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.

 
 
 
 

17 things the proposed “Tulip” skyscraper that London mayor Sadiq Khan just scrapped definitely resembled

Artist's impression. See if you can guess which one The Tulip is. Image: Foster + Partners.

Sadiq Khan has scrapped plans to build a massive glass thing in the City of London, on the grounds it would knacker London’s skyline. The “Tulip” would have been a narrow, 300m skyscraper, designed by Norman Foster’s Foster & Partners, with a viewing platform at the top. Following the mayor’s intervention, it now won’t be anything of the sort.

This may be no bad thing. For one thing, a lot of very important and clever people have been noisily unconvinced by the design. Take this statement from Duncan Wilson, the chief executive of Historic England, from earlier this year: “This building, a lift shaft with a bulge on top, would damage the very thing its developers claim they will deliver – tourism and views of London’s extraordinary heritage.”

More to the point, the design was just bloody silly. Here are some other things that, if it had been built, the Tulip would definitely have looked like.

1. A matchstick.

2. A drumstick.

3. A cotton ear bud.

4. A mystical staff, of the sort that might be wielded by Gandalf the Grey.

5. A giant spring onion.

6. A can of deodorant, from one of the brands whose cans are seemingly deliberately designed in such a way so as to remind male shoppers of the fact that they have a penis.

7. A device for unblocking a drain.

8. One of those lights that’s meant to resemble a candle.

9. A swab stick, of the sort sometimes used at sexual health clinics, in close proximity to somebody’s penis.

10.  A nearly finished lollipop.

11. Something a child would make from a pipe cleaner in art class, which you then have to pretend to be impressed by and keep on show for the next six months.

12. An arcology, of the sort seen in classic video game SimCity 2000.

13. Something you would order online and then pray will arrive in unmarked packaging.

14. The part of the male anatomy that the thing you are ordering online is meant to be a more impressive replica of.

15. A building that appears on the London skyline in the Star Trek franchise, in an attempt to communicate that we are looking at the FUTURE.


14a. Sorry, the one before last was a bit vague. What I actually meant was: a penis.

16. A long thin tube with a confusing bulbous bit on the end.

17. A stamen. Which, for avoidance of doubt, is a plant’s penis.

One thing it definitely does not resemble:

A sodding tulip.

Anyway, it’s bad, and it’s good the mayor has blocked it.

That’s it, that’s the take.

(Thanks to Anoosh Chakelian, Jasper Jackson, Patrick Maguire for helping me get to 17.)

Jonn Elledge is editor of CityMetric and the assistant editor of the New Statesman. He is on Twitter as @jonnelledge and on Facebook as JonnElledgeWrites.

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