Does a given vector-matrix pair correspond to a PH distribution?

The analysis of practical queueing problems benefits if realistic distributions can be used as parameters. Phase type (PH) distributions can approximate many distributions arising in practice, but their practical applicability has always been limited when they are described by a non-Markovian vector-matrix pair. In this case it is hard to check whether the non-Markovian vector-matrix pair defines a non-negative matrix-exponential function or not. In this paper we propose a numerical procedure for checking if the matrix-exponential function defined by a non-Markovian vector-matrix pair can be represented by a Markovian vector-matrix pair with potentially larger size. If so, then the matrix-exponential function is non-negative. The proposed procedure is based on O’Cinneide’s characterization result, which says that a nonMarkovian vector-matrix pair with strictly positive density on (0,∞) and with a real dominant eigenvalue has a Markovian representation. Our method checks the existence of a potential Markovian representation in a computationally efficient way utilizing the structural properties of the applied representation transformation procedure.