Tools for Phase-Type Distributions (butools.ph)¶

Phase-type and matrix-exponential distributions¶

Continuous time phase-type (PH) distributions are characterized by two parameters, the initial probability vector \(\alpha\) and the transient generator matrix of a continuous time (transient) Markov chain \(\alpha\). The PH distribution represents the absorption time of this transient Markov chain starting from \(\alpha\). The cummulative distribution function (cdf) of PH distributions is

where \(\mathbf{1}\) is the column vector of ones.

Matrix exponential (ME) distributions are the generalizations of PH distributions. Formally, the cdf and all the related formulas for the statistical quantities are very similar to the ones of PH distributions

however, \(\mathbf{b},B\) and \(\mathbf{e}\) can hold general numbers, the entries do not have to be valid probabilities or transition rates. ME distributions therefore lack the simple stochastic interpretation that PH distributions have. Vector \(\mathbf{b}\) is called “starting operator”, matrix \(B\) is the “process rate operator” and vector \(\mathbf{e}\) is the “summing operator” [1].

BuTools provides several tools for both distribution classes in the ph package. However, BuTools uses a special form of ME distributions, where the summing operator is a vector of ones, thus \(\mathbf{e}=\mathbf{1}\). This is not a restriction, as all ME distributions defined with general summing operator can be easily transformed to this representation. Assume we have an ME distribution in the general form with parameters \(\mathbf{b}, B, \mathbf{e}\). The necessary similarity transform is obtained by calling the T = TransformToOnes(e) procedure of the BuTools reptrans package. The parameters of the ME distribution used by all related BuTools tools can be calculated by \(\mathbf{b'}=\mathbf{b}T^{-1}\) and \(B'=T\,B\,T^{-1}\).

Simple statistical properties and tools¶

Inverse characterization tools¶

Representation transformation methods¶

Minimal representations¶

References¶