Moments of a batch-birth linear-death process
= Moments of a batch-birth linear-death process
For $m=\mathbb E N$ and $v=\operatorname{var}N$, the Markov jump-process generator applied to $n$ and $n^2$ gives
$$
m'=k\lambda-\beta m,
\qquad
v'=k^2\lambda+\beta m-2\beta v.
$$
Consequently
$$
m\longrightarrow\frac{k\lambda}{\beta},
\qquad
v\longrightarrow\frac{k(k+1)\lambda}{2\beta}.
$$