  We characterize absorption-time distributions for birth-death Markov chains
with an absorbing boundary. For "extinction-prone" chains (which drift on
average toward the absorbing state) the asymptotic distribution is Gaussian,
Gumbel, or belongs to a family of skewed distributions. The latter two cases
arise when the dynamics slow down dramatically near the boundary. Several
models of evolution, epidemics, and chemical reactions fall into these classes;
in each case we establish new results for the absorption-time distribution.
Applications to African sleeping sickness are discussed.
