Asymptotic expansions for the moments of the boundary functionals of the renewal reward process with a discrete interference of chance (Q710424)

Summary: 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 \(\{\zeta_n\}\), \(n\ge 0\), 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 \((EN^k_1)\) of the boundary functional \(N_1\) are tested by Monte Carlo simulation method.