5.16 Linking travel times infinite regions to spatial Poisson processes In this problem we wish to explore the validity of the spatial Poisson process model as we increase the number of independently, uniformly located response units in our area.

  1. If a unit n is uniformly distributed over a square region of area N and if the incident is located at the center of the square, find the probability density function of Dn, the travel distance for unit n to reach the incident. (Assume that we have right-angle response parallel to the sides of the square.)

  2. Assume that there are N such units in the region, indexed from n = 1 to n = N. The minimum travel time to an incident is
    RN = Min [D1, D2, . . ., DN]
    But the spatial Poisson assumption implies that as N gets large, the pdf for RN approaches a Rayleigh with parameter 2. The cumulative distribution function for a Rayleigh random variable with parameter 2 is

    Thus, if the spatial Poisson model is correct for large N, we must have

    Prove (*).

  3. Explain briefly how your analysis in part (b) is modified if you do not condition on the position of the incident being at the center of the square.

  4. How might we use this result in developing an approximate model for travel time in a finite homogeneous region with N response units, demand rate (incidents/hour), average service time -1, and a service discipline that assigns units completing service to the closest waiting call ?