  A well-known theorem for an irreducible skip-free chain with absorbing state
$d$, under some conditions, is that the hitting (absorbing) time of state $d$
starting from state 0 is distributed as the sum of $d$ independent geometric
(or exponential) random variables. The purpose of this paper is to present a
direct and simple proof of the theorem in the cases of both discrete and
continuous time skip-free Markov chains. Our proof is to calculate directly the
generation functions (or Laplace transforms) of hitting times in terms of the
iteration method.
