classification and properties of acyclic discrete phase-type distributions based on geometric and shifted geometric distributions

Acyclic phase-type distributions form a versatile model, serving as approximations to many probability distributions in various circumstances. they exhibit special properties and characteristics that usually make their applications attractive. compared to acyclic continuous phase-type (acph) distributions, acyclic discrete phase-type (adph) distributions and their subclasses (adph family) have received less attention in the literature. in this paper, we present the definition, properties, characteristics and ph representations of adph distributions and their subclasses with finite state space. based on the definitions of geometric and shifted geometric distributions, we propose a distinct classification for the adph subclasses analogous to acph family. we develop the ph representation for each adph subclass and prove them through their closure properties. the advantage of our proposed classifications is in applying precise representations of each subclass and preventing miscalculation of the probability mass function, by computing the adph family based on geometric and shifted geometric distributions.