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In a simply connected Markov renewal problem, each state is either transient under all policies or an element of a single chain under some policy. This property is easily verified; it implies invariance of the maximal long-term average return (gain) with respect to the initial state, which in turn assures convergence of Odoni's bounds in the damped value-iteration algorithm due to Schweitzer, even when the maximal-gain process is multiple-chained and/or periodic.

Technical Note—Improved Conditions for Convergence in Undiscounted Markov Renewal Programming