Graph-theoretic computation of characteristic function based on representation of phase-type distribution

We derive a unified graph-theoretic approach to continuous and discrete phase-type distributions. The algorithms are given to obtain the signal-flow graph directly from either the matrix representation of the distribution or from the transition diagram of the underlying Markov chain. The transfer function of the signal-flow graph, easily computable using Mason’s rule, gives the characteristic function of the phase-type distribution in a symbolic form. The proposed approach intrinsically includes non-trivial initial probabilities of the states. Moreover, in the continuous case, it results in graphs that are simpler to obtain than those found in the literature. Finally, we show that the approximate discrete counterpart of the continuous phase-type distribution can be viewed as the forward difference (Euler) mapping between continuous and discrete time domains.