Asymptotic expansions for the moments of the boundary functionals of the renewal reward process with a discrete interference of chance

Abstract

In this study, two boundary functionals $N_1$ and $\tau _1$ of the renewal reward process with a discrete interference of chance (X(t) ) are investigated. A relation between the moment generating function ($\psi _N (Z)$ ) of the boundary functional $N_1$ and the Laplace transform ( $\Phi _\tau \mu $ ) of the boundary functional $\tau _1$ is obtained. Using this relation, the exact formulas for the first four moments of the boundary functional $\tau _1$ are expressed by means of the first four moments of the boundary functional $N_1$ . Moreover, the asymptotic expansions for the first four moments of these boundary functionals are established when the random variables{$\zeta_n$}, n ≥ 0 , which describe a discrete interference of chance, have an exponential distribution with parameter λ > 0 . Finally, the accuracy of the approximation formulas for the moments ($EN^k_1$) of the boundary functional $N_1$ are tested by Monte Carlo simulation method.