Cars arrive at a gate. Each car is of random length L having distribution functi

Question

Cars arrive at a gate. Each car is of random length L having distribution function F. The first car arrives and parks against the gate. Each succeeding car parks behind the previous one at a distance that is random according to a uniform distribution [0,1]. Consider the number of cars Nx that are lined up within a total distance x of the gate. Determine for F a degenerate distribution of length c, i.e., it is deterministic, and also for the case that the car length is exponentially distributed with F(l) = 1 - e-l . Determine the limiting probability that a location is empty.

Explanation / Answer

a. 2/(1+2c)

b. 2/3

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