  In this article, we provide different representations for a time-fractional
birth and death process $N_{\alpha}(t)$, whose transition probabilities are
governed by a time-fractional system of differential equations. More
specifically, we present two equivalent characterizations for its trajectories:
the first one as a time-changed classic birth and death process, whereas the
second one is a Markov renewal process. Also, we provide results for the
asymptotic behavior of the process conditioned not to be killed. The most
important is that the concept of quasi-limiting distribution and
quasi-stationary distribution do not coincide, which is a consequence of the
long-memory nature of the process. As an application example, we revisit the
linear case to show the consequences of our main theorems.
