The class of L boolean AND D and Its application to renewal reward process

Abstract

The class of L boolean AND D is generated by intersection of two important subclasses of heavy tailed distributions: the long tailed distributions and dominated varying distributions. This class itself is also an important member of heavy tailed distributions and has some principal application areas especially in renewal, renewal reward and random walk processes. the aim of this study is to observe some well and less known results on renewal functions generated by the class of L boolean AND D and apply them into a special renewal reward process which is known in the literature a semi Markovian inventory model of type (s, S). Especially we focused on Pareto distribution which belongs to the L boolean AND D subclass of heavy tailed distributions. As a first step we obtained asymptotic results for renewal function generated by Pareto distribution from the class of L boolean AND D using some well-known results by Embrechts and Omey [1]. Then we applied the results we obtained for Pareto distribution to renewal reward processes. As an application we investigate inventory model of type (s, S) when demands have Pareto distribution from the class of L boolean AND D. We obtained asymptotic expansion for ergodic distribution function and finally we reached asymptotic expansion for nth order moments of distribution of this process.