Limit theorem for a semi - Markovian stochastic model of type (s,S)

Abstract

In this study, a semi-Markovian inventory model of type $(s,S)$ is considered and the model is expressed by means of renewal-reward process $(X(t))$ with an asymmetric triangular distributed interference of chance and delay. The ergodicity of the process $X(t)$ is proved and the exact expression for the ergodic distribution is obtained. Then, two-term asymptotic expansion for the ergodic distribution is found for standardized process $W(t)\equiv (2X(t)) / (S-s)$. Finally, using this asymptotic expansion, the weak convergence theorem for the ergodic distribution of the process $W(t)$ is proved and the explicit form of the limit distribution is found.