Ladies in Waiting

(updated  31st August 2007)

Keywords: simulating people, simulating crowds, simulating crowd dynamics

Article from the New Scientist - July 29th 2000 - Robert Mathews
A simple formula could bring relief to scores of women this summer

It's one of those immutable laws of the cosmos: you can't travel faster than light, like charges repel--and there's always a queue for the women's loos.

From the beachside to the bar, pop concert to theme park, it's a familiar sight every summer. Men stroll nonchalantly in and out of their conveniences, while women stand in a serpentine queue for theirs, going nowhere fast.

Everyone knows why, of course: women spend more time in the loo. As worldwide studies show, women typically take 89 seconds to use the loo--more than twice as long as the 39 seconds required by the average man.

But hang on: doesn't that mean the queue for the women's loos should only be about twice as long as the one for the men's? So how come they're usually longer--far longer? This was the question I found myself pondering recently as I waited (and waited) for my better half to get to the front of her queue at a stately home. And in my search for an answer to the Great Loo Mystery, I found myself drawn into the byzantine world of queueing theory.

There are two things worth knowing about queueing theory. The first is that its name contains the longest unbroken run of vowels of any word in the English language. The second thing, slightly more relevant to the Great Loo Mystery, is that queueing theory is based on that most paradoxical branch of mathematics, probability. And whenever probability is involved, you can be sure surprising things are just around the corner.

We humans are just not wired up to reason about probability at all accurately. Think you can? Well try this: if an average of seven accidents take place each week on a stretch of road, what are the chances of any week having exactly one accident per day? 50:50? 1 in 10? In fact, the correct answer is less than 1 in 100. You've been fooled by that deceptive term "average": the inevitable random variation about the average means that 99.4 per cent of all weeks will feature clusters of accidents, with two or more taking place on the same day.

Just such counter-intuitive results routinely rear their heads in queueing theory--and sometimes catch out even the experts. For instance, there was a time when queueing theorists thought that the queue would stay the same length if all the people in it were dealt with at the same average rate as they arrived. It seems sensible enough--but in fact it leads to queues growing infinitely long. Once again, it's the random variations about the average that cause all the trouble. In a shop, say, there may well be an average of 100 people an hour turning up over a week, but at any particular moment there could be no one waiting to be served--and then a whole bus-load, who are still being dealt with when the next crowd arrives. This means that unless you leave some slack in the system, you're heading for trouble (which is something hospital managers looking to minimise the number of "under-utilised staff" might like to bear in mind before the next flu epidemic).

It was just this kind of phenomenon I suspected might hold the key to the Great Loo Mystery. Perhaps random variation could explain why a small difference in the amount of time people spend in the loo added up to a big difference in the length of the queue. The only way to find out why was to mug up on some queueing theory.

So it was down to the university library, where I found a stackful of books on the subject. But when I flipped through them, I was amazed to find nothing at all about queues in public toilets. For some reason, the authors of all these books on queueing theory, who lavished whole chapters on the arcana of "job scheduling" and "optimal queue disciplines" hadn't included anything on the single most irksome queueing phenomenon of them all. But then, all the authors were men.

Faced with this glaring omission, I realised there was nothing else for it: I'd have to work out the theory of loo queues for myself. Fortunately, there's a kind of master formula which can solve all kinds of queueing problems, like a mathematical Swiss army knife. For example, if you plug in the average rate at which people form a queue, the rate at which they are served, and the numbers of servers dealing with them, this master formula spits out the average length of queue you can expect.

One thing about the formula struck me immediately, though. It looked like a dog's breakfast, with powers and factorials everywhere. Put slightly more mathematically, standing in line is a pretty non-linear phenomenon, with just a small change in one of the factors involved potentially making a huge difference to the final length of the queue.

The impact of all this on the Loo Queue Mystery turned out to be dramatic. According to the master formula, if women spend X times longer in the loo than men, then the average length of their queue will not just be X times longer, but at least X2 longer.

So, plugging in X = 2.3, the figure that emerges from all those dodgy-sounding timing studies--means the average queue for the women's loo will be at least 2.32, that is, five times longer than the queue for the gents.

It's a figure that certainly fits with my own experiences--or, more accurately, those of my better half. And, oddly enough, it's an experience that becomes more exasperating the more cubicles are provided for both sexes. The master formula shows that if each sex is provided with lots of cubicles, then when things get busy the relative lengths of their queues goes up even faster than X2 (roughly speaking, because the trouble one gets with just one cubicle is repeated with all the others).

So what can be done to make things fairer for women? Unisex loos are one possibility. But now another paradox raises its head. It turns out that this could make things worse for both sexes, as men lose their, um, natural advantage, while women find themselves queueing behind men as well as other women. Things only improve if priority is given to men, so they can accelerate the speed of the unisex queue--and you need a lot of faith in human generosity to see that working well.

In the end, all the sums have led me to conclude that there's only one solution to the Great Loo Mystery: positive discrimination. Women should simply get more cubicles than men. How many more? The master formula gives the answer, and for once it's pretty intuitive: X times as many as men (that is, at least twice as many).

It's a conclusion that flies in the face of public convenience design since at least Victorian times. From state-of-the-art facilities to the grottiest portable toilets at a pop concert, the same rule holds sway: what's good for the gander is good for the goose--and both get the same number of loos.

There are moves afoot to change things, in Britain at least. Julie Morgan, Labour MP for Cardiff North, the London-based Women's Design Service and the British Toilet Association have all begun campaigning for a change in the law, which would compel local authorities to provide more loos just for women. Every woman--and every man who's ever stood around waiting for her--can only hope they succeed.

Robert Matthews is a Visiting Fellow at the Department of Information Engineering, Aston University, Birmingham. Published in New Scientist 29th July 2000 number 2249