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.

 
 
 
 

Barcelona’s car-free “superblocks” could extend lives. So will they catch on elsewhere?

Barcelona. Image: Getty.

The world’s biggest cities have larger populations and higher economic outputs than some countries. But as they grow in size and complexity, cities are also facing thorny challenges that threaten the health and happiness of residents. Congestion, pollution and a lack of community spaces have become major drags on people’s aspirations and experiences of urban living.

In response, cities must manage their resources and priorities to create sustainable places for visitors and residents, and foster innovation and growth. Enter Barcelona – the capital of Catalonia, in Spain – where a bold stroke of urban planning first introduced “superblocks” in 2016.

Image: ISGlobal/FAL.

Superblocks are neighbourhoods of nine blocks, where traffic is restricted to major roads around the outside, opening up entire groups of streets to pedestrians and cyclists. The aim is to reduce pollution from vehicles, and give residents much-needed relief from noise pollution. They are designed to create more open space for citizens to meet, talk and do activities.


Health and well-being boost

There are currently only six superblocks in operation, including the first, most prominent one in Eixample. Reports suggest that – despite some early push back – the change has been broadly welcomed by residents, and the long-term benefits could be considerable.

A recent study carried out by the Barcelona Institute for Global Health estimates that if, as planned, 503 potential superblocks are realised across the city, journeys by private vehicle would fall by 230,000 a week, as people switch to public transport, walking or cycling.

The research suggests this would significantly improve air quality and noise levels on the car-free streets: ambient levels of nitrogen dioxide (NO₂) would be reduced by a quarter, bringing levels in line with recommendations from the World Health Organisation (WHO).

The plan is also expected to generate significant health benefits for residents. The study estimates that as many as 667 premature deaths from air pollution, noise and heat could be prevented each year. More green spaces will encourage people to get outdoors and lead a more active lifestyle.

This, in turn, helps to reduce obesity and diabetes and ease pressure on health services. The researchers claim that residents of Barcelona could expect to live an extra 200 days thanks to the cumulative health benefits, if the idea is rolled out across the city.

Space to play. Imag: Mosa Moseneke/Unsplash.

There are expected to be benefits to mental health, as well as physical health. Having access to such spaces can stave off loneliness and isolation – especially among elderly residents – as communities form stronger bonds and become more resilient.

Stumbling blocks

It was Salvador Rueda, director of the Urban Ecology Agency of Barcelona, who first championed the introduction of superblocks – and he argues that the idea could be used in any city. Even so, authorities looking to expand the concept in Barcelona or beyond will need to be mindful of some concerns.

Changes like these require capital investment. Even as the car-free streets are transformed with urban furniture and greenery, the remaining major roads will likely have to accommodate heavier traffic.

Nothing comes for free. Image: Zvileve/Flickr/creative commons.

Further investments in local infrastructure – such as improving surrounding roads to deal with more traffic, or installing smart traffic management system – could be required to prevent serious congestion. Then the question remains, how to finance such investments – a higher tax rate is unlikely to be popular.


What’s more, whenever a location becomes more desirable, it leads to an increase in property demand. Higher prices and rent could create pockets of unaffordable neighbourhoods. This may lead to use of properties for investment purposes and possibly, displacement of local residents.

It’s also worth noting that Barcelona is an old and relatively well-planned European city. Different challenges exist in emerging global cities across Asia, Africa and Latin America – and in younger cities in the US and Australia. There is a great deal of variation in scale, population density, urban shape and form, development patterns and institutional frameworks across the cities. Several large cities in the developing world are heavily congested with uncontrolled, unregulated developments and weak regulatory frameworks.

Replicating what’s been done in Barcelona may prove difficult in such places, and will require much greater transformations. But it’s true that the basic principles of superblocks – that value pedestrians, cyclists and high quality public spaces over motor vehicles – can be applied in any city, with some adjustments.

Leading the way

Over the history of human civilisation, great cities have been at the forefront of innovation and social progress. But cities need a robust structure of governance, which is transparent and accountable, to ensure a fair and efficient use of resources. Imposing innovation from the top down, without consultations and buy-in, can go squarely against the idea of free market capitalism, which has been a predominant force for modern economies and can lead push-back from citizens and local businesses.

Citizens must also be willing to change their perspectives and behaviour, to make such initiatives work. This means that “solutions” to urban living like superblocks need to have buy-in from citizens, through continuous engagement with local government officials.

A man speaks at a public consultation on the Eixample superblock in Barcelona. Image: Ajuntament Barcelona/Flickr/creative commons.

Successful urban planning also needs strong leadership with a clear and consistent vision of the future, and a roadmap of how that vision can be delivered. The vision should be co-developed with the citizens and all other stakeholders such as local businesses, private and public organisations. This can ensure that everybody shares ownership and takes responsibility for the success of local initiatives.

There is little doubt that the principles and objectives of superblocks are sound. The idea has the potential to catch on around the world – though it will likely take a unique and specific form in every city.

The Conversation

Anupam Nanda, Professor of Urban Economics and Real Estate, University of Reading.

This article is republished from The Conversation under a Creative Commons license. Read the original article.