A Note on Unicyclic Representations of Phase Type Distributions

In this note, we study the unicyclic representation introduced in O'Cinneide [21] O'Cinneide , C. A. Phase-type distributions: open problems and a few properties . Stochastic Models 1999 , 15 ( 4 ), 731 – 757 .[Taylor & Francis Online] , [Google Scholar]. First, we present a counterexample to the conjecture that every PH-representation has an equivalent unicyclic representation of the same order. Then we show that the conjecture holds if the order of the PH-representation is 3. We also introduce an algorithm for computing a unicyclic generator of order 3, which PH-majorizes the original PH-generator, for any PH-generator of order 3. For the general case, we develop a nonlinear program for computing unicyclic representations for PH-distributions.