The relation between phase-type representation and positive system realization in both the discrete and continuous time is discussed. Using the Perron-Frobenius theorem of nonnegative matrix theory, a transformation from positive realization to phase-type realization is derived under the excitability condition. In order to explain the connection, some useful properties and characteristics such as irreducibility, excitability, transparency, and order reduction for positive realization and phase-type representation are discussed. In addition, the connection between the phase-type renewal process and the feedback positive system is discussed in the stabilization concept.
Kyungsup Kim. "On the Relation between Phase-Type Distributions and Positive Systems." Abstr. Appl. Anal. 2015 1 - 7, 2015. https://doi.org/10.1155/2015/731261