Linear birth process with immigration
= Linear birth process with immigration
The pure birth process with rates $\lambda_n=\alpha n+\beta$, where $\alpha,\beta>0$, combines linear reproduction with immigration. Starting from zero, its mean satisfies
$$
m'(t)=\alpha m(t)+\beta,
\qquad
m(0)=0,
$$
and hence
$$
\mathbb EN(t)=\frac{\beta}{\alpha}(e^{\alpha t}-1).
$$