  In this paper one presents the extension of the transient analysis of the
class of continuous-time birth and death processes defined on non-negative
integers with special transitions from and to the origin. From the origin
transitions can occur to any state. But being in any other state, besides
ordinary transitions to neighbouring states, a transition to the origin can
occur. All possible transition intensities are assumed to be non-random
functions of time and may depend on the state of the process. We improve
previously known ergodicity and truncation bounds for this class of processes
which were known only for the case when transitions from the origin decay
exponentially (other intensities must have unique uniform upper bound). We show
how the bounds can be obtained the decay rate is slower than exponential.
Numerical results are also provided.
