Non-Uniqueness of the Minimal Realization Problem Representation of the Markovian Arrival Process of Order 2 and Moment Fitting

Markovian arrival processes (MAPs) can be represented in many different ways such as the Markovian representation, Laplace transform, Jordan representation, and minimal realization problem (MRP) representation, to name a few. The MRP representation is given in two real-valued matrices (K’, R’) and can be used to determine the Markovian representation (D0, D1) by similarity transformation. The MRP representation has been claimed to be unique but redundant. In this paper, we show that the MRP representation is not unique and then provide a non-redundant MRP representation of MAP(2)s. We also present closed-form formulas for the transformation from the MRP representation to the Markovian representation (D0, D1) for MAP(2)s.