= Solution
For $\lambda_n=\alpha n+\beta$,
$$
\sum_{n=0}^{\infty}\frac1{\alpha n+\beta}=\infty,
$$
so part (b) ensures nonexplosion. The <Markov jump-process generator> acts on the identity function $f(n)=n$ as
$$
(Lf)(n)=(\alpha n+\beta)\bigl((n+1)-n\bigr)
=\alpha n+\beta.
$$
Consequently $m(t)=\mathbb EN(t)$ satisfies
$$
m'(t)=\mathbb E[(Lf)(N(t))]
=\alpha m(t)+\beta,
\qquad m(0)=0.
$$
Solving this linear <ordinary differential equation> gives the mean of the <linear birth process with immigration>:
$$
\boxed{\mathbb EN(t)=\frac{\beta}{\alpha}(e^{\alpha t}-1)}.
$$
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