## 3.9.1 Spread Process Yielding the Binomial PMF
Rogers has studied two particular processes--one spread and one
clustered--that have appealing time-Poisson process interpretations and that have been
found useful in analyzing the locations of retail trade [ROGE 74]. We consider first
Rogers's spread process, the binomial process. Imagine that entities enter the cell of
interest over some time interval [0, t], initially with 0 entities in the cell. We are
interested in the number of entities at time t, N(t). Being a spread process, each time
that another entity enters the cell the rate at which new entities enter the cell
diminishes. Thus, suppose initially that entities enter the cell as a time-Poisson process
at rate c per unit time. Then, after the first enters the cell, the cell becomes
"less attractive," so the new Poisson arrival rate is c - b. In general, after k
arrivals, the Poisson arrival rate is reduced to c - kb. Thus, the cell becomes less
attractive in a linear manner with the number of entities already in the cell. We assume
that c/b is integer, so that there exists some maximum k, k_{max} = c/b, at which
the Poisson arrival rate is reduced to c - k_{max}b = 0 Thus, the maximum number
of entities in a cell is k_{max} = c/b. This pure birth process is characterized
by the state-transition diagram shown in Figure 3.36.
Proceeding as in Chapter 2 for the Poisson process, this process is
governed by the following set of coupled differential equations:
which is always less than unity (which is what we want with a spread process).
While the "diminishing attractiveness" interpretation of this
birth process is perfectly valid, and quite appealing as a description of the dynamics of
a spread process, it is not the only interpretation of the process. Alternatively, one
might imagine a population fixed with n = c/b individuals. Each one will eventually locate
within the cell, but the time until such location is an exponentially distributed random
variable with mean 1/b. All n such random variables are *mutually independent*. Thus,
at time t = 0, n "Poisson generators" are turned on, yielding a rate of
transition nb from state 0 to 1; after the first transition, (n - 1) Poisson generators
remain turned on with a net rate of occurrence equal to (n - 1)b. This "fixed
population" interpretation of cell occupancy also yields the binomial distribution,
and it could imply markedly different policy decisions in practice than the
"diminishing attractiveness" interpretation. The two equally plausible
interpretations provide a good example that any particular probability law may have two,
three, or even a greater number of plausible underlying explanations. Thus, just because a
probability law assumes a particular form does not assure us that one underlying causal
model is *the* model explaining the process dynamics. |