Poisson Traffic Clumps and Spatial Gaps

Cars pass a fixed point on a long highway according to a Poisson process with rate $\lambda$ (cars per unit time). Define a clump as a maximal consecutive run of cars whose inter-arrival times are all at most $T$ ; a new clump starts immediately after an inter-arrival time exceeding $T$ . (a) What is the distribution of the spatial gap between successive clumps if each car travels at constant speed $v$ (so time gaps map to space gaps)? (b) Using the Poisson clumping heuristic, approximate the expected number of clumps per mile.

$G$ is uniform on $[0,vT]$ , and the expected clumps per mile are about $1/(vT)$ .

$G \sim \operatorname{Exp}(\lambda e^{-\lambda T}/v)$ , and the expected clumps per mile are about $\lambda e^{-\lambda T}/v$ .

$G \sim \operatorname{Exp}(\lambda/v)$ , and the expected clumps per mile are about $\lambda/v$ .

$G \sim \operatorname{Exp}(e^{-\lambda T}/v)$ , and the expected clumps per mile are about $e^{-\lambda T}/v$ .