= Linear birth process with catastrophes
{title2=$q_{n,n+1}=\lambda(n+1),\quad q_{n,0}=\mu\ (n\ge1)$}
Add independent rate-$\mu$ resets to zero to a <linear birth process with immigration>. The resulting irreducible <continuous-time Markov chain> is nonexplosive by domination of births and finiteness of Poisson reset counts. When $\mu=\lambda$, an excursion from one reaches $n$ with probability $2/(n+1)$, so its <jump chain> is recurrent with infinite expected jump count. The elapsed return-cycle time instead has mean
$$
\lambda^{-1}+\sum_{n\ge1}\frac{2}{\lambda(n+1)(n+2)}=2/\lambda.
$$
Thus the continuous-time chain is positive recurrent while its jump chain is null recurrent. Its stationary probabilities are $\pi_0=1/2$ and $\pi_n=1/[(n+1)(n+2)]$ for $n\ge1$.
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