Future directions

Our moment matching algorithms (or the ones recently proposed by Bobbio et al. [22]) by no mean provide perfect solution for mapping a general distribution into a PH distribution. Interesting future directions include extension to matching more moments and extension to a sequence of correlated PH distributions. Our moment matching algorithms match the first three moments of an input distribution, but it is desirable to match more moments. Also, a PH distribution can be used to model a sequence of independent random variables (such as a sequence of job sizes and a sequence of interarrival times); however, in some applications, it is desirable to capture the correlation in the sequence of random variables. The Markovian arrival process (see Section 3.2) generalizes the PH distribution, and it represents a sequence of correlated PH distributions. However, how we should map a sequence of correlated random variables into a Markovian arrival process is not well understood (see e.g. [72] for this direction of work).

An interesting future direction in characterizing PH distributions is to characterize the set of distributions that are well-represented by an $n$-phase general PH distribution, not limited to an acyclic ones. Such a characterization could be used to prove the minimality of the number of phases used in our closed form solutions in a stronger sense. Although our experience via numerical experiments suggests that allowing cycles in the PH distribution does not help to reduce the number of phases used to match the three moments, a rigorous characterization is not known to date. Note, however, that an acyclic PH distribution has a computational advantage over a cyclic one, since the generator matrix of the Markov chain whose absorption time defines an acyclic PH distribution is upper triangular. Therefore, in some applications, one might prefer an acyclic PH distribution with more phases to a cyclic PH distribution with less phases. Another interesting future direction is to characterize the set of distribution whose first $k$-moments can be matched by an $n$-phase (acyclic or general) PH distribution for $k>3$. Such a characterization would be useful in designing moment matching algorithms that match the first $k>3$ moments of an input distribution.