Get involved:
It鈥檚 a popular clich茅 to say that you wait ages for a bus, and then three turn up at once. Is there any truth to this? Or is it a false impression formed because we notice coincidences more than other events? If true, are there laws governing this behaviour, and are there any natural phenomena that obey the same principles?
鈥� The technical term for several buses arriving at once is 鈥渂unching鈥�, and the reason for it is quite simple.
Advertisement
If for some reason a bus is delayed by a few minutes, there will probably be more people waiting for it than on average. This is especially true when the frequency of buses on a particular route is high enough (one every 10 minutes, say) that passengers tend to arrive at stops randomly rather than according to the timetable.
Any late running will therefore increase the time a bus has to pause to pick up passengers at a stop, especially if the bus driver has to sell or validate tickets on entry. The late bus is therefore made slightly later still. This effect is compounded at each stop, causing more and more people to be waiting, delaying it even more.
Meanwhile, the next bus on that route is getting a pretty quick run because many of the passengers it would have picked up are on the late-running bus. Eventually it catches up the bus in front and, if it doesn鈥檛 overtake it, we are left with two buses trundling along together, with the next service behind catching up on them too.
鈥淓ventually the next bus catches up on the late-runner and we are left with two buses trundling along鈥�
There are a couple of solutions to this. The most obvious and widely practised is to include 鈥渢iming points鈥� along the route 鈥� stops where a bus is scheduled to wait for a few minutes before continuing. A late-running bus may ignore this wait and so make up a few minutes, but on the downside a bus that is on time will put passengers through an unnecessary delay whenever it reaches a timing point.
Another solution is to give a bus priority at traffic lights (or some other priority over other vehicles), but only if it is running late. This allows the bus to make up time, after which it no longer requires privileged treatment. While this helps public transport run to schedule, it too has its disadvantages, including the cost of implementation and the disruption to schemes designed to optimise overall traffic flow.
Dean Purkis, Eltham North, Victoria, Australia
鈥� Inevitably at some point on the route there will be an unusually large number of passengers waiting; after all, they do not arrive at stops at a constant rate. The first bus along will have to deal with the crowd as best it can, while the bus behind it will have a smaller load to pick up and can go faster. As their journeys progress, this disparity becomes more pronounced until eventually the bus behind catches up.
This is a mathematical inevitability, not a matter of friendly drivers attempting to travel in packs. For further information you may want to read Why Do Buses Come in Threes? The hidden mathematics of everyday life by Robert Eastaway and Jeremy Wyndham (Robson Books/Wiley, 2005) or the briefer explanation in my book What Are The Chances? Voodoo deaths, office gossip and other adventures in probability (Johns Hopkins University Press, 2002).
Bart Holland, New York City, US