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An (upward) skip-free Markov chain with the set of nonnegative integers asstate space is a chain for which upward jumps may be only of unit size; thereis no restriction on downward jumps. In a 1987 paper, Brown and Shaodetermined, for an irreducible continuous-time skip-free chain and any d, thepassage time distribution from state 0 to state d. When the nonzero eigenvaluesnu_j of the generator are all real, their result states that the passage timeis distributed as the sum of d independent exponential random variables withrates nu_j. We give another proof of their theorem. In the case ofbirth-and-death chains, our proof leads to an explicit representation of thepassage time as a sum of independent exponential random variables. Diaconis andMiclo recently obtained the first such representation, but our construction ismuch simpler.  We obtain similar (and new) results for a fastest strong stationary time T ofan ergodic continuous-time skip-free chain with stochastically monotonetime-reversal started in state 0, and we also obtain discrete-time analogs ofall our results.  In the paper's final section we present extensions of our results to moregeneral chains.

On hitting times and fastest strong stationary times for skip-free and more general chains