## MatheMUSEments

Articles for kids about math in everyday life, written by Ivars Peterson for Muse magazine.

## May 5, 2007

### Bunching Buses

You're standing at a bus stop, waiting for a bus to arrive. You wait and you wait. There's supposed to be a bus every 10 minutes, but you haven't seen one for at least 20 minutes. Finally, a bus arrives. It's full of people. Just as you try to squeeze yourself in, you spot another bus coming down the street, and another one behind it!

What's going on? Why do buses always seem to come in bunches instead of at regular intervals?

Some people claim that bus bunching doesn't happen very often. They say that passengers tend to remember the few times when more than one bus arrives at a stop and to forget the many more times when a bus arrives alone.

Mathematicians who study traffic, however, say that bunching really can happen.

The problem is the people at the stops. If there were always the same number of people, the buses would keep to their schedule. But usually there are many people at a few stops and no one at others.

Suppose many people happen to gather at a particular stop. It takes longer than usual for the passengers to board the first bus that arrives, so it gets delayed and the bus behind it catches up a bit. When the second bus arrives at the same stop, there has been less time for passengers to assemble, so the bus goes on its way quickly. Meanwhile, the first bus arrives at its next stop a little later than usual, so there's more time for passengers to join the crowd, and so on. After several more stops, the second bus catches up with the first.

Once one bus catches up with the other bus, the two buses end up traveling together. If the route is a long one, a third bus could eventually catch up with the first two.

Check it out the next time you hop a bus!

Muse, December 2001, p. 39.