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.

 
 
 
 

12 things we learned by reading every single National Rail timetable

Some departure boards, yesterday. Image: flickr.com/photos/joshtechfission/ CC-BY-SA

A couple of weeks ago, someone on Twitter asked CityMetric’s editor about the longest possible UK train journey where the stations are all in progressive alphabetical order. Various people made suggestions, but I was intrigued as to what that definitive answer was. Helpfully, National Rail provides a 3,717 page document containing every single timetable in the country, so I got reading!

(Well, actually I let my computer read the raw data in a file provided by ATOC, the Association of Train Operating Companies. Apparently this ‘requires a good level of computer skills’, so I guess I can put that on my CV now.)

Here’s what I learned:

1) The record for stops in progressive alphabetical order within a single journey is: 10

The winner is the weekday 7.42am Arriva Trains Wales service from Bridgend to Aberdare, which stops at the following stations in sequence:

  • Barry, Barry Docks, Cadoxton, Cardiff Central, Cardiff Queen Street, Cathays, Llandaf, Radyr, Taffs Well, Trefforest

The second longest sequence possible – 8 – overlaps with this. It’s the 22:46pm from Cardiff Central to Treherbert, although at present it’s only scheduled to run from 9-12 April, so you’d better book now to avoid the rush. 

  • Cardiff Central, Cardiff Queen Street, Cathays, Llandaf, Radyr, Taffs Well, Trefforest, Trehafod

Not quite sure what you’ll actually be able to do when you get to Trehafod at half eleven. Maybe the Welsh Mining Experience at Rhondda Heritage Park could arrange a special late night event to celebrate.

Just one of the things that you probably won't be able to see in Trehafod. Image: Wikimedia/FruitMonkey.

There are 15 possible runs of 7 stations. They include:

  • Berwick Upon Tweed, Dunbar, Edinburgh, Haymarket, Inverkeithing, Kirkcaldy, Leuchars
  • Bidston, Birkenhead North, Birkenhead Park, Conway Park, Hamilton Square, James Street, Moorfields
  • Bedford, Flitwick, Harlington, Leagrave, Luton, St Albans City, St Pancras International

There is a chance for a bit of CONTROVERSY with the last one, as you could argue that the final station is actually called London St Pancras. But St Pancras International the ATOC data calls it, so if you disagree you should ring them up and shout very loudly about it, I bet they love it when stuff like that happens.

Alphabetical train journeys not exciting enough for you?

2) The longest sequence of stations with alliterative names: 5

There are two ways to do this:

  • Ladywell, Lewisham, London Bridge, London Waterloo (East), London Charing Cross – a sequence which is the end/beginning of a couple of routes in South East London.
  • Mills Hill, Moston, Manchester Victoria, Manchester Oxford Road, Manchester Piccadilly – from the middle of the Leeds-Manchester Airport route.

There are 20 ways to get a sequence of 4, and 117 for a sequence of 3, but there are no train stations in the UK beginning with Z so shut up you at the back there.

3) The longest sequence of stations with names of increasing length: 7

Two of these:

  • York, Leeds, Batley, Dewsbury, Huddersfield, Manchester Victoria, Manchester Oxford Road
  • Lewes, Glynde, Berwick, Polegate, Eastbourne, Hampden Park, Pevensey & Westham

4) The greatest number of stations you can stop at without changing trains: 50

On a veeeeery slow service that calls at every stop between Crewe and Cardiff Central over the course of 6hr20. Faster, albeit less comprehensive, trains are available.

But if you’re looking for a really long journey, that’s got nothing on:

5) The longest journey you can take on a single National Rail service: 13 hours and 58 minutes.

A sleeper service that leaves Inverness at 7.17pm, and arrives at London Euston at 9.15am the next morning. Curiously, the ATOC data appears to claim that it stops at Wembley European Freight Operations Centre, though sadly the National Rail website makes no mention of this once in a lifetime opportunity.

6) The shortest journey you can take on a National Rail service without getting off en route: 2 minutes.

Starting at Wrexham Central, and taking you all the way to Wrexham General, this service is in place for a few days in the last week of March.

7) The shortest complete journey as the crow flies: 0 miles

Because the origin station is the same as the terminating station, i.e. the journey is on a loop.

8) The longest unbroken journey as the crow flies: 505 miles

Taking you all the way from Aberdeen to Penzance – although opportunities to make it have become rarer. The only direct service in the current timetable departs at 8.20am on Saturday 24 March. It stops at 46 stations and takes 13 hours 20 minutes. Thankfully, a trolley service is available.

9) The shortest station names on the network have just 3 letters

Ash, Ayr, Ely, Lee, Lye, Ore, Par, Rye, Wem, and Wye.

There’s also I.B.M., serving an industrial site formerly owned by the tech firm, but the ATOC data includes those full stops so it's not quite as short. Compute that, Deep Blue, you chess twat.

10) The longest station name has 33 letters excluding spaces

Okay, I cheated on this and Googled it – the ATOC data only has space for 26 characters. But for completeness’ sake: it’s Rhoose Cardiff International Airport, with 33 letters.

No, I’m not counting that other, more infamous Welsh one, because it’s listed in the database as Llanfairpwll, which is what it is actually called.

 

This sign is a lie. Image: Cyberinsekt.

11) The highest platform number on the National Rail network is 22

Well, the highest platform number at which anything is currently scheduled to stop at, at least.

12) if yoU gAze lOng into an abYss the abySs alSo gazEs into yOu

Image: author's own.

“For I have seen God face to face, and my life is preserved”, said Thomas.

Ed Jefferson works for the internet and tweets as @edjeff.

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