ABSTRACT
Before proceeding any further, it should be mentioned that the somewhat smaller class 'P1-l of phase-type distributions (Neuts [1 5] ; see Section 2 for more detail) has become increasingly popular in recent years and is now the computational vehicle of much of modern applied probability, serving to extend the algorithmic tractability of models which can be solved explicitly when the underlying distributions are exponential. Even if the class 'P1-l is slightly smaller than K, the current view in applied probability is that by working there one circumvents many of the objection which has been raised against the classical approach: one gets transform-free solutions ; one avoids algebra involving complex numbers ; matrix calculations take the role of rootfinding in the complex plane based on Rouche's theorem; and finally, but not least, the set-up allows a probabilistic interpretation based on Markov processes with finitely many states which is not inherent at all in the classical approach. In some sense our paper may be viewed as a revisit to the class K, with many of the results inspired from the phase-type literature rather than the classical papers in the area. One of our conclusions is that , maybe somewhat contradictory to common belief, some, basic phase-type methodology extends to the class K as well. The advantage is not only the added generality, but also that even for phase type distributions there may be some gain in the matrix-exponential point of view because it may sometimes be possible to use a matrix-exponential representation of much smaller dimension than the phase-type representation.
