Ehrenfest urn model ($P_{i,i+1}=(k-i)/k,P_{i,i-1}=i/k$)

The Ehrenfest urn model moves one uniformly chosen ball between two boxes at each step. For $k\ge 1$ balls, the count in one box is a Markov chain on $\{0,\dots ,k\}$. Its stationary distribution is $2^{-k}\left(\genfrac{}{}{0ex}{}{k}{i}\right)$, and detailed balance makes it a reversible Markov chain. The chain has period two, but its mean recurrence time to the all-in-one-box state is still $2^k$.

Stop the four-ball Ehrenfest urn model upon reaching count zero or four. From count $r$, the transitions are to $r-1$ with probability $r/4$ and to $r+1$ with probability $(4-r)/4$. First-step analysis with absorbing boundary values gives the displayed vector $h$ of probabilities of absorption at zero and vector $t$ of expected hitting times of either endpoint. For instance, $t_1=1+3t_2/4$ and $t_2=1+(t_1+t_3)/2$, with $t_1=t_3$, give $t_1=7$. From any interior state, absorption has probability at least $1/4$ within two steps; the resulting geometric tail bound proves both almost-sure absorption and finiteness of the expectations.