Spectral conditions for Phase-type representations

Abstract

Current paper tries to find appropriate similarity transformation that could convert a given ME (Matrix Exponential) representation to a more favorable PH (Phase-type) representation. As the main result of this paper, we give necessary conditions for the existence of such a representation. We also give methods for the search and provide conjectures on necessary and sufficient conditions too. PH distribution is the distribution of the time until absorption into the absorbent state in a Markov chain. If the arrival and service time distributions are PH distributions in a queuing system, we can use simple linear algebraic methods to derive the most important features or to perform simulation. Robust methods exist that can approximate any distribution with a ME distribution (with respect to a given measure and matrix order), but the PH transformation have not been sufficiently examined yet. This transformation is the object of the current presentation.