 Two-lane Markov chain with a deterministic restart

Take states $0,n,n^{\prime }$ for $n\ge 1$, with $0\to 1^{\prime }$ deterministic and transitions $n\to n-1$ with probability $q$, $n\to (n+1{)}^{\prime }$ with probability $p=1-q$, $n^{\prime }\to n$ with probability $q$, and $n^{\prime }\to (n+1{)}^{\prime }$ with probability $p$. The Markov chain is transient for $q<q_c$, null recurrent for $q=q_c$, and positive recurrent for $q>q_c$. To see the Markov chain recurrence threshold, eliminate the primed-state hitting probabilities to obtain $p(1+q)a_{n+1}-(1+pq)a_n+q{a}_{n-1}=0$, with characteristic roots $1$ and $q/(1-q^2)$. Boundedness forces return with probability one when the second root is at least one. When it is below one, the minimal nonnegative solution gives return probability $q^2/p<1$ from $0$. In the positive recurrent case, put $t=(1-q^2)/q$ and $D=q^2+q-1$. The stationary distribution is ${\pi }_0=D/[q(q+2)]$, ${\pi }_n=({\pi }_0/q)t^{n-1}$ and ${\pi }_{n^{\prime }}={\pi }_0{t}^{n-1}$. It follows from balance across each level cut and sums to one; at criticality the invariant weights cannot be normalized.