(I asked this question on Reddit a while ago, but nobody could come up with an answer.)
Given a one-lane road of length D, and given that your initial velocity is V, and given that cars enter the one-lane road with a frequency density f(t), and given that a random car's uninterrupted speed u follows a probability density g(u), what is the expected amount time for a driver to drive the length of the road? Assume that one car cannot pass another. Assume also that there are no intersections on the road, and that all cars enter the road at the same point, in the same direction. Also, you start driving at time t=0. Keep in mind that your speed is affected only by slower cars which started driving before you.
To clarify: The probability that a car's speed is 0 is assumed to be 0. And when a faster car meets up with a slower car, it travels at the slower car's speed: The speed of a car doesn't change unless this happens.
I thought of this problem while driving along a one-lane road. I've been thinking about it for a long time now, and I can attest that it's a deceptively hard problem. I'm pretty sure that the answer will be in the form of a triple or quadruple integral. If any of you can figure it out, please let me know.