Investigation of Boundary Functionals for Renewal-Reward Process with a Generalized Reflecting Barrier

Abstract

In this study, renewal - reward process with a generalized reflecting barrier (X(t)) and its three boundary functionals are mathematically constructed. Next, the asymptotic expansions are obtained for the first four moments of these boundary functionals of the process X(t).

Keywords

• Renewal - reward process, Boundary functional, Asymptotic expansion,Reflecting barrier.

References

1. Aliyev, R., Okur Bekar, N., Khaniyev, T. and Unver, I. Asymptotic expansions for the moments of the boundary functionals of the renewal reward process with a discrete interference of chance, Mathematical and Computational Applications 15, 117 ˆu 126, 2010.
2. Beyer, D., Sethi, S.P. and Taksar, M. Inventory Models with Markovian Demands and Cost Functions of Polynomial Growth, Journal of Optimization Theory and Application 98 (2), 281-323, 1998.
3. Borovkov, A.A. Stochastic Processes in Queuing Theory, (Spinger-Verlag, New York, 1976).
4. Brown, M. and Solomon, H.A. Second-order approximation for the variance of a renewalreward process, Stochastic Processes and Their Applications 3 , 301-314, 1975.
5. Chen, F. and Zheng, Y. Waiting time distribution in (T,S) inventory systems, Operations Research Letters 12 , 145-151, 1992.
6. Federyuk, M.V. Asymptotics for Integrals and Series,(Nauka, Moscow, 1984).
7. Feller, W. Introduction to Probability Theory and Its Applications II,(John Wiley, New York, 1971).
8. Khaniyev, T. A., About moments of generalized renewal process, Transactions of NAS of Azerbaijan, Series of Phys. Tech. and Math. Sciences, 25 (1), 95 ˆu 100, 2005

9. Khaniev, T.A., Unver, I. and Maden, S., On the semi-Markovian random walk with two reflecting barriers, Stochastic Analysis and Applications, 19(5), 799–819, 2001.
10. Lotov, V.I. On some boundary crossing problems for Gaussian random walks, Annals of Probability 24 (4), 2154-2171, 1996.
11. Patch, B., Nazarathy, Y. and Taimre, T., A correction term for the covariance of multivariate renewal-reward processes, Statistics and Probability Letters, 102, 1–7, 2014.
12. Prabhu, N.U. Stochastic Storage Processes, (Springer-Verlag, New York, 1981).7
13. Rogozin, B.A., On the distribution of the first jump, Theory of Probability and Its Applications, 9(3), 450–465, 1964.
14. Ross, S.M. Introduction to Probability Models, (Academic Press, INC., Boston, 1989).
15. Smith, W.L. Renewal Theory and Its Ramification, Journal of the Royal Statistical Society. Series B (Methodological) 20 (2) , 243-302, 1958.