Branching Processes with Negative Poisson Offspring Distribution

A branching process is a mathematical description of the growth of a population for which the individual produces offsprings according to stochastic laws. A typical problem would be of the following form. Considering a population of individuals developing from a single progenitor, the initial individual. The initial individual produces a random number of offsprings, each of them in turn produces a random number of offsprings; and so the process continues as long as there are live individuals in the population. An interesting problem is to find the probability that the population survives, (or, extincts). The branching process was first proposed by Galton [4], and the extinction probability was first obtained by Watson [14] by considering the probability generating function of the number of children in the nth generation. This mathematical model was known as Galton-Watson branching process, and had been studied thoroughly in literature, for example, [12, 2, 5, 6, 9]. For the interesting details on the early history of branching processes, please consult [8]. Another model for the branching process was based on the interpretation of the random walk Sn − n and the branching process in terms of queuing theory which is due to Kendall [7]. Here Sn = X1+X2+· · ·+Xn, whereXi’s are independent random variables with the identical distribution as the offsprings. This model can be formulated as follows. Let Yt be the number of individuals at time t. Initially there is only one individual, i.e., Y0 = 1. At each time unit, we select a living individual, and replace it by its children. In formula,