try again later) until there is once again free space available in the store. This is also the difference between the arrival rate and the effective arrival rate, where the arrival rate roughly corresponds to the rate of which customers arrive at the store, whereas the effective arrival rate corresponds to the rate of which customers enter the store. In a system with an infinite size and no loss, the two are however equal.