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Math. Comput. Appl. 2010, 15(1), 117-126; doi:10.3390/mca15010117

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

1
Department of Statistics and Computer Sciences, Karadeniz Technical University, 61080, Trabzon, Turkey
2
Institute of Cybernetics of Azerbaijan National Academy of Sciences, F. Agayev str.9, AZ 1141, Baku, Azerbaijan
3
Department of Mathematics, Karadeniz Technical University, 61080, Trabzon, Turkey
4
Department of Industrial Engineering, TOBB University of Economics and Technology, Sogutozu, 06560, Ankara, Turkey
*
Author to whom correspondence should be addressed.
Published: 1 April 2010
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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.
Keywords: Renewal Reward Process; Discrete Interference of Chance; Boundary Functional; Laplace Transform; Asymptotic Expansion; Monte Carlo Method Renewal Reward Process; Discrete Interference of Chance; Boundary Functional; Laplace Transform; Asymptotic Expansion; Monte Carlo Method
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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MDPI and ACS Style

Aliyev, R.; Bekar, N.O.; Khaniyev, T.; Unver, I. Asymptotic Expansions for the Moments of the Boundary Functionals of the Renewal Reward Process with a Discrete Interference of Chance. Math. Comput. Appl. 2010, 15, 117-126.

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Math. Comput. Appl. EISSN 2297-8747 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
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