My companera and I were recently waiting for the 42 on Spruce Street. A long time went by before we noticed the sign announcing the detour. Seems the 42 was being detoured down to Baltimore Ave because the U/Penn kids were moving in for the new semester. And we were forced to walk three blocks down the street to catch it.

Half a block away, though, we saw the 42 come barreling down Spruce Street, detour be damned. We raced to catch up with it.

"I thought you were on detour," I asked the driver.

"S'posed to be. But I could get through, and figured people wouldn't want to walk down to Baltimore," she replied.

Good for us, I guess, but sorry about the folks who obeyed the detour signs and were waiting down on Baltimore.

But that gave me a chance to think about this latest bus research I came across. Seems that some scientists at California Institute of Technology and Harvard recently took on the big question of whether it was better to wait for a bus in one spot or walk down the street to try to catch up with it later.

First, the problem:

Recently, the first author (Justin) had to walk to the second author (Scott)’s house to work on a problemset. There is a bus route which covers the distance directly, but the bus arrives sporadically.

So, he faced a conundrum: walk the distance or wait for the bus? Being lazy, he would always rather ride the bus, if possible. Being punctual, however, he will always choose the option which gets him to his destination as quickly as possible.

Formally, this problem can be stated as follows: Justin has to travel a distance of d miles along a bus route. (The units are arbitrary, but will ground our discussion.) Along this route, there are n bus stops i, each spaced at a distance of di from the starting point. Without loss of generality, we assume the starting point is the first bus stop, so that d1 = 0 and that the destination is the last bus stop, so that dn = d. At each bus stop, Justin is faced with a choice: to walk or to wait. If he walks on, he can still catch a bus at the next bus stop.

Justin can walk at a velocity of vw miles per hour and the bus drives at a velocity of vb > vw miles per hour. He must make a decision at the starting point at time t = 0. He expects that a bus will arrive at the starting point at time t with probability density function p(t), independently distributed across t.

And what did the find? Well, as any (lazy) bus rider will tell you, it's better to wait for the bus in one place.

Now let's see if someone can do a study about the merits of waiting for the BSL express vs. taking the local now arriving.

You can read the three-page study here.

(image credit: flickr user timailius.)