On the weak convergence of the ergodic distribution for an inventory model of type (s,S)

In this study, a renewal - reward process with a discrete interference of chance is constructed. This process describes in particular a semi-Markovian inventory model of type (s, S). The ergodic distribution of this process is expressed by a renewal function, and a second-order approximation for the ergodic distribution of the process is obtained as S - s -> infinity when the interference has a triangular distribution. Then, the weak convergence theorem is proved for the ergodic distribution and the limit distribution is derived. Finally, the accuracy of the approximation formula is tested by the Monte Carlo simulation method.