On count data models based on Bernstein functions or their inverses

We present a class of positive discrete random variables extending the Conway--Maxwell-Poisson distribution. This class emerges in a natural way from an application in queueing theory and contains distributions exhibiting quite different features. Some of these distributions are characterized by the presence of Bernstein and inverse Bernstein functions. As a byproduct, we give some results on these inverses for which the existing literature is limited. Moreover, we investigate dispersion properties for these count data models, giving necessary and/or sufficient conditions to obtain both over and underdispersion. We also provide neat expressions for the factorial moments of any order. This furnishes us with a compact form also in the case of the Conway--Maxwell-Poisson.