Why do you wait ages for a bus, then three come along at once?

You wait ages for a bus, then three come at once (Photo by State Records NSW - CC-BY)

You wait ages for a bus, then three come along at once. It sounds like a joke, but there’s an element of truth to it.

Imagine a busy bus route, with a service frequent enough that passengers arrive at the bus stop at random times rather than to meet a specific bus. Normally you don’t need to wait long before a bus arrives. If there is a bus every ten minutes, your average wait will be five minutes.

Suppose the next bus encounters a slight delay early in its route, which puts it a couple of minutes late. At the next stop, there will be 20% more passengers (because they have arrived during the delay, after the time at which the bus was due). Those extra passengers increase the time needed for embarking, paying the fare, and disembarking, so the bus becomes even later. And as the bus becomes later, the time needed at each stop increases further.

Soon the delay is such that the next bus is just behind it. We might say colloquially that the next bus has “caught up”, but it is just on its regular schedule.

If the roads are wide, the bus in front will stop to pick up passengers while the one behind overtakes it. In this way the buses will “share the load” and won’t be likely to lose further time. But if there is any congestion, or if the road is too narrow for easy overtaking, or if bus company policy requires the buses to stay in the correct sequence, the bus in front will still be picking up more passengers than average while the one behind is delayed by it.

It won’t be long until these two buses are so delayed that another one has caught up behind them.

And so, the people who found themselves waiting “ages” for a bus were the ones who were met by “three coming at once”.

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