Queueing theory is the mathematical study of waiting lines, or queues.

Wikipedia – Queueing theory

This is timely, as the big news of the day is that our Prime Minister, Rishi Sunak, wants all pupils to study maths to age 18. So maybe this is one of the things people should be learning about (in addition to bank balances, household budgets, taxes, interest rates etc.).

An example

Imagine a busy Accident & Emergency department that only has one bay free. Patients take an average of just 10 minutes to triage and treat, and they arrive at the rate of 5.8 per hour.

What will the expected waiting time be?

With only one bay open, patients will have to wait nearly five hours on average before they are seen.

What happens if you’re able to free up another bay?

If you open a second bay, the average waiting time is not just cut in half; it goes down to about 3 minutes. The waiting time is reduced by a factor of 93x.

Why was the wait so long with one bay?

There’s not enough slack in the system.

Patients are arriving every 10.3 minutes on average and are taking 10 minutes to see on average. If patient arrivals were exactly evenly spaced and each took exactly 10 minutes to treat, there would be no problem. Each patient would be seen before the next arrived. No waiting.

BUT… the service and arrival times have to be very close to their average values to avoid a line, and in reality that’s not how things happen. On average there will be a long line, 28 people. But with a second bay, it’s not likely that even two people will arrive before one of the bays is free.

Adapted from What happens when you add a new teller? by John D. Cook

Why does A&E only have one bay free?

Back pressure.

There’s a queue of ambulances outside of A&E because at any given moment all the bays are busy (and the corridors are already full of trolleys).

A&E can’t move patients onto wards because their beds are full.

Wards can’t discharge patients who are fit to leave because there’s no social care available (whether they should go to a care home, or get a care package in their own home).

Nobody is accountable end to end

If the scope of these problems was (say) a factory, then we could pull out Theory of constraints and go looking for bottlenecks and resolving them. But that sort of approach assumes that there’s somebody who can look at then fix the end to end flow.

But as we look at the back pressure, it builds up across the dividing lines between organisations, with nobody empowered or accountable to free up capacity for the stage before them.

This isn’t just about hospitals

Across the public sector and private sector the relentless pursuit of efficiency has resulted in services that might be approaching 100% efficient, but are also approaching 0% effective. Everybody is waiting, nobody is getting served.

We need more slack.

You’re efficient when you do something with minimum waste. And you’re effective when you’re doing the right something.

The Knowledge Project blog – Efficiency is the Enemy

It might seem good for profits (or spending of precious taxpayer money) to be sweating every last asset, but what about the effect on the overall economy? If everybody’s busy doing nothing, because they’re waiting on hold or whatever, then overall productivity has taken a nosedive.