Abstract: This paper deals with downward properties of two kinds of birth and death processes with state 0 as their reflecting and quasi-leap-reflecting barriers before explosion. By first proposing a system of infinite dimensional linear equations with one boundary, we obtain some precise expressions of average time when the processes move from state k(k≥i) to state i-1 in several cases. On the other hand, we not only give a new defination of the characteristic number e_i^a but also show its probability meaning.