Time and Ratio Expected Average Cost Optimality for Semi-Markov Control Processes on Borel Spaces Fernando Luque-V´asquez and Oscar Vega-Amaya April 24, 2003 Abstract We deal with semi-Markov control models with Borel state and con- trol spaces, and unbounded cost functions under the ratio and the time expected average cost criteria. Under suitable growth conditions on the costs and the mean holding times together with stability conditions on the embedded Markov chains, we show the following facts: (i ) the ratio and the time average costs coincide in the class of the stationary policies; (ii ) there exists an stationary policy which is optimal for both criteria. Moreover, we provide a generalization of the classical Wald’s Lemma to semi-Markov processes. These results are obtained combining the exis- tence of solutions of the average cost optimality equation and the Optional Stopping Theorem. 1 Introduction This paper deals with semi-Markov control models (SMCMs) with Borel state and control spaces, unbounded costs and holding mean times. We consider the two main expected average cost criteria studied in SMCMs, namely, the ratio expected average cost (ratio-EAC) criterion and the time expected average cost (time-EAC) criterion. It is well-known that these criteria in general differ even for the case of finite state and control spaces (see [4]). In the present paper under stability conditions and suitable growth assumption on the cost we show that these criteria coincide when the processes are controlled by stationary policies and also that there exists an stationary policy which is optimal for both criteria; thus, the optimality criteria defined by the ratio-EAC criterion and the time- EAC criterion are equivalent. As a by-product we also obtain a generalization of the classical Wald’s Lemma in renewal theory to semi-Markov processes which is interesting by itself. The ratio-EAC criterion has been widely studied in many works (see, e.g. [3], [4], [5], [12], [19], [21], [23], [24],[25] and [29]) but there are only few authors that consider the time-EAC (see, e.g. [4], [15], [17], [23], [24], [25] and [30]) and most of them deal with the countable state space and/or use bounded costs. 1