  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.
