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AbstractThis paper aims to collect a set of properties of bilateral phase type (BPH) distributions that helps their practical use. Similar to the relation of the phase type (PH) and the acyclic phase type (APH) distributions, it turns out that restricting the underlying Markov chain to be acyclic results in a more tractable class of distributions, the acyclic bilateral phase type (ABPH) distributions. Here we present a canonical representation, moment bounds and moment matching methods for ABPH distributions.