Mixed Poisson process with Min-U-Exp mixing variable

Abstract: This work continues the research done in Jordanova and Veleva (2023) where the history of the problem could be found. In order to obtain the structure distribution of the newly-defined Mixed Poisson process, here the operation "max" is replaced with "min". We start with the definition of Min-U-Exp distribution. Then, we compute its numerical characteristics and investigate some of its properties. The joint distribution of the inter-arrival times (which are dependent) is the Multivariate Exp-Min-U-Exp distribution of $II^{-nd}$ kind. Its univariate and multivariate versions are described, and the formulae for their numerical characteristics are obtained. The distribution of the moments of arrival of different events is called Erlang-Min-U-Exp. Different properties of these distributions are obtained, and their numerical characteristics are computed. Multivariate ordered Mixed Poisson-Min-U-Exp distribution describes the joint distribution of the time-intersection of a Mixed Poisson process with Min-U-Exp mixing variable. The corresponding distribution of the additive increments (which are also dependent) is the Mixed Poisson-Min-U-Exp one. The considered relations between these distributions simplify their understanding.