Immigration chain with complete population resets (${\pi }_0=p/(1+p),{\pi }_j=p(1-p{)}^{j-1}/(1+p)$)

From state zero the chain always goes to one. From a positive integer $j$, it resets to zero with probability $p$ and goes to $j+1$ otherwise. The stationary equations give ${\pi }_1={\pi }_0$ and ${\pi }_j=(1-p{)}^{j-1}{\pi }_0$, and normalization gives the displayed distribution. The chain is irreducible and positive recurrent; its return lengths two and three also make it aperiodic. The long-run proportion of time above a positive integer $K$ is $(1-p{)}^K/(1+p)$.