The goal of this research is to develop algorithms for identifying a low-order phase-type distribution which matches, exactly or as nearly as possible, given specifications such as moments. The motivation for the research is the extensive literature on stochastic models based on phase-type distributions. Many of these models may be solved by straight forward algorithms. If these are to be used in applications, then methods are needed for specifying the phase-type distributions involved. There is much research underway on fitting phase-type distributions, motivated by the algorithms available for them. This research emphasizes fitting low-order distributions. The reason for this emphasis is that the size of the state space of a model, and hence also the computational burden in solving it, typically depend multiplicatively on the orders of the phase-type distributions used in the model. Thus it is important to keep the order as small as possible. This direction in fitting phase-type distributions has not been explored systematically to date. The techniques are largely based on recent developments in the theory of phase-type distributions. Bringing these ideas to bear on the practical problem of fitting low-order phase-type distributions is an important step in the development of the phase methodology.
