  In this paper, we discuss the computation of first passage moments of a
time-homogeneous semi-Markov process (SMP) with finite state space to certain
of its states that possess the property of universal accessibility (UA). A UA
state is one which is accessible from any other state of the SMP, but which may
or may not connect back to one or more other states. An important
characteristic of UA is that it is the state-level version of the oft-invoked
process-level property of irreducibility. We adapt existing results for
irreducible SMPs to the derivation of an analytical matrix expression for the
first passage moments to a single UA state of the SMP. In addition, consistent
estimators for these first passage moments are given.
