Abstract
The problem studied is that of controlling a Markov chain so as to minimize the long run expected cost per unit time. Three results are obtained. First, a necessary and sufficient condition for optimality is given. The second gives for any strategy <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</tex> , an easily computable bound <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B (u) \geq J(u)-J^{\ast}</tex> , where J <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*</sup> is the minimum cost. The third result consists of an algorithm which, starting with any strategy, successively generates alternative strategies so that the bound <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B (u)</tex> decreases monotonically to zero.
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