5.17 Coverage problem for urban service systems Consider a collection of response units in the plane whose positions are distributed according to a spatial Poisson process with parameter A(S), where A(S) denotes the area of the region S. Each unit is available for dispatch with probability (1 - ), independent of the status of all other units. Units have different response speeds: the (Euclidean) distance that a randomly chosen available unit can travel in a time T is determined by the probability density function fR(r, T) (T fixed). Show that the number of available response units which can travel to a random incident in the time T is a Poisson random variable with parameter

Hint: Define the family of random variables C(r, T) the number of available response units that can get to an incident in time T and that are located at a distance less than r from the incident. Show that the family C(r, T) determines a time-varying Poisson process where the time variable is taken to be the distance r. To do this, prove that a "Poisson event" occurring in a ring between r and r + dr centered at the incident has probability fR(x, T) dx, and events occurring over disjoint intervals constitute independent random variables. Then show that C(r, T) is a variable-time (nonhomogeneous) Poisson process with parameter