Batch-birth linear-death process
= Batch-birth linear-death process
A batch-birth linear-death process has transitions
$$
n\longrightarrow n+k\quad\hbox{at rate }\lambda,
\qquad
n\longrightarrow n-1\quad\hbox{at rate }\beta n.
$$
Writing $P_n(t)=\mathbb P(N_t=n)$ and taking $P_j=0$ for $j<0$, its master equation is
$$
\dot P_n=\lambda P_{n-k}+\beta(n+1)P_{n+1}-(\lambda+\beta n)P_n.
$$