Asymptotic results for an inventory model of type (s, S) with a generalized beta interference of chance

In this study, asymptotic expansion for ergodic distribution of an inventory control model of type (s, S) with generalized beta interference of chance is obtained, when S − s → ∞. Moreover, weak convergence theorem is proved for ergodic distribution. Finally, the accuracy of the asymptotic expansion is examined with Monte Carlo simulation method.Publisher's Versio

Modeling disease transmission dynamics with random data and heavy tailed random effects: the Zika case

This work was supported by Research Fund of the Recep Tayyip Erdogan University.In this study, we investigate a compartmental model of Zika Virus transmission under random effects. Random effects enable the analysis of random numerical characteristics of transmission, which cannot be modeled through deterministic equations. Data obtained from Zika studies in the literature are used along with heavy tailed random effects to obtain new random variables for the parameters of the deterministic model. Finally, simulations of the model are carried out to analyze the random dynamics of Zika Virus transmission. Deterministic results are compared with results from the simulations of the random system to underline the advantages of a random modeling approach. It is shown that the random model provides additional results for disease transmission dynamics such as results for standard deviation and coefficients of variation, making it a valuable alternative to deterministic modeling. Random results suggest around 90% - 120% coefficient of variation for the random model underlining the fact that the randomness should not be ignored for the transmission of this disease.Publisher's Versio

Limit theorem for a semi-Markovian random walk with general interference of chance

A semi-Markovian random-walk process with general interference of chance was constructed and investigated. The key point of this study is the assumption that the discrete interference of chance has a general form. Under some conditions, it is proved that the process is ergodic, and the exact forms of the ergodic distribution and characteristic function of the process are obtained. By using basic identity for random walks, the characteristic function of the process is expressed by the characteristic function of a boundary functional. Then, two-term asymptotic expansion for the characteristic function of the standardized process is found. Using this asymptotic expansion, a weak convergence theorem for the ergodic distribution of the standardized process is proved, and the limiting form for the ergodic distribution is obtained. The obtained limit distribution coincides with the limit distribution of the residual waiting time of the renewal process generated by a sequence of random variables expressing the discrete interference of chance

Asymptotic expansions for the ergodic moments of a semi-markovian random walk with a generalized delaying barrier

In this study, a semi-Markovian random walk process (X(t)) with a generalized delaying barrier is considered and the ergodic theorem for this process is proved under some weak conditions. Then, the exact expressions and asymptotic expansions for the first four ergodic moments of the process X(t) are obtained.This research is supported partially by TUBITAK, under Project 110T559.Publisher's Versio

Limit theorem for a semi - Markovian stochastic model of type (s,S)

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

On the Moments of a Semi-Markovian Random Walk with Gaussian Distribution of Summands

In this article, a semi-Markovian random walk with delay and a discrete interference of chance (X(t)) is considered. It is assumed that the random variables (n), n=1, 2,..., which describe the discrete interference of chance form an ergodic Markov chain with ergodic distribution which is a gamma distribution with parameters (, ). Under this assumption, the asymptotic expansions for the first four moments of the ergodic distribution of the process X(t) are derived, as 0. Moreover, by using the Riemann zeta-function, the coefficients of these asymptotic expansions are expressed by means of numerical characteristics of the summands, when the process considered is a semi-Markovian Gaussian random walk with small drift .[Aliyev, Rovshan] Baku State Univ, Dept Probabil Theory & Math Stat, Baku, Azerbaijan; [Khaniyev, Tahir] TOBB Univ Econ & Technol, Dept Ind Engn, Ankara, Turkey; [Aliyev, Rovshan; Khaniyev, Tahir] Azerbaijan Natl Acad Sci, Inst Cybernet, Baku, Azerbaija

Asymptotic Results for an Inventory Model of Type (s, S) with Asymmetric Triangular Distributed Interference of Chance and Delay

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.[Hanalioglu, Zulfiye] Karabuk Univ, Dept Actuarial Sci & Risk Management, Karabuk, Turkey; [Khaniyev, Tahir] TOBB Univ Econ & Technol, Dept Ind Engn, TR-06560 Ankara, Turke

Had Teorem untuk Jalan Rawak Semi-Markovan dengan Kemungkinan Gangguan Umum

Proses jalan rawak semi-Markovan dengan kemungkinan gangguan umum telah dibangunkan dan dikaji. Isi utama kajian ini adalah andaian bahawa kemungkinan gangguan diskrit mempunyai bentuk umum. Dalam beberapa keadaan, terbukti bahawa prosesnya ergodik dan bentuk asal taburan ergodik serta fungsi pencirian prosesnya diperoleh. Dengan menggunakan identiti asas untuk jalan rawak, fungsi pencirian prosesnya diungkapkan oleh fungsi pencirian sempadan fungsian. Kemudian, pengembangan asimptotik dua penggal untuk fungsi pencirian piawai prosesnya ditemui. Dengan menggunakan pengembangan asimtotik ini, teorem penumpuan yang lemah untuk taburan ergodik daripada proses piawai dibuktikan dan bentuk pembatasan untuk taburan ergodik diperoleh. Taburan had yang diperoleh bertepatan dengan had taburan sisa masa menunggu proses pembaharuan yang dihasilkan oleh jujukan pemboleh ubah rawak yang mengungkapkan kemungkinan gangguan diskrit.A semi-Markovian random-walk process with general interference of chance was constructed and investigated. The key point of this study is the assumption that the discrete interference of chance has a general form. Under some conditions, it is proved that the process is ergodic, and the exact forms of the ergodic distribution and characteristic function of the process are obtained. By using basic identity for random walks, the characteristic function of the process is expressed by the characteristic function of a boundary functional. Then, two-term asymptotic expansion for the characteristic function of the standardized process is found. Using this asymptotic expansion, a weak convergence theorem for the ergodic distribution of the standardized process is proved, and the limiting form for the ergodic distribution is obtained. The obtained limit distribution coincides with the limit distribution of the residual waiting time of the renewal process generated by a sequence of random variables expressing the discrete interference of chance

A novel replacement policy for a linear deteriorating system using stochastic process with dependent components

In this study, a mechanical system with linear deterioration and preventive maintenance is considered. The state of the system over time is represented by a semicontinuous stochastic process with dependent components. The system cycles through on and off periods during its lifetime. The state of the system deteriorates linearly as a function of the usage time during on periods. When the system is offline, preventive maintenance is conducted, which improves the system state by a random amount. The system's on and off times and random improvement amounts are assumed to have general distributions. For such a system, our objective is to determine the expected value and variance for the number of preventive maintenance activities needed during the system lifetime and to propose a novel replacement policy for the system based on delay-time modeling. Finally, the effectiveness of the obtained asymptotic results and the proposed replacement policy are tested through simulation

Weak convergence theorem for ergodic distribution of stochastic processes with discrete interference of chance and generalized reflecting barrier

In this paper, a stochastic process with discrete interference of chance and generalized reflecting barrier (X (t)) is constructed and the ergodicity of this process is proved. Using basic identity for random walk processes, a characteristic function of the ergodic distribution is written with the help of characteristics of the boundary functional S-N1(x). Moreover, a weak convergence theorem for the ergodic distribution of the standardized process Y-lambda(t) = X(t)/lambda is proved, as lambda -> infinity and the limit form of the ergodic distribution is found.[Aliyev, R.] Baku State Univ, Dept Probabil Theory & Math Stat, Baku, Azerbaijan; [Aliyev, R.; Khaniyev, T.] Azerbaijan Natl Acad Sci, Inst Cybernet, Baku, Azerbaijan; [Khaniyev, T.; Gever, B.] TOBB Univ Econ & Technol, Dept Ind Engn, Ankara, Turke