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.

中文翻译：

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连续时间生死链的首次通过时间和相关时刻

考虑了连续时间的生死链的首次通过时间问题。根据势系数，明确获得了首次出站时刻和首次通过时刻的递归公式。利用当前的概率，确定了无限制和受限制的连续时间生死链的转移概率之间的一些函数关系。最后，考虑了两个恒定速率的连续生死链。对于他们，明确获得了首次出站时间和首次通过时间上的封闭式结果。