2.II.22H

Consider a Markov chain on the state space $S=\{0,1,2, \ldots\} \cup\left\{1^{\prime}, 2^{\prime}, 3^{\prime}, \ldots\right\}$ with transition probabilities as illustrated in the diagram below, where $0<q<1$ and $p=1-q$.

For each value of $q, 0<q<1$, determine whether the chain is transient, null recurrent or positive recurrent.

When the chain is positive recurrent, calculate the invariant distribution.

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