||On the relation between matrix-geometric and discrete phase-type distributions
||Nielsen, Bo Friis (Mathematical Statistics, Department of Informatics and Mathematical Modeling, Technical University of Denmark, DTU, DK-2800 Kgs. Lyngby, Denmark)
||Technical University of Denmark, DTU, DK-2800 Kgs. Lyngby, Denmark
||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 , 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) = Tn−1t, n 2 N.
If we take a probability density of the same form, but not necessarily require
, 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.
Creation date: 2009-06-24
Update date: 2010-08-25