Acyclic Minimality by Construction---Almost

Acyclic phase-type distributions are phase-type distributions with triangular matrix representations. They constitute a versatile modelling tool in many circumstances. The size of their matrix representation has a strong influence on computational efforts needed when analyzing these distributions. This representation, however, is not unique, and two representations of the same distribution can differ drastically in size. This paper explores an effective algorithm to reduce the size of the matrix representation without altering the distribution. This proceeds via a symbolic representation of the Laplace-Stieltjes transform of the phase-type distribution. The algorithm is of cubic complexity in the size of the given representation. We clarify the circumstances under which the algorithm returns a minimal representation, and discuss in how far it can keep representations minimal when they are constructed via any of three operations---convolution, minimum, and maximum. A case study modelling delay propagation in railway networks demonstrates the practicality of the approach.