On the distributions of a renewal reward process and it’s additive functional

Abstract

In this study, a renewal reward process with a discrete interference of chance (X(t) ) is constructed and distribution of the process X(t) is investigated. Onedimensional distribution of the process X(t) is given by means of the probability characteristics of the renewal processes $\{T_n\}$ and \{S_n\}. Moreover, one dimensional distribution function of the additive functional $J_f(t)$ of the process X(t) is expressed by the probability characteristics of the initial sequences of the random variables $\{\xi_n\}$ and $\{\eta_n\}$