# The Universe of Discourse

Tue, 14 Aug 2018

A few years ago Katara was very puzzled by traffic jams and any time we were in a traffic slowdown she would ask questions about it. For example, why is traffic still moving, and why does your car eventually get through even though the traffic jam is still there? Why do they persist even after the original cause is repaired? But she seemed unsatisfied with my answers. Eventually in a flash of inspiration I improvised the following example, which seemed to satisfy her, and I still think it's a good explanation.

Suppose you have a four-lane highway where each lane can carry up to 25 cars per minute. Right now only 80 cars per minute are going by so the road is well under capacity.

But then there is a collision or some other problem that blocks one of the lanes. Now only three lanes are open and they have a capacity of 75 cars per minute. 80 cars per minute are arriving, but only 75 per minute can get past the blockage. What happens now? Five cars more are arriving every minute than can leave, and they will accumulate at the blocked point. After two hours, 600 cars have piled up.

But it's not always the same 600 cars! 75 cars per minute are still leaving from the front of the traffic jam. But as they do, 80 cars have arrived at the back to replace them. If you are at the back, there are 600 cars in front of you waiting to leave. After a minute, the 75 at the front have left and there are only 525 cars in front of you; meanwhile 80 more cars have joined the line. After 8 minutes all the cars in front of you have left and you are at the front and can move on. Meanwhile, the traffic jam has gotten bigger.

Suppose that after two hours the blockage is cleared. The road again has a capacity of 100 cars per minute. But cars are still arriving at 80 per minute, so each minute 20 more cars can leave than arrived. There are 600 waiting, so it takes another 30 minutes for the entire traffic jam to disperse.

This leaves out some important second-order (and third-order) effects. For example, traffic travels more slowly on congested roads; maximum safe speed decreases with following distance. But as a first explanation I think it really nails the phenomenon.