@MASTERSTHESIS\{IMM2009-06944,
    author       = "S. Greeuw",
    title        = "On the relation between matrix-geometric and discrete phase-type distributions",
    year         = "2009",
    school       = "Technical University of Denmark, Department of Applied Mathematics and Computer Science",
    address      = "Richard Petersens Plads, Building 324, {DK-}2800 Kgs. Lyngby, Denmark, compute@compute.dtu.dk",
    type         = "",
    note         = "Supervisor: Bo Friis Nielsen, bfni@dtu.dk",
    url          = "http://www.compute.dtu.dk/English.aspx",
    abstract     = "A discrete phase-type distribution describes the time until absorption in a discrete-time Markov chain with a finite number of transient states and one absorbing state. The density f(n) of a discrete phase-type distribution can be expressed by the initial probability vector {T,} the transition probability matrix T of the transient states of the Markov chain and the vector t containing the probabilities of entering the absorbing state from the transient states: f(n) = aTn1t, nEN.

If we take a probability density of the same form, but not necessarily require a, T and t to have the probabilistic Markov-chain interpretation, we obtain the density of a matrix-geometric distribution. Matrix-geometric distributions can equivalently be defined as distributions on the non-negative integers that have a rational probability generating function.

In this thesis it is shown that the class of matrix-geometric distributions is strictly larger than the class of discrete phase-type distributions. We give an example of a set of matrix-geometric distributions that are not of discrete phasetype. We also show that there is a possible order reduction when representing a discrete phase-type distribution as a matrix-geometric distribution.

The results parallel the continuous case, where the class of matrix-exponential distributions is strictly larger than the class of continuous phase-type distributions, and where there is also a possible order reduction."
}