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.

 
 
 
 

As EU funding is lost, “levelling up” needs investment, not just rhetoric

Oh, well. Image: Getty.

Regional inequality was the foundation of Boris Johnson’s election victory and has since become one of the main focuses of his government. However, the enthusiasm of ministers championing the “levelling up” agenda rings hollow when compared with their inertia in preparing a UK replacement for European structural funding. 

Local government, already bearing the brunt of severe funding cuts, relies on European funding to support projects that boost growth in struggling local economies and help people build skills and find secure work. Now that the UK has withdrawn its EU membership, councils’ concerns over how EU funds will be replaced from 2021 are becoming more pronounced.

Johnson’s government has committed to create a domestic structural funding programme, the UK Shared Prosperity Fund (UKSPF), to replace the European Structural and Investment Fund (ESIF). However, other than pledging that UKSPF will “reduce inequalities between communities”, it has offered few details on how funds will be allocated. A public consultation on UKSPF promised by May’s government in 2018 has yet to materialise.

The government’s continued silence on UKSPF is generating a growing sense of unease among councils, especially after the failure of successive governments to prioritise investment in regional development. Indeed, inequalities within the UK have been allowed to grow so much that the UK’s poorest region by EU standards (West Wales & the Valleys) has a GDP of 68 per cent of the average EU GDP, while the UK’s richest region (Inner London) has a GDP of 614 per cent of the EU average – an intra-national disparity that is unique in Europe. If the UK had remained a member of the EU, its number of ‘less developed’ regions in need of most structural funding support would have increased from two to five in 2021-27: South Yorkshire, Tees Valley & Durham and Lincolnshire joining Cornwall & Isles of Scilly and West Wales & the Valley. Ministers have not given guarantees that any region, whether ‘less developed’ or otherwise, will obtain the same amount of funding under UKSPF to which they would have been entitled under ESIF.


The government is reportedly contemplating changing the Treasury’s fiscal rules so public spending favours programmes that reduce regional inequalities as well as provide value for money, but this alone will not rebalance the economy. A shared prosperity fund like UKSPF has the potential to be the master key that unlocks inclusive growth throughout the country, particularly if it involves less bureaucracy than ESIF and aligns funding more effectively with the priorities of local people. 

In NLGN’s Community Commissioning report, we recommended that this funding should be devolved to communities directly to decide local priorities for the investment. By enabling community ownership of design and administration, the UK government would create an innovative domestic structural funding scheme that promotes inclusion in its process as well as its outcomes.

NLGN’s latest report, Cultivating Local Inclusive Growth: In Practice, highlights the range of policy levers and resources that councils can use to promote inclusive growth in their area. It demonstrates that, through collaboration with communities and cross-sector partners, councils are already doing sterling work to enhance economic and social inclusion. Their efforts could be further enhanced with a fund that learns lessons from ESIF’s successes and flaws: a UKSPF that is easier to access, designed and delivered by local communities, properly funded, and specifically targeted at promoting social and economic inclusion in regions that need it most. “Getting Brexit done” was meant to free up the government’s time to focus once more on pressing domestic priorities. “Getting inclusive growth done” should be at the top of any new to-do list.

Charlotte Morgan is senior researcher at the New Local Government Network.