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transformed model, it is no restriction to assume that for each stationary policy the associated Markov chain {X n } is aperiodic By choosing the constant strictly less than mini,a i (a), we always have pii (a) > 0 for all i, a and thus the required aperiodicityALGORITHMS FOR AN OPTIMAL POLICY In this section we outline how the algorithms for the discrete-time Markov decision model can be extended to the semi-Markov decision model Policy-iteration algorithm The policy-iteration algorithm will be described under the unichain assumption This assumption requires that for each stationary policy the embedded Markov chain {Xn } has no two disjoint closed sets By data transformation, it is directly veri ed that the value-determination equations (632) for a given stationary policy R remain valid provided that we replace g by g i (Ri ) The policy-improvement procedure from Theorem 621 also remains valid when we replace g by g i (R i ) Suppose that g(R) and i (R), i I , are the average cost and the relative values of a stationary policy R If a stationary policy R is constructed such that, for each state i I , ci (R i ) g(R) i (R i ) +j I pij (R i ) j (R) i (R),(721)then g(R) g(R) Moreover, g(R) < g(R) if the strict inequality sign holds in (721) for some state i which is recurrent under R These statements can be veri ed by the same arguments as used in the second part of the proof of Theorem 621 Under the unichain assumption, we can now formulate the following policyiteration algorithm: Step 0 (initialization) Choose a stationary policy R Step 1 (value-determination step) For the current rule R, compute the average cost g(R) and the relative values i (R), i I , as the unique solution to the linear equations i = ci (Ri ) g i (Ri ) +