= Immigration chain with complete population resets
{title2=$\pi_0=p/(1+p),\quad\pi_j=p(1-p)^{j-1}/(1+p)$}
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 $(1-p)^K/(1+p)$.
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