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Title: Asymptotic Rate for Weak Convergence of the Distribution of Renewal-Reward Process with a Generalized Reflecting Barrier
Authors: Khaniyev, Tahir
Gever, Başak
Hanalioğlu, Zülfiye
Issue Date: 2016
Publisher: Springer-Verlag Berlin
Source: Khaniyev, T., Gever, B., & Hanalioglu, Z. (2016). Asymptotic Rate for Weak Convergence of the Distribution of Renewal-Reward Process with a Generalized Reflecting Barrier. In Intelligent Mathematics II: Applied Mathematics and Approximation Theory (pp. 313-331). Springer, Cham.
Abstract: In this study, a renewal-reward process (X(t)) with a generalized reflecting barrier is constructed mathematically and under some weak conditions, the ergodicity of the process is proved. The explicit form of the ergodic distribution is found and after standardization, it is shown that the ergodic distribution converges to the limit distribution R(x), when lambda -> infinity, i. e., QX (lambda x) [GRAPHICS] P{X(t) <= lambda x} -> R(x) 2/m(2) [GRAPHICS] [1 - F(u)]dudv. Here, F(x) is the distribution function of the initial random variables {eta(n)}, n = 1, 2,..., which express the amount of rewards and m(2) = E(eta(2)(1)). Finally, to evaluate asymptotic rate of the weak convergence, the following inequality is obtained: vertical bar QX (lambda x) - R(x)vertical bar <= 2/lambda vertical bar pi(0)(x) - R(x)vertical bar. Here, pi(0)(x) = (1/m(1)) [GRAPHICS] (1 - F(u))du is the limit distribution of residual waiting time generated by {eta(n)}, n = 1, 2,..., and m(1) = E(eta(1)).
[Khaniyev, Tahir; Gever, Basak] TOBB Univ Econ & Technol, Ankara, Turkey; [Khaniyev, Tahir] Azerbaijan Natl Acad Sci, Inst Control Syst, Baku, Azerbaijan; [Hanalioglu, Zulfiye] Karabuk Univ, Karabuk, Turkey
ISBN: 978-3-319-30322-2; 978-3-319-30320-8
ISSN: 2194-5357
Appears in Collections:Endüstri Mühendisliği Bölümü / Department of Industrial Engineering
Scopus İndeksli Yayınlar Koleksiyonu / Scopus Indexed Publications Collection
WoS İndeksli Yayınlar Koleksiyonu / WoS Indexed Publications Collection

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