Abstract: This paper first deals with some properties of birth and death processes
with state 0 as their leap-reflecting barriers before explosion. By proposing
a system of linear equations and shift operator, the authors obtain some precise expressions of both probability and average time when the processes move from state i to state n(i ≤ n ≤∞ or n≤ i) in several cases. On the other
hand, the authors not only give a new definition of the characteristic numbers
such as m_i,e_i,N_i,R etc, but also show their probability meanings.

Key words: Q^a₀ process, Birth and death process, Leap-reflecting barrier, Before explosion