Yazar "Hanalioglu, Zulfiye" seçeneğine göre listele

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    Approximation results for the moments of random walk with normally distributed interference of chance
    (Tubitak Scientific & Technological Research Council Turkey, 2023) Hanalioglu, Zulfiye; Poladova, Aynura; Khaniyev, Tahir
    In this study, a random walk process (X (t)) with normally distributed interference of chance is considered. In the literature, this process has been shown to be ergodic and the limit form of the ergodic distribution has been found. Here, unlike previous studies, the moments of the X (t) process are investigated. Although studies investigating the moment problem for various stochastic processes (such as renewal-reward processes) exist in the literature, it has not been considered for random walk processes, as it requires the use of new mathematical tools. Therefore, in this study, firstly, the exact formulas for the first four moments of the ergodic distribution of the X (t) process, which is a modification of the random walk process, are found. Due to the extremely complex mathematical structure of the exact formulas, in the second part of the study, three-term asymptotic expansions are attained for these moments. Based on the asymptotic expansions, simple and useful approximation formulas, for the moments of the process X (t) are proposed. In order to show that the approximate formulas are close enough to the exact formulas, a special example is given at the end of the study and the accuracy of the approximate formulas is examined on this example.
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    Asymptotic Rate for Weak Convergence of the Distribution of Renewal-Reward Process with a Generalized Reflecting Barrier
    (Springer-Verlag Berlin, 2016) Khaniyev, Tahir; Gever, Basak; Hanalioglu, Zulfiye
    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)).
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    Asymptotic Results for an Inventory Model of Type (s, S) with Asymmetric Triangular Distributed Interference of Chance and Delay
    (Gazi Univ, 2018) Hanalioglu, Zulfiye; Khaniyev, Tahir
    In this study, a semi - Markovian inventory model of type (s, S) is considered and the model is expressed by a modification of a renewal - reward process (X(t)) with an asymmetric triangular distributed interference of chance and delay. The ergodicity of the process X(t) is proved under some weak conditions. Additionally, exact expressions and three - term asymptotic expansions are found for all the moments of the ergodic distribution. Finally, obtained asymptotic results are compared with exact results for a special case.
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    Limit theorem for a semi - Markovian stochastic model of type (s,S)
    (Hacettepe Univ, Fac Sci, 2019) Hanalioglu, Zulfiye; Khaniyev, Tahir
    In this study, a semi-Markovian inventory model of type (s,S) is considered and the model is expressed by means of renewal-reward process (X(t)) with an asymmetric triangular distributed interference of chance and delay. The ergodicity of the process X(t) is proved and the exact expression for the ergodic distribution is obtained. Then, two-term asymptotic expansion for the ergodic distribution is found for standardized process W(t) equivalent to (2X(t))/(S - s). Finally, using this asymptotic expansion, the weak convergence theorem for the ergodic distribution of the process W(t) is proved and the explicit form of the limit distribution is found.