3.8.2 TimeSpace Poisson ProcessSuppose that in some region of space S with area A(S) events occur in time as a Poisson process with rate A(S) per unit time. Then utilizing the foregoing ideas about multidimensional Poisson processes, the probability that k events occur in S in time t is Problem 3.26 applies this concept. Example 15: Distribution of Travel Distance ("Nearest Neighbor") Suppose that emergency response units are distributed throughout a large region as a twodimensional Poisson process with intensity parameter y units per square mile. We wish to know the pdf of the travel distance D between an incident, whose position is selected independently of response unit positions, and the nearest response unit. Assume Euclidean travel distance. (This is sometimes known as a "nearestneighbor" problem; in threedimensional space this problem has been used to determine the distribution of distance between stars in a galaxy.) Solution We use the neverfail cumulative distribution method in conjunction with our new knowledge of spatial Poisson processes. This is a Rayleigh pdf with parameter Thus, the mean and variance are Question: How could you extend these ideas to obtain other interesting properties of the system? Example 16: Nearest Neighbor with RightAngle Travel Distance If travel distance is rightangle, rather than Euclidean, the analysis in Example 15 follows straight through, except instead of a circle of radius r we have a square rotated at 45°, centered at (x, y), with area equal to 2r^{2} . Following the same steps in the solution, This is a Rayleigh pdf with parameter The mean and variance are Question: In Example 4 in this chapter we derived that in an isotropic environment a responseunit traveling according to the rightangle distance metric travels 4/ = 1.273 times farther (on the average) than a unit traveling "as the crow flies." Thus, one might be tempted to think that the ratio of the mean rightangle to Euclidean distances computed in Examples 15 and 16 would be 1.273. In fact, the ratio is . Why? Hint: See Problems 3.9 and 3.10. Further work: Problems 3.25 and 3.26.
