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.
