Today’s question: Why do buses on central city routes always come in bunches of two or three at a time.
Here’s an intuitive proof.
It seems reasonable to argue, that in the absence of traffic and any other reason for cars or buses to stop on the road, buses should arrive with roughly the same timing that they started on their route. So it has to be some characteristic of a city route, that violates the above condition, that causes the clustering of buses.
One such characteristic immediately comes to mind: traffic lights. It is plainly observable that — even for ordinary sized vehicles — traffic lights tend to bunch them into groups. The lights break the flow of traffic artificially at several points within the flow, causing several cars to stop together at the same point. However, in the case of ordinary cars, those at the head of the traffic flow, tend to stay at the head, and those at the rear, likewise tend to stay at the rear. If we were to simply sprinkle some buses into the mix, then the buses should stay roughly equidistant from each other in the flow of traffic.
So what is it about buses that is different from ordinary cars. I would say a couple things:

1. They can’t avoid traffic obstacles as easily. They are big and slow, so seldom do you see them passing a slower vehicle. If there is some kind of traffic obstruction, normal cars can easily evade it, but often buses will get stuck behind.
2. Buses on the same route don’t pass each other. This is more of an artificial rule of bus management. Because they are on a schedule, they believe it does not make sense for a later bus to pass an earlier one, so if one bus gets behind, then the probability of other buses to get stuck behind it only increases.

So in combination, here’s the basic thesis:

• Traffic control has a intrinsic clustering effect.
• Buses tend to get slowed down for reasons that normal cars don’t have to deal with, or can deal with more efficiently.
• Buses can’t pass other slow buses in front of them, because of their schedule.

These three combine to result in the phenomenon that a bus that normally comes every ten minutes, will be 30 minutes late, but then 3 empty buses will arrive at the same time.