Mittag - Leffler function distribution - a new generalization of hyper-Poisson distribution

In this paper a new generalization of the hyper-Poisson distribution is proposed using the Mittag-Leffler function. The hyper-Poisson, displaced Poisson, Poisson and geometric distributions among others are seen as particular cases. This Mittag-Leffler function distribution (MLFD) belongs to the generalized hypergeometric and generalized power series families and also arises as weighted Poisson distributions. MLFD is a flexible distribution with varying shapes and has a unique mode at zero or it is unimodal with one/two non-zero modes. It can be under-, equior overdispersed. Various distributional properties like recurrence relation for probability mass function, cumulative distribution function, generating functions, formulas for different type of moments, their recurrence relations, index of dispersion and its classification, log-concavity, reliability properties like survival, increasing failure rate, unimodality, and stochastic ordering with respect to hyper-Poisson distribution are discussed. A particular case of the distribution is shown to arise as the steady state probability of a queuing system under state dependent service rate. The distribution has been found to fare well when compared with the hyper-Poisson and COM-Poisson type negative binomial distributions in its suitability in empirical modeling of differently dispersed count data. It is therefore expected that the proposed MLFD with its interesting features and flexibility will be a useful addition as a model for count data.