3.9.2 Clustered Process Yielding the Negative Binomial PMF

Rogers's clustered spatial process gives rise to the negative binomial distribution. In this model we assume that a cell becomes more attractive with each additional entity that locates there. In particular, if there are m entities there at time t, new entities arrive in a (time) Poisson manner at rate c + bm (c > 0, b > 0). The state-transition diagram for this infinite-state pure birth process is shown in Figure 3.37. Proceeding as usual, the set of coupled differential equations governing this process are

This is the negative binomial pmf [See, for example, A. W. Drake, Fundamentals of Applied Probability, McGraw-Hill, Inc., New York, 1967, pp. 129-130,153.] with mean

                      E[N(t)] = (c/b)(ebt - 1    (3.117)

and variance

                      2N(t) = (c/b)(ebt - 1)ebt  (3.118)

The ratio of the mean to the variance is

                              r = ebt		 (3.119)

which is always greater than unity (which is what we want with a clustered process).
    Although the "increasing attractiveness" interpretation of this process is appealing for a clustering process, there are other plausible system dynamics yielding the same negative binomial pmf.
    For comparative purposes, we have sketched the mean value E[N(t)] for each of the three cell occupancy laws- Poisson, binomial, negative binomial--in Figure 3.38. Note that the binomial (spread) process reaches a

"saturation" population, P_sat, whereas the Poisson process grows linearly in time and the negative binomial (clustered) process "explodes" at an exponential rate. Figure 3.39 illustrates each process over a 10- by 10-kilometer city.