3.25 Spatial Poisson process Suppose that response units are distributed throughout the city as a homogeneous spatial Poisson process, with an average of y response units per square mile. Assume that the travel time between (x1, y1) and (X2, Y2) is

pg173b.gif (2314 bytes)

where vx. and vy are travel speeds in the directions of the abscissa and ordinate, respectively.
    Assume that an incident occurs somewhere in the city, independent of the locations of the response units.

a. Find the pdf for Tk, where

Tk = travel time to the kth nearest response unit, kth = 1, 2, . . .

Hint: The set of points that are a given travel time from the incident is given by a diamond centered at the incident.

b. Find E[Tk] and pg172c.gif
(1014 bytes). and note their functional dependence on k.