A generalized Conway-Maxwell-Poisson distribution which includes the negative binomial distribution

The Conway-Maxwell-Poisson (COM-Poisson) distribution with two parameters was originally developed as a solution to handling queueing systems with state-dependent arrival or service rates. This distribution generalizes the Poisson distribution by adding a parameter to model over-dispersion and under-dispersion and includes the geometric distribution as a special case and the Bernoulli distribution as a limiting case. In this paper, we propose a generalized COM-Poisson (GCOM-Poisson) distribution with three parameters, which includes the negative binomial distribution as a special case, and can become a longer-tailed model than the COM-Poisson distribution. The new parameter plays the role of controlling length of tail. The GCOM-Poisson distribution can become a bimodal distribution where one of the modes is at zero and is applicable to count data with excess zeros. Estimation methods are also discussed for the GCOM-Poisson distribution.