# Why do three buses always come along at once? This game explains

Typical. Bloody typical. Two London buses serving the same route, bunched up in the Clapton area. Image: Felix O, via Flickr.

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.

# How can cities become more bike friendly? The Netherlands offers useful lessons

(Aurore Belot/AFP via Getty Images)

It might seem like cycling is in the DNA of the Netherlands, a country where even the prime minister takes his bicycle to work. But the Dutch haven’t always lived as one with their bikes. In the Amsterdam of the early 1970s, cars were considered the wave of the future. They can be seen filling up squares and streets in historical photographs, and killed an average of over two Amsterdammers per week, including many children.

It is nothing more than an “accident of history” that the Netherlands embraced cycling, says Marco te Brömmelstoet, the director of the Urban Cycling Institute in Amsterdam and a man better known as the city’s cycling professor. Today’s bike rider’s paradise was created after parents and activists took to the streets to protest “child murder” by car. A Saudi oil embargo, rising gas prices, concerns about pollution and anger about the destruction of entire neighbourhoods to build motorways did the rest.

Amsterdam, 1958. Not a cyclist's paradise. (Keystone/Getty Images)

What’s important about this history is that it can be replicated in other cities, too. Of course, the Netherlands has certain advantages – it’s flat as a pancake, for example. But in the eyes of traffic reformers, the rise of e-bikes (and even cargo bikes) means there’s no excuse for prioritising cars everywhere.

So how can cities, flat or not, follow Amsterdam’s path to creating places where cycling is a pleasant, safe and common way to get around? The Dutch have some tips.

## Separate bikes from car traffic

Any city could start painting dedicated bike lanes on the streets. But in the Netherlands, those white marks indicating space for cyclists are considered just a minor first step.

“A line on the road is not enough. Motorists will ignore it,” says Frans Jan van Rossem, a civil servant specialising in cycling policy in Utrecht. If other cities want their residents to choose bikes instead of cars when dodging pandemic-era public transport, protecting them from fast-moving car traffic must be the priority, Van Rossem says.

The Dutch research institute CROW developed a widely praised design manual for bicycle infrastructure, full of tips for creating these protected lanes: A row of vertical white posts or a curb can serve as a physical separator, for example. Still, cyclists tend to feel safest in a "solitary" path, separated from the road by grass, trees, or an elevated concrete island.

“The main bottleneck, the main reason why people don’t cycle, is that they don’t feel safe,” Van Rossem notes. “To start, construct separate paths.”

## Turn those bike paths into a network

Many cities may have some bike lanes on some streets, but leave cyclists to roll the dice everywhere else. Will conditions still be safe when they turn left or right? Often they have to continue their way without any protected facilities for cyclists.

“In many cases, cities take fast action, without thinking it through very well,” says Lucas Harms. He leads the Dutch Cycling Embassy, a partnership between the Dutch government and several companies, which promotes Dutch bike knowhow globally. “Don’t build small pieces of bike lane from nothing to nowhere. Think about a network of cycling infrastructure.”

Utrecht aims to have cyclists within 200 to 300 metres of a connected path anywhere in the city, Van Rossem says. Avoid constructing those paths in sketchy industrial areas, he warns. “A connection through an unattractive area may be fast, but won’t be used a lot.”

## Embrace the ‘fietsstraat’, a street where bikes come first

On some streets, drivers have to give up their privileges. (Rick Nederstigt/AFP via Getty Images)

A peculiar Dutch invention called "fietsstraat" (cycling street) holds strong potential for the rest of the world, Kevin Krizek says. He’s a transportation professor from Colorado who spent three years at Radboud University in Nijmegen.

On cycling streets, cars are “guests”, restricted by a speed limit of 30 kilometres per hour. Drivers are not allowed to pass, so cyclists comfortably dominate the road. In the Netherlands the fietsstraat is usually paved with red asphalt, to resemble a bike path and notify drivers of their secondary status. But creating a cycling street can be easy. “All you need to do is put signs at intersections,” Krizek says. The effect is revolutionary in his view. Drivers have to give up their privileges, and cyclists can take the lead.

Some Dutch traffic experts worry the cycling street won’t work if a city doesn’t also have a robust cycling culture. In the Netherlands, drivers are aware of the perils of urban cycling because they too use bicycles. Moreover, Dutch cities use sophisticated “circulation plans” to direct cars away from city centres and residential areas, onto a few main routes.

Without “calming” traffic this way, the cycling street could be a step too far, Harms says. “In a city like New York, where all roads are equally accessible and full, it’s better to separate bicycles and cars,” he says.

If cyclists have to cross intersections “at the mercy of the Gods”, you’re not there yet, says Harms. When he travels abroad, he often finds clumsily designed crossings. As soon as cars turn, cyclists may fear for their lives.

Harms recommends placing physical barriers between cars and bikes in places where they must cross. The Dutch build elevated islands to direct traffic into separate sections. The golden rule: cars wait behind bicycles. That way, drivers can see cyclists clearly at all times. Barriers also force Dutch cyclists to turn left in the safest way possible. They cross the street first and wait for their turn again before making their way left.

“You can create that with simple temporary measures,” Harms says. Planters work fine, for example. “They must be forgiving, though. When someone makes a mistake, you don’t want them to get seriously injured by a flower box’s sharp edge.”

Professor Krizek points out how the Dutch integrated cycling routes into roundabouts. Some are small; some are big and glorious, like the Hovenring between Eindhoven and Veldhoven, where cyclists take a futuristic-looking roundabout lifted above the highway. Most of those traffic circles move high volumes of cars and cyclists through intersections efficiently and safely. For a simpler solution, the Dutch manual suggests guiding cyclists to quieter streets – crossing a block up or down may be safer. “Nobody knows how to do intersections better than the Dutch,” says Krizek.

## Ban cars, or at least discourage them

A man rides down from a three-level bicycle parking garage near Amsterdam's main train station. (Timothy Clary/AFP via Getty Images)

The quickest, most affordable way to make a city more bikeable is to ban cars, says Ria Hilhorst, cycling policy advisor for the City of Amsterdam. It will make streets remarkably safe – and will most likely enrage a significant amount of people.

Amsterdam doesn’t outlaw cars, but it does deliberately make their owners feel unwelcome in the historic city’s cramped streets. Paid parking is hugely effective, for example. Many car owners decide to avoid paying and use bicycles or public transportation for trips into the city. Utrecht, meanwhile, boasts the world’s largest bicycle parking garage, which provides a dizzying 12,500 parking spots.

To further discourage drivers from entering the city’s heart, Amsterdam will soon remove more than 10,000 car-parking spaces. Strategically placed barriers already make it impossible to cross Amsterdam efficiently by car. “In Amsterdam, it is faster to cross the city on a bike than by car,” Harms says. “That is the result of very conscious policy decisions.”

## Communicate the benefits clearly

Shopkeepers always fear they will lose clients when their businesses won’t be directly accessible by car, but that’s a myth, says Harms. “A lot of research concludes that better access for pedestrians and cyclists, making a street more attractive, is an economic boost.”

Try replacing one parking space with a small park, he recommends, and residents will see how it improves their community. Home values will eventually rise in calmer, bike-friendlier neighbourhoods without through traffic, Van Rossem says. Fewer cars mean more room for green spaces, for example.

“I often miss the notion that cycling and walking can contribute a lot to the city. One of the greatest threats to public health is lack of exercise. A more walkable and bikeable city can be part of the solution,” says Ria Hilhorst. “But in many countries, cycling is seen as something for losers. I made it, so I have a car and I’m going to use it, is the idea.

“Changing this requires political courage. Keep your back straight, and present a vision. What do you gain? Tranquility, fewer emissions, health benefits, traffic safety, less space occupied by vehicles.”

Again, she points to Amsterdam’s history. “It is possible; we were a car city too.”

Karlijn van Houwelingen is a journalist based in New York City.