Week 3: Variability is the tax
A counter running at ninety-five percent of capacity is not a well-run counter. It is one bad morning from collapse.
8 March 2027
Week 2 gave you averages to work with. This week is really about why averages on their own are not enough, and in our experience it is the week most likely to change how you look at a workplace.
Picture two counters serving exactly the same number of customers an hour. At one of them the wait comes out at ninety seconds, while at the other it is eleven minutes, and nothing whatsoever about the average will explain the difference to you. What explains it is that the second counter’s customers arrive in clumps and take wildly different amounts of time, and a queue punishes both.
The punishment is not linear, which is the part that surprises people. Push a system from eighty percent utilisation to ninety and the wait does not go up by an eighth. It roughly doubles. Push it to ninety-five and it doubles again. Every organisation that has ever been told to sweat its assets has walked into this, and most of them concluded they had a staff problem.
Spare capacity is not waste. It is what you are bought instead of a queue.
The shape of the wall#
Take the utilisation term on its own and measure it against a counter running at half capacity. This is arithmetic rather than measurement, so it is exact, and it is also only half the story, because the variability term is the multiplier that sits on top of it.
| Utilisation | Waiting time, relative to 50% |
|---|---|
| 50% | 1x |
| 80% | 4x |
| 90% | 9x |
| 95% | 19x |
| 99% | 99x |
The last row is the one to sit with. Going from 95 to 99 percent utilisation buys you four percent more throughput and costs you five times the wait.
Kingman’s formula#
Kingman’s formula is an approximation rather than an identity, and it is the most quietly devastating thing in the course.
W ≈ (ρ / (1 − ρ)) × variability × service time
Waiting time is the utilisation term, multiplied by how uneven things are, multiplied by how long a customer takes.
- ρ
- utilisation, the fraction of capacity in use
- ρ / (1 − ρ)
- the term that goes to infinity as you approach full capacity
- variability
- how uneven arrivals and service times are
An approximation for a single queue with one server. You may use it, and you may not use it without saying you know it is one.
Three consequences we work through on the board.
- Spare capacity is not waste. It is what you are buying instead of a queue.
- Reducing variability is usually cheaper than adding servers, and almost nobody tries it first.
- A system with no variability can run at ninety-nine percent forever. This is why a production line and a walk-in clinic are different animals.
The honest caveat#
Kingman is an approximation for a single queue with one server, and we will spend ten minutes on where it stops being trustworthy. You are allowed to use it in The Measured Queue. You are not allowed to use it without saying you know it is an approximation.
After this lecture#
You can explain why the queue got worse after the efficiency drive.
Teaching team
- Amos Redfern, Lecturer