  This paper is a continuation of the study on the stability speed for Markov
processes. It extends the previous study of the ergodic convergence speed to
the non-ergodic one, in which the processes are even allowed to be explosive or
having general killings. At the beginning stage, this paper is concentrated on
the birth-death processes. According to the classification of the boundaries,
there are four cases plus one having general killings. In each case, some dual
variational formulas for the convergence rate are presented, from which the
criterion for the positivity of the rate and an approximating procedure of
estimating the rate are deduced. As the first step of the approximation, the
ratio of the resulting bounds is usually no more than 2. The criteria as well
as basic estimates for more general types of stability are also presented. Even
though the paper contributes mainly to the non-ergodic case, there are some
improvements in the ergodic one. To illustrate the power of the results, a
large number of examples are included.
