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 \(\{\zeta_{n}\}\), \(n \geq 0\), 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 (EN\(_{1}^{k}\)) of the boundary functional N\(_{1}\) are tested by Monte Carlo simulation method.