First-passage time problems for continuous-time birth-death chains are considered. Recursive formulas for the moments of the first-exit time and of the first-passage time in terms of the potential coefficients are explicitly obtained. Making use of the probability current, some functional relations between transition probabilities for unrestricted and restricted continuous-time birth-death chains are determined. Finally, two continuous-time birth-death chains with constant rates are taken in account; for them, closed form results on the first-exit time and on the first-passage time are explicitly obtained.
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