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 lambda > 0. Finally, the accuracy of the approximation formulas for the moments (EN1k) of the boundary functional N-1 are tested by Monte Carlo simulation method.