The Bus Problem

Having waited for many buses the question arose of why it always seems that a bus comes first on the other side of the road.

Assuming that it is the same buses running in a circle on the route then we know that the average probability of a bus on either side is the same (call it p).

However the fun occurs if there is any synchrony. Suppose that there are no delays and the buses are scheduled to arrive at 10mins past the hour (PTH) on the other side and 20 PTH on your side. In order to see your bus come first you need to arrive between 10 and 20 PTH which is only 1/6th of the time. At such a bus stop you will experience an apparent “unfairness”.

In reality buses set off on time from the depot but synchrony happens en route. The reason is that the number of passenger to board a bus is proportional to the time since the last bus, and the time spent at the stop is proportional to the number of passengers. Any delays thus feedback positively. A large party of people turning up at the stop, or someone looking for their ticket will delay the bus. It means they have more people to pick up at the next stop, and the bus behind will have caught up a bit and so have less people to pick up and so very quickly it will be following the bus in front. This is why buses never come singly (especially toward the end of the day or the bus route). This means that there is a different grouping of buses for people waiting on the outward and return routes of a bus route.

2Do…

I still need to do the maths here…

The other thing I wanted to do was use appropriate distributions to show the exact probability of n buses passing on the other side…