Week 2: Little's Law
You can work out how long people are waiting without timing a single one of them.
1 March 2027
This is easily the single most useful thing in the course, and it is also the only equation you will ever be asked to reach for without warning, so it is worth getting comfortable with early.
The length of a queue equals the rate at which people arrive multiplied by the time each of them spends in it. Written down like that it looks almost insultingly simple, and quite a few people assume there must be more to it. In practice it means you can stand at a distance, count heads, count arrivals for five minutes, and tell a manager how long their customers are waiting without ever having followed one.
It holds for any stable system. It does not care what order you serve people in, how many servers you have, how the arrivals are distributed, or whether the thing in the queue is a person, a packet, a claim form or a kidney. That generality is the reason it is worth an hour, and the reason people distrust it when they first meet it.
What it actually claims#
L = λ W
The number of people in the queue equals how fast they arrive multiplied by how long each one stays.
- L
- average number in the system, which you count
- λ
- average arrival rate, which you count over a known time
- W
- average time in the system, which is the thing you wanted
Holds for any stable system, any queue discipline, any arrival distribution. Every letter is an average, and a system that is filling up is not stable.
The word doing the work is average. Little’s Law is a statement about the long run of a system that is not filling up and not draining. Point it at a queue that is growing without bound and it will tell you something true and useless.
The three ways it goes wrong#
We spend the second half on failures, because you will hit all three in week 5.
- The system is not stable, and the average you computed is a snapshot of a backlog.
- Your boundary is wrong. People who balked, reneged or were served out of sight are in one of your counts and not the other.
- You measured arrivals during a rush and residence time during a lull.
After this lecture#
You can put a number on a wait you never timed. That is the whole of the field session on Wednesday and most of The Measured Queue.
Teaching team
- Amos Redfern, Lecturer