= Solution
The additional transition is $n\to n+k$ at rate $\epsilon(n)$. Its gain term at state $n$ comes from state $n-k$, so the new <master equation> is
$$
\boxed{
\frac{dp_n}{dt}
=\lambda(p_{n-1}-p_n)
+\beta\bigl((n+1)p_{n+1}-np_n\bigr)
+\epsilon(n-k)p_{n-k}-\epsilon(n)p_n,}
$$
where the gain term is zero when $n<k$. In the generating-function equation the new contribution is
$$
(s^k-1)\sum_{n\geq0}\epsilon(n)p_n s^n.
$$
For a general state-dependent $\epsilon$, this is not closed in $\phi$. Even when it is closed, $s^k-1$ is not proportional to $s-1$ for $k>1$, so the Poisson ansatz used in part (a) is not preserved.
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