On the moments for ergodic distribution of an inventory model of type (s; S) with regularly varying demands having infinite variance

Abstract

In this study a stochastic process X(t) which represents a semi Markovian inventory model of type (s,S) has been considered in the presence of regularly varying tailed demand quantities. The main purpose of the current study is to investigate the asymptotic behavior of the moments of ergodic distribution of the process X(t) when the demands have any arbitrary distribution function from the regularly varying subclass of heavy tailed distributions with infinite variance. In order to obtain renewal function generated by the regularly varying random variables, we used a special asymptotic expansion provided by Geluk [14]. As a first step we investigate the current problem with the whole class of regularly varying distributions with tail parameter 1 < ? < 2 rather than a single distribution. We obtained a general formula for the asymptotic expressions of n th order moments (n = 1, 2, 3, . . .) of ergodic distribution of the process X(t). Subsequently we consider this system with Pareto distributed demand random variables and apply obtained results in this special case.