Some Properties of Presence Detectors

Presence detectors on highways record whether or not there are any cars on some nonzero length section of highway. The absence of cars is used as a criterion for switching a vehicle-actuated signal. The problem considered here is the following. Cars pass the detector according to a Poisson process, at velocities that are independent, identically distributed random variables. Starting at some arbitrary time origin, how long will it be before the detector of length L is empty for the first time. It is shown that, in most cases, this time is nearly the same as if all cars had the same velocity q/k, q = flow, k = spacial density, and is rather insensitive to the distribution of the velocities.