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This chapter is based on Cheng and Sethi (1999b) and Beyer and Sethi (2005). We have proved a verification theorem for the average cost optimality equation, which we have used to establish the existence of an optimality state-dependent (s,S) policy. As with the discounted cost models, the optimality of an (s,S) policy for a lost sales case is established only under the condition of zero ordering leadtime. With nonzero leadtime, the results for models with backlog do not generalize to the lost sales case. Specifically, an (s,S) policy is no longer optimal, and the form of the optimal policy is more complicated; (e.g., Zipkin (2008a) and Huh et al. (2008). In this project we have carried out a rigorous analysis of the average cost stochastic inventory problem with Markovin demand, fixed ordering cost and lost sales. We have proved a verification theorem for the average cost optimality equation, which we have used to establish the existence of an optimal state-dependent (s,S)-policy. Throughout the analysis, we have assumed that the inventory and shortage cost functions are of linear growth. Further research should address extending this analysis to more general cost functions.