One-lane Highway

You have N cars that are all traveling the same direction on an infinitely long one-lane highway. Unfortunately, they are all going different speeds, and cannot pass each other. Eventually the cars will clump up in one or more traffic jams.
In terms of N, what is the expected number of clumps of cars?

My Solution

We have N cars, the expected location of the slowest car is in the middle thus forming one group from the back up to the mid point (we have at least 1 group), consider all the cars ahead of this group as the new subset M, perform the same operation and we'll have another group that goes up to the 3/4 N mark from the back for 2 groups. We can do this as far as till the next subset X is at least 1 car long. So for N starting cars we'd expect X number of clumps where 2^X = N giving X = lnN/ln2. (This makes more sense for N being a large number and doesn't make sense at all for N = 2 as in that situation we'd expect 1.5 groups where as my formula would give 1) -- Dominic Leung 31 Aug 2009