  In this paper, we demonstrate through the use of matrix calculus a
transparent analysis of fractional inhomogeneous Markov models for life
insurance where transition matrices commute. The resulting formulae are
intuitive matrix generalizations of their single-state counterparts, and the
absorption times are matrix versions of well-known scalar distributions. A
further advantage of this approach is that it allows extending the analysis to
the non-Markovian case where sojourns are Mittag-Leffler distributed, and where
the absorption times are fractional phase-type distributed. Considering
deterministic time transforms gives rise to fractional inhomogeneous phase-type
distributions as absorption times. The latter underlying processes are an
example of a regime where not only the present but also the history of a
policyholder influences its future evolution. The sub-exponential nature of
stable distributions translates into the multi-state insurance model as a
random longevity risk at any given state of the chain.
