I did my graduate work at the University of Chicago, and lived in Hyde Park. On occasion I would take the bus (the #6 Jeffery Express) to downtown. Although the buses were scheduled to run every 15 minutes, I would invariably end up waiting a half hour. Sometimes more. Often in the freezing cold, or the sweltering heat. Most infuriatingly, when the bus finally arrived, there was always another one immediately behind it! The buses inevitably came in pairs. Sometimes even in triples or quads.

Let’s assume that the buses are supposed to arrive every 15 minutes. If the buses adhered to their schedule, and I showed up at a random time, I should generally have to wait roughly half the mean bus arrival time: 7.5 minutes. If the buses were totally random, then I would have to wait the average time between bus arrivals: 15 minutes (if you haven’t thought about this before, this statement should sound crazy; perhaps I’ll do a future post on it). So the question is: why did I always end up waiting roughly 30 minutes or more?

I always assumed that the Universe was conspiring against me. This is a common feeling in graduate school. However….

I just stumbled across a blog post of a friend of mine from graduate school, Alex Lobkovsky. In it, he discusses precisely this problem, and presents various reasons for the bunching of buses. I have no doubt that he was inspired from similar suffering. Perhaps at the very same bus stop.

At the end of the day, there’s a fairly straightforward solution. Imagine all of the buses are roughly on time. Now imagine that one bus (call it bus S) happens to fall behind. Because S is running behind, more time has elapsed since the previous bus has passed. This means that more waiting passengers have accumulated, at more bus stops. This in turn means that bus S has to stop more often, and has to pick up more people at each stop. Hence, bus S falls even farther behind. Which means even more people accumulate at each stop. Which means the bus falls even farther behind. And so on. In short: a slow bus gets slower and slower.

Now let us consider the bus behind bus S; we’ll call it bus F. Bus F starts out roughly on schedule. But because bus S is running late, less time than average has elapsed between when bus S last passed and when bus F arrives. This means fewer people have accumulated, at fewer stops. Which means bus F makes fewer stops, and picks up fewer people. Which means that it starts to run faster than average. Which means even fewer people accumulate. Which means it runs even faster. And so on. In short: a fast bus gets faster and faster.

Putting this all together: if a random fluctuation creates a slow bus, then it will get slower and slower, and the bus behind it will get faster and faster, until the two buses meet up. At this point, the buses stick together, and are essentially incapable of separating. Thus, in general, buses will bunch up. This will usually happen in pairs, though on occasion triples and even quads may occur. This argument predicts that the arrival of buses will be random, with pairs of buses arriving more often than not, being separated by on average double the mean bus separation. And this is precisely what I discovered, the hard way, shivering at the corner of 55th St. and Hyde Park Boulevard. (N.B. I spent a year in Berlin. There, the buses are fermions, and always arrive exactly on time. It’s the stereotype, but it turns out to be true.)

After writing this post, I found that wikipedia has already figured it all out. Regardless, it’s nice to know that my suffering was due to statistics, and not because the Universe is out to get me.