Consider the Markov chain with state space $S = \{0, 1, 2, \ldots \}$ and transition probabilities: \[ p(i, j) =\begin{cases} p, &j = i + 1\\ q, &j = 0\\ 0, &\text{otherwise} \end{cases} \] where $p, q > 0$ and $p + q = 1.1$. This Markov chain counts the lengths of runs of heads in a sequence of independent coin tosses. Let \[ T_y = \min\{n\ge 1: X_n=y\} \] be the time of the first return to $y$.