  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.
