 Geometric-jump countdown Markov chain

On the nonnegative integers, every positive state decreases by one, while zero jumps to $j\ge 1$ with probability $p{q}^{j-1}$, $p=1-q\in (0,1)$. Its return time from zero is one plus this geometric jump, so it has mean $(1+p)/p$. Return lengths two and three occur, establishing aperiodicity; deterministic downward motion and positive jumps establish irreducibility.