Moment based approximations for semi-markovian inventory models of type (s,S)

This research aims to develop moment-based approximations for a semi-Markov inventory model of type (s,S), which is one of the important inventory control models in the literature. There are various asymptotic expansions for this model in the literature. All these studies are basically based on the condition of knowing the structure of the distribution function F(t) of the demand random variables and obtaining the asymptotic expansion of the renewal function generated in accordance with this requirement. However, obtaining the renewal function may be difficult for certain families of distributions, and even if the renewal function is obtained, the mathematical structure of this function may create difficulties in practice. In many cases, it is simple to compute the moments of F(t). Therefore, in this study, certain properties of the (s,S) stochastic control model are investigated based on the condition that only the first three moments of the demand random variables are known. With this method, simple and compact approximations are obtained for the ergodic distribution of the stochastic process expressing the stochastic control model of type (s,S) and for the nth-order moments of the ergodic distribution. In this study, we use the approach proposed by Kambo et al. (2012) for the renewal function based on the first three moments of the random variables constituting the renewal function.