3.9.2 Clustered Process Yielding the Negative Binomial PMFRogers'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 statetransition diagram for this infinitestate 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, McGrawHill, Inc., New York, 1967, pp. 129130,153.] with mean E[N(t)] = (c/b)(e^{bt}  1 (3.117) and variance ^{2}_{N}(t) = (c/b)(e^{bt}  1)e^{bt} (3.118) The ratio of the mean to the variance is r = e^{bt} (3.119) which is always greater than unity (which is what we want with a clustered process). "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 10kilometer city.
