Tahir Khaniyev

Asymptotic Expansions for the Moments of the Semi-Markovian Random Walk with Gamma Distributed Interference of Chance

In this study, a semi-Markovian random walk with a discrete interference of chance (X(t)) is considered and under some weak assumptions the ergodicity of this process is discussed. The exact formulas for the first four moments of the ergodic distribution of the process X(t) are obtained when the random variable ζ1, which describes a discrete interference of chance, has a gamma distribution with parameters (α, λ), α > 1, λ > 0. Based on these results, the asymptotic expansions are obtained for the first four moments of the ergodic distribution of the process X(t), as λ → 0. Furthermore, the asymptotic expansions for the skewness and kurtosis of the ergodic distribution of the process X(t) are established. Finally, it is discussed that the alternative estimations for the stationary characteristics of this process can be offered by using obtained asymptotic expansions.

Stochastic approximations for optimal buffer capacity of many-station production lines

Probabilistic processing times, times between breakdowns and repair times make the amount of stock in buffers between stations in production lines behave as a stochastic process. Too much or too little buffer stock reduces system economy and efficiency, respectively. We obtain optimum buffer capacities and initial stock levels for production lines employing a mathematical random walk approach based on the maximum and minimum values of a stochastic process in a time window. Two approximations are developed, each useful under different risk-acceptance assumptions. Simulation results populate the equations. A motivating case study from a discrete part manufacturing line, including an example of using regression on the simulated results, is presented.

Ergodic distribution for a fuzzy inventory model of type (s,S) with gamma distributed demands

In this study, a stochastic process (X(t)), which describes a fuzzy inventory model of type (s,S) is considered. Under some weak assumptions, the ergodic distribution of the process X(t) is expressed by a fuzzy renewal function U(x). Then, membership function of the fuzzy renewal function U(x) is obtained when the amount of demand has a Gamma distribution with fuzzy parameters. Finally, membership function and alpha cuts of fuzzy ergodic distribution of this process is derived by using extension principle of L. Zadeh.

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 ζ 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 (α, λ). Under this assumption, the asymptotic expansions for the first four moments of the ergodic distribution of the process X(t) are derived, as λ → 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 β.

Asymptotic approach for a renewal-reward process with a general interference of chance

ABSTRACT In this study, a renewal-reward process with a discrete interference of chance is constructed and considered. Under weak conditions, the ergodicity of the process X(t) is proved and exact formulas for the ergodic distribution and its moments are found. Within some assumptions for the discrete interference of chance in general form, two-term asymptotic expansions for all moments of the ergodic distribution are obtained. Additionally, kurtosis coefficient, skewness coefficient, and coefficient of variation of the ergodic distribution are computed. As a special case, a semi-Markovian inventory model of type (s, S) is investigated.

Investigation of fuzzy inventory model of type (s, S) with Nakagami distributed demands

In this study, a fuzzy inventory model of type (s, S) is considered under Nakagami distribution of demands. We first obtain the membership function of the fuzzy renewal function when the amount of demand has Nakagami distribution with a fuzzy spread parameter. By using fuzzy renewal function, we obtain the fuzzy ergodic distribution of this process. Also some numerical results are obtained with the use of this membership function.

Approximation Formulas for the Ergodic Moments of Gaussian Random Walk with a Reflecting Barrier

In this study, Gaussian random walk process with a generalized reflecting barrier is constructed mathematically. Under some weak conditions, the ergodicity of the process is discussed and exact form of the first four moments of the ergodic distribution is obtained. After, the asymptotic expansions for these moments are established. Moreover, the coefficients of the asymptotic expansions are expressed by means of numerical characteristics of a residual waiting time.

Approximation formulas for the moments of the boundary functional of a Gaussian random walk with positive drift by using Siegmund's formula

Abstract In this study, a boundary functional () are mathematically constructed for a Gaussian random walk (GRW) with positive drift β and first four moments of the functional are expressed in terms of ladder variables based on Dynkin Principle. Moreover, approximation formulas for first three moments of ladder height are proposed based on the formulas of Siegmund (1979) when β ↓0. Finally, approximation formulas for the first four moments of the boundary functional are obtained by using Siegmund formulas and meta modeling, when β ∈[0.1, 3.6].

Deterministic stability and random behavior of a Hepatitis C model

The deterministic stability of a model of Hepatitis C which includes a term defining the effect of immune system is studied on both local and global scales. Random effect is added to the model to investigate the random behavior of the model. The numerical characteristics such as the expectation, variance and confidence interval are calculated for random effects with two different distributions from the results of numerical simulations. In addition, the compliance of the random behavior of the model and the deterministic stability results is examined.

Three-term asymptotic expansion: A semi-Markovian random walk with a generalized beta distributed interference of chance

ABSTRACT A semi-Markovian random walk process (X(t)) with a generalized beta distribution of chance is considered. The asymptotic expansions for the first four moments of the ergodic distribution of the process are obtained as E(ζn) → ∞ when the random variable ζn has a generalized beta distribution with parameters (s, S, α, β);  α, β > 1, 0  ⩽ s < S < ∞. Finally, the accuracy of the asymptotic expansions is examined by using the Monte Carlo simulation method.