Two particles start out at random positions on a unit-circumference circle.
Each has a random speed (distance per unit time) moving counterclockwise uniformly distributed
within $[0,1]$. How long until they occupy the same position? In the example below,
the red particle catches the green particle at $t=5.9$, i.e., nearly six times around the circle:

The distribution of overtake-times is quite skewed, indicating perhaps
the mean could be $\infty$. For example, in one simulation run, it
took more than $3$ million times around the circle before one particle finally caught
the other. So I don't trust the means I am seeing (about $25$).

What is the distribution of overtake-times?

I was initially studying $n$ particles on a circle, but $n=2$ seems already somewhat interesting...

**Update** (*2Dec12*).
Alexandre Eremenko concisely established that the expected overtake-time (the mean)
is indeed $\infty$.
But I wonder what is the median, or the mode? Simulations suggest the median is about $1.58$ and
the mode of rounded overtake-times is $1$, reflecting a distribution highly skewed toward rapid overtake. (The median is suspiciously close to $\pi/2$ ...)

**Update** (*3Dec12*). Fully answered now with Vaughn Climenhaga's derivation of the
distribution, which shows that the median is
$1 + \frac 1{\sqrt{3}} \approx 1.577$.