Asymptotic Expansions for the Moments of the Boundary Functionals of the Renewal Reward Process with a Discrete Interference of Chance

Mathematical and Computational Applications

Aliyev, Rovshan¹ 

关键词: Renewal Reward Process;    Discrete Interference of Chance;    Boundary Functional;    Laplace Transform;    Asymptotic Expansion;    Monte Carlo Method;   

DOI  :  10.3390/mca15010117

学科分类：计算数学

来源: mdpi

【 摘 要 】

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.

【 授权许可】

CC BY