# Q: Why do buses come in threes? A: Mathematical inevitability

That's the problem with the internet in summer, isn't it? Nothing but repeats.

Anyway, here's something we originally published in June 2015. But it's dead good, honest.

The old joke that you wait ages for one bus, then three come along at once, is bordering on cliché. But it's also, as it turns out, true – not just because of bad planning, but also because of maths.

The phenomenon is so common, in fact, that it has a choice of names. Bus bunching, clumping, convoying, platooning – all relate to the depressing reality that, over any length of time, buses serving a single route are likely to end up tootling along directly behind each other.

The reasons why this should be can be difficult to get your head around – so Lewis Lehe, a postgrad working on a PhD in transport engineering at the University of California, Berkeley, has built a game of sorts to demonstrate. You can play it here. But if, like us, you are both lazy and impatient, here's how it works, with a few helpful screenshots to illustrate.

The game features two buses, serving a circular route with four stops. At the start of the game, the two buses are evenly spaced, at opposing ends of the loop. Passenger flows at each stop are identical to those opposite: when one bus has to pause for a set period at one stop, the other is pausing for the same length of time across the map.

The result is that the two buses are always carrying the same number of passengers. Left to their own devices, they will remain equally spaced forever, in perfect equilibrium.

And we know this, because at this point we got distracted, had a conversation on Twitter, went to make a coffee, and then suddenly remembered we were in the middle of doing something about buses, flicked back to the game in a panic, and discovered that it looked like this:

Our game was still ticking along nicely, the two buses now holding seven passengers apiece.

Eternal equilibrium is really boring, though, so let's mess things up a bit. What happens if we delay Bus One, just for a single second?

Err, well, this as it turns out. This is a couple of moments after the unexpected delay.

Two things to notice here. One is that the gap between the two buses has closed, slightly: that momentary delay allowed Bus Two to catch up, slightly. The other thing is that Bus One now has more passengers.

Those two things are actually features of a single phenomenon. In those few moments in which Bus One was delayed, Bus Two started to catch up – not by much, but for enough for it to have an effect. The consequence is that it's now reaching stops which last saw a bus relatively recently – and where there are fewer passengers for it to pick up.

But here's the insidious thing: that process is self-perpetuating.

Think this through for a moment. If Bus Two is serving stops that have relatively fewer passengers waiting, it can get moving again faster. Bus One, by contrast, is serving stops that haven't seen a bus in a while (it was delayed, after all), and so it's picking up more people. That means it needs to stop for longer, both to let its larger contingent of people off, and to collect the next lot.

And so, Bus Two gets closer...

...and then catches up altogether.

Look at the distribution of passengers now. Bus One is now arriving at stops with populations bigger than some small Chinese cities. Bus Two, just behind, arrives moment later to find them deserted. Without intervention, this situation will persist, essentially, forever.

In real life, of course, there would be an intervention (Bus Two would almost certainly overtake, as soon as there’s room). And real life is messier than this model in other ways, too. Bus routes are served by more than two buses; passenger flows aren't so evenly spread.

But it’s this inevitable messiness that provides that slight delay that kicks the whole process off. A slight disparity in passenger numbers, a traffic jam, bad luck with the lights; whatever the cause, one bus will be delayed, and those behind it will start to catch up.

Until suddenly you get three buses showing up at once and everyone makes tutting noises.

There are ways of mitigating this feedback loop. One is setting maximum or minimum stopping times in advance, to regulate the service. Another is building in waiting time at one end of a route, so that a bus doesn't immediately turn around and set off again (in effect, doing the second half of a circular route), thus bringing a measure of predictability to the time it takes to do an entire circuit.

These things can help – but the ubiquity of bus bunching suggests that they can’t elimate the problem altogether.

So the next time someone in your hearing asks why it is that buses all come along at once, you can reply, "Mathematical inevitability". Then you can look smug about it, as they, presumably, decide that they’d rather walk.

# CityMetric is now City Monitor! Come see us at our new home

City Monitor is now live in beta at citymonitor.ai.

CityMetric is now City Monitor, a name that reflects both a ramping up of our ambitions as well as our membership in a network of like-minded publications from New Statesman Media Group. Our new site is now live in beta, so please visit us there going forward. Here’s what CityMetric readers should know about this exciting transition.

Regular CityMetric readers may have already noticed a few changes around here since the spring. CityMetric’s beloved founding editor, Jonn Elledge, has moved on to some new adventures, and a new team has formed to take the site into the future. It’s led by yours truly – I’m Sommer Mathis, the editor-in-chief of City Monitor. Hello!

My background includes having served as the founding editor of CityLab, editor-in-chief of Atlas Obscura, and editor-in-chief of DCist, a local news publication in the District of Columbia. I’ve been reporting on and writing about cities in one way or another for the past 15 years. To me, there is no more important story in the world right now than how cities are changing and adapting to an increasingly challenging global landscape. The majority of the world’s population lives in cities, and if we’re ever going to be able to tackle the most pressing issues currently facing our planet – the climate emergency, rising inequality, the Covid-19 pandemic ­­­– cities are going to have to lead the way.

That’s why City Monitor is now a global publication dedicated to the future of cities everywhere – not just in the UK (nor for that matter just in the US, where I live). Our mission is to help our readers, many of whom are in leadership positions around the globe, navigate how cities are changing and discover what’s next in the world of urban policy. We’ll do that through original reporting, expert opinion and most crucially, a data-driven approach that emphasises evidence and rigorous analysis. We want to arm local decision-makers and those they work in concert with – whether that’s elected officials, bureaucratic leaders, policy advocates, neighbourhood activists, academics and researchers, entrepreneurs, or plain-old engaged citizens – with real insights and potential answers to tough problems. Subjects we cover include transportation, infrastructure, housing, urban design, public safety, the environment, the economy, and much more.

The City Monitor team is made up of some of the most experienced urban policy journalists in the world. Our managing editor is Adam Sneed, also a CityLab alum where he served as a senior associate editor. Before that he was a technology reporter at Politico. Allison Arieff is City Monitor’s senior editor. She was previously editorial director of the urban planning and policy think tank SPUR, as well as a contributing columnist for The New York Times. Staff writer Jake Blumgart most recently covered development, housing and politics for WHYY, the local public radio station in Philadelphia. And our data reporter is Alexandra Kanik, whose previous roles include data reporting for Louisville Public Media in Kentucky and PublicSource in Pittsburgh, Pennsylvania.

Our team will continue to grow in the coming weeks, and we’ll also be collaborating closely with our editorial colleagues across New Statesman Media Group. In fact, we’re launching a whole network of new publications, covering topics such as the clean energy transition, foreign direct investment, technology, banks and more. Many of these sectors will frequently overlap with our cities coverage, and a key part of our plan is make the most of the expertise that all of these newsrooms combined will bring to bear on our journalism.

As for CityMetric, some of its archives have already been moved over to the new website, and the rest will follow not long after. If you’re looking for a favourite piece from CityMetric’s past, for a time you’ll still be able to find it here, but before long the whole archive will move over to City Monitor.

On behalf of the City Monitor team, I’m thrilled to invite you to come along for the ride at our new digs. You can follow City Monitor on LinkedIn and on Twitter. If you’re interested in learning more about the potential for a commercial partnership with City Monitor, please get in touch with our director of partnerships, Joe Maughan.

I want to thank and congratulate Jonn Elledge on a brilliant run. Everything we do from here on out will be building on the legacy of his work, and the community that he built here at CityMetric. Cheers, Jonn!

To our readers, on behalf of the City Monitor team, thank you from all of us for being such loyal CityMetric fans. We couldn’t have done any of this without you.

Sommer Mathis is editor-in-chief of City Monitor.