THE SEMI-MARKOV DECISION MODEL

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This discrete-time Markov decision model has the same class of stationary policies as the original semi-Markov decision model For each stationary policy R, let g i (R) denote the long-run average cost per time unit in the discrete-time model when policy R is used and the initial state is i Then it holds for each stationary policy R that i I (713) gi (R) = g i (R), This result does not require any assumption about the chain structure of the Markov chains associated with the stationary policies However, we prove the result (713) only for the unichain case Fix a stationary policy R and assume that the embedded Markov chain {Xn } in the semi-Markov model has no two disjoint closed sets Denote by Xn the state at the nth decision epoch in the transformed discretetime model It is directly seen that the Markov chain {Xn } is also unichain under policy R The equilibrium probabilities j (R) of the Markov chain {X n } satisfy the equilibrium equations j (R) =

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Hence, letting uj = j (R)/ j (Rj ) and dividing by , we nd that uj =

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These equations are precisely the equilibrium equations for the equilibrium probabilities j (R) of the embedded Markov chain {Xn } in the semi-Markov model The equations determine the j (R) uniquely up to a multiplicative constant Thus, for some constant > 0, j (R) = j (R) , j (Rj ) j I

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Since j I j (R) = 1, it follows that = (713) now follows easily We have g(R) =

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SEMI-MARKOV DECISION PROCESSES

and so, by Theorem 711, g(R) = g(R) Thus we can conclude that an average cost optimal policy in the semi-Markov model can be obtained by solving an appropriate discrete-time Markov decision model This conclusion is particularly useful with respect to the value-iteration algorithm In applying value iteration to the 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 aperiodicity

ALGORITHMS 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 ) +