On some models leading to the generalized poisson distribution

A new generalization of the Poisson distribution was given by Consul and Jain (1970, 73). Since then more than twenty papers, written by various researchers, have appeared on this model under the titles of Generalized Poisson Distribution (GPD), Lagrangian Poisson distribution or modified power series distribution. Here the author provides two physical models, based on differential-difference equations, which lead to the GPD. A number of axioms are given for a steady state point process which produce the generalized Poisson process. Also, the GPD is derived as the limiting distribution of the two quasi-binomial distributions based on urn models.