Tuesday, September 22, 2009

The Long Island Expressway and Brownian Motion

I recently moved back to the east coast and now have to commute to work from New York City out to my college on Long Island. Contrary to expectations, driving against rush hour traffic has not proven to be smooth. Despite leaving the city in the morning and returning in the evening, I have to press my way through 70-90 minutes of dense, mostly highway traffic. I’ve had both the time and the motivation (frustration) to ponder what’s going wrong.
Accidents (probably an average of one a day) snarl traffic, even on the side where they didn’t occur, due to rubber-necking delays (one of my favorite phrases to explain to a foreigner; I have yet to encounter another language with the same concept). But accidents are not the only, or even the main, problem, I think. A kind of coordination problem is. And this brings me to Brownian Motion – i.e. random movements. Of course, the traffic is not moving randomly. That would be truly unpleasant. Not only are cars moving in the same direction, but generally speaking cars in the left hand lane are moving fastest, those in the middle next fastest, and those on the right slowest. But on the margins, random fluctuations play a major role in gumming things up. The fluctuations seem to occur in two forms. I admit that the first kind is not truly random, at least not initially, but I think random fluctuations do play a role after the first disturbance has occurred. Namely, somebody who doesn’t understand the left-lane-is-the-fast-lane rule sits in the left lane, blocking all the speedsters (like myself) who can see beautiful open road ahead, but just can’t get there. It suffices if, say, just one in 20 drivers does not understand this crucial rule in order to slow things down for miles. But this is not a truly random effect; rather, it’s the result of incompetence. However, subsequent reactions do reflect elements of randomness. Namely, speeding drivers, once they come upon the slow poke, can respond in several ways. In an ideal world, each speeding driver would slow down merely to the speed of the slow poke. Many do do this. But with others, random variation kicks in. Some don’t break enough and hit the slow poke. This is, of course, the worst possible outcome (and I’m only speaking of the other drivers here, not of those involved in the collision). Others, no doubt the majority, brake too hard, slowing to, say, five mph below the slow poke’s speed. The chain reaction unleashed by a series of cars behaving this way (i.e. -5-5-5...) leads, ultimately, to standstill, those apparently mysterious cases where everyone comes to a halt, but then after things get going again you can’t figure out why. This first process, then, we might call incompetence-inspired Brownian Motion.
The second is a purer form. Namely, the faster cars will be in the left lane, cruising along at the same high speed. But then the driver of a car somewhere in the chain will let his attention lapse - and this happens not because of incompetence, but because of the basic, random flightiness of our attention-spans - and his speed will either increase or decrease. The results will then mirror what happens in the first case: either an accident, or overreactions by the following cars. The net result is the same.
Everything could flow much more smoothly on the LIE, which would make everyone happier. It's incredibly frustrating to witness - to be stuck in - these unnecessary slow-downs and traffic jams. I bet I could cut my journey from 70-90 minutes down to 50 if they didn't occur. All we’d need to do would be to eliminate incompetent drivers and eliminate Brownian Motion among all drivers. My guess is that the first part of the solution, as hard as it may seem to accomplish, would be easier to achieve than the second. Traffic delays on the LIE may just be an inescapable feature of the universe’s architecture. So maybe I shouldn't feel frustrated, after all.(Or would there be some way to set up Biased Brownian Motion on the highways?)