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The importance of the order two Markovian arrival process (MAP(2)) comes from its compactness, serving either as arrival or service process in applications, and from the nice properties which are not available for higher order MAPs. E.g., for order two processes the acyclic MAP(2) (AMAP(2)), the MAP(2) and the order two matrix exponential process (MEP(2)) are equivalent. Additionally, MAP(2) processes can be represented in a canonical form, from which closed form moments bounds are available. In this paper we investigate possible fitting methods utilizing the special nice properties of MAP(2). We present two fitting methods. One of them partitions the exact boundaries of the MAP(2) class into bounding subsurfaces reducing the numerical inaccuracy of the optimization based moment fitting. Without knowing the objective function. The characterizing new feature of the other one is that it considers the distance of joint density functions of infinitely many arrivals.