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The policy-iteration algorithm and the linear programming formulation both require that in each iteration a system of linear equations of the same size as the state space is solved In general, this will be computationally burdensome for a large state space and makes these algorithms computationally unattractive for large-scale Markov decision problems In this section we discuss an alternative algorithm which avoids solving systems of linear equations but uses instead the recursive solution approach from dynamic programming This method is the value-iteration algorithm which computes recursively a sequence of value functions approximating the minimal average cost per time unit The value functions provide lower and upper bounds on the minimal average cost and under a certain aperiodicity condition these bounds converge to the minimal average cost The aperiodicity condition is not restrictive, since it can be forced to hold by a simple data transformation The value-iteration algorithm endowed with these lower and upper bounds is in general the best computational method for solving large-scale Markov decision problems This is even true in spite of the fact that the value-iteration algorithm does not have the robustness of the policy-iteration algorithm: the number of iterations is problem dependent and typically increases in the number of states of the problem under consideration Another important advantage of value iteration is that it is usually easy to write a code for speci c applications By exploiting the structure of the particular application one usually avoids computer memory problems that may be encountered when using policy iteration Value iteration is not only a powerful method for controlled Markov chains, but it is also a useful tool to compute bounds on performance measures in a single Markov chain; see Example 661 In this section the value-iteration algorithm will be analysed under the weak unichain assumption from Section 65 Under this assumption the minimal average cost per time unit is independent of the initial state Let g = the minimal long-run average cost per time unit The value-iteration algorithm computes recursively for n = 1, 2, the value function Vn (i) from pij (a)Vn 1 (j ) , i I, (661) Vn (i) = min ci (a) + a A(i).

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starting with an arbitrarily chosen function V0 (i), i I The quantity Vn (i) can be interpreted as the minimal total expected costs with n periods left to the time horizon when the current state is i and a terminal cost of V0 (j ) is incurred when the system ends up at state j ; see Denardo (1982) and Derman (1970) for a proof

Intuitively, one might expect that the one-step difference Vn (i) Vn 1 (i) will come very close to the minimal average cost per time unit and that the stationary policy whose actions minimize the right-hand side of (661) for all i will be very close in cost to the minimal average cost However, these matters appear to be rather subtle for the average cost criterion due to the effect of possible periodicities in the underlying decision processes Before explaining this in more detail, we investigate an operator which is induced by the recursion equation (661) The operator T adds to each function v = (vi , i I ) a function T v whose ith component (T v)i is de ned by (T v)i = min ci (a) + pij (a)vj , i I (662) a A(i)

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