The Queue: Little's law and the lying-in-state

Introduction

In September 2022 the coffin of Queen Elizabeth II lay in state in Westminster Hall, open 24 hours a day from 5 pm on 14 September to 6.30 am on 19 September. The queue to file past it became famous in its own right. At its longest it stretched about 10 miles (16 km), back as far as Southwark Park. People waited more than 24 hours. On 16 September it was temporarily declared closed. An estimated 250,000 people filed past.

Ten miles of people sounds like chaos. It was arithmetic. Three numbers and one law tell you how long that queue had to be, and the same law tells you what the organisers could and could not control.

Little’s law

Stand at the back of a queue that is moving steadily, and ask how many people are in front of you. Everyone ahead of you will be served before you, and you will wait W hours. If the queue is served at μ people an hour, then μ × W people will go through in that time: exactly the people in front of you. So the queue holds μW people.

That is Little’s law: the average number of people in a system equals the rate at which they pass through it times the average time each one spends there, L = λW. It holds for any queue that is not growing or shrinking without limit, whatever the pattern of arrivals, so long as you use averages over a long enough period. For a queue running flat out, people leave at the service rate, μ: the number of people a queue’s exit can process per unit time, here the rate at which people filed past the coffin.

Figure 1 shows the same thing with cumulative curves: the total number of people who have arrived, and the total who have been served, plotted against time. You met them in Module 1, lesson 5. The vertical gap between the curves is the queue. The horizontal gap is the wait. When the service line is straight, with slope μ, the vertical gap is μ times the horizontal one. That is Little’s law, read off a chart.

The 2022 queue in three lines of arithmetic

The real hourly rates were never published, so the first step is to estimate one from the facts.

1. Throughput. From 5 pm on 14 September to 6.30 am on 19 September is 109½ hours. About 250,000 people filed past. That averages 250,000 ÷ 109.5 ≈ 2,300 people an hour, about 38 a minute. This is our arithmetic, not a published figure.
2. People in the queue. A wait of 24 hours at 2,300 an hour means 24 × 2,300 ≈ 55,000 people were queuing. Every one of them would be served in the next day.
3. Length. Suppose the queue was 2.5 m wide and people stood at 1.5 per square metre. Each metre of queue then holds 2.5 × 1.5 = 3.75 people, and 55,000 people need 55,000 ÷ 3.75 ≈ 14,700 m: about 14.7 km, or 9 miles.

The reported maximum was about 10 miles, or 16 km. Our estimate lands within a couple of kilometres of it, which is as close as you should expect, because the width and density are our assumptions, not reported figures. A density of 1.5 p/m² is Fruin’s queuing Level of Service D (1.4–3.3 p/m²): close enough to shuffle forward, not close enough to touch.

Arrivals against service

While people arrive faster than they are served, the queue grows at the difference between the two rates. If 3,000 people an hour arrive and 2,300 are served, the queue gains 700 an hour: 7,000 more after 10 hours, and each newcomer then faces about 7,000 ÷ 2,300 ≈ 3 hours more wait. When arrivals fall below the service rate, overnight for example, the queue drains at the difference. A queue like this is a running total, not a fixed object, and that is how the model below works.

A model of the queue

The model steps through the five days a tenth of an hour at a time. Each step it adds the people who arrive, subtracts the people served, and works out the wait (queue ÷ service rate) and the length (queue ÷ density ÷ width). Arrivals follow a daily pattern: busy from 9 am to 9 pm, quieter after midnight. Two rules close the queue, described in the next section.

Closing a queue is a control action

Closing the queue to newcomers is often reported as something going wrong. In queueing terms it is the organisers using one of the only two levers they have. The service rate is limited by the space and time it takes people to file past; the number of people allowed to join is not. The model uses two closure rules.

• Temporary closure. When the wait passes a limit (24 hours by default), stop admitting people until it falls below a lower one (20 hours). The queue stops growing, so by Little’s law the wait stops rising. It does not get any shorter for those already in it: they are still served at the same rate.
• Final closure. Stop admitting people once the wait equals the time left before the hall closes. Anyone who joins later cannot be served. Without this rule, the model leaves tens of thousands of people queuing when the doors shut at 6.30 am on the 19th.

The third lever is information. Telling people the expected wait changes who decides to come, which changes the arrival rate. Warnings of 24-hour waits were issued in 2022. In the model, lowering the daytime arrival rate is the closest you can get to that, and it shows why a published wait is a control action in its own right: it is the cheapest way to lower λ.

Altenburg and the queue: what is reported

The queue appears in the Start Right story. The World Marathon Majors magazine reports that Marcel Altenburg’s prediction for it came “within 50 metres and 14 seconds”. That claim appears only in that article, a promotional feature, and the article does not say what was predicted: a length, a passing time or something else. Manchester Metropolitan University’s own news mentions only “crowd control for the Queen’s funeral”. Treat it as reported by one source, not established. Lesson 2 explains how to weigh claims like this.

What you can say with confidence is that the arithmetic in this lesson is the kind any planner, Altenburg included, would start from: a service rate, a wait and Little’s law give the size of the queue, and a density and a width turn that into a distance.