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/1 Queueing Models

by Alexander Zeifman, Yacov Satin and Alexander Sipin

Mathematics 2021, 9(15), 1752; https://doi.org/10.3390/math9151752 - 25 Jul 2021

Cited by 3 | Viewed by 2150

Abstract

We apply the method of differential inequalities for the computation of upper bounds for the rate of convergence to the limiting regime for one specific class of (in)homogeneous continuous-time Markov chains. Such an approach seems very general; the corresponding description and bounds were [...] Read more.

We apply the method of differential inequalities for the computation of upper bounds for the rate of convergence to the limiting regime for one specific class of (in)homogeneous continuous-time Markov chains. Such an approach seems very general; the corresponding description and bounds were considered earlier for finite Markov chains with analytical in time intensity functions. Now we generalize this method to locally integrable intensity functions. Special attention is paid to the situation of a countable Markov chain. To obtain these estimates, we investigate the corresponding forward system of Kolmogorov differential equations as a differential equation in the space of sequences

l_{1}

. Full article

(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications II)

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18 pages, 1279 KB

Open AccessArticle

Entropy-Based Classification of Elementary Cellular Automata under Asynchronous Updating: An Experimental Study

by Qin Lei, Jia Lee, Xin Huang and Shuji Kawasaki

Entropy 2021, 23(2), 209; https://doi.org/10.3390/e23020209 - 8 Feb 2021

Cited by 12 | Viewed by 3983

Abstract

Classification of asynchronous elementary cellular automata (AECAs) was explored in the first place by Fates et al. (Complex Systems, 2004) who employed the asymptotic density of cells as a key metric to measure their robustness to stochastic transitions. Unfortunately, the asymptotic density seems [...] Read more.

Classification of asynchronous elementary cellular automata (AECAs) was explored in the first place by Fates et al. (Complex Systems, 2004) who employed the asymptotic density of cells as a key metric to measure their robustness to stochastic transitions. Unfortunately, the asymptotic density seems unable to distinguish the robustnesses of all AECAs. In this paper, we put forward a method that goes one step further via adopting a metric entropy (Martin, Complex Systems, 2000), with the aim of measuring the asymptotic mean entropy of local pattern distribution in the cell space of any AECA. Numerical experiments demonstrate that such an entropy-based measure can actually facilitate a complete classification of the robustnesses of all AECA models, even when all local patterns are restricted to length 1. To gain more insights into the complexity concerning the forward evolution of all AECAs, we consider another entropy defined in the form of Kolmogorov–Sinai entropy and conduct preliminary experiments on classifying their uncertainties measured in terms of the proposed entropy. The results reveal that AECAs with low uncertainty tend to converge remarkably faster than models with high uncertainty. Full article

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16 pages, 562 KB

Open AccessArticle

Investigations of the Potential Application of k-out-of-n Systems in Oil and Gas Industry Objects

by Vladimir V. Rykov, Mikhail G. Sukharev and Victor Yu. Itkin

J. Mar. Sci. Eng. 2020, 8(11), 928; https://doi.org/10.3390/jmse8110928 - 16 Nov 2020

Cited by 23 | Viewed by 3289

Abstract

The purpose of this paper was to demonstrate the possibilities of assessing the reliability of oil and gas industry structures with the help of mathematical models of k-out-of-n systems. We show how the reliability of various structures in the oil and [...] Read more.

The purpose of this paper was to demonstrate the possibilities of assessing the reliability of oil and gas industry structures with the help of mathematical models of k-out-of-n systems. We show how the reliability of various structures in the oil and gas complex can be described and investigated using k-out-of-n models. Because the initial information about the life and repair time of components of systems is only usually known on the scale of one and/or two moments, we focus on the problem of the sensitivity analysis of the system reliability indices to the shape of its components repair time distributions. To address this problem, we used the so-called markovization method, based on the introduction of supplementary variables, to model the system behavior with the help of the two-dimensional Markov process with discrete-continuous states. On the basis of the forward Kolmogorov equations for the time-dependent process’ state probabilities, relevant balance equations for the process’ stationary probabilities are presented. Using these equations, stationary probabilities and some reliability indices for two examples from the oil and gas industry were calculated and their sensitivity to the system component’s repair time distributions was analyzed. Calculations show that under “rare” component failures, most system reliability indices become practically insensitive to the shape of the components repair time distributions. Full article

(This article belongs to the Special Issue Safe, Secure and Sustainable Oil and Gas Drilling, Exploitation and Pipeline Transport Offshore)

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15 pages, 492 KB

Open AccessArticle

On Probability Characteristics for a Class of Queueing Models with Impatient Customers

by Yacov Satin, Alexander Zeifman, Alexander Sipin, Sherif I. Ammar and Janos Sztrik

Mathematics 2020, 8(4), 594; https://doi.org/10.3390/math8040594 - 15 Apr 2020

Cited by 8 | Viewed by 2886

Abstract

In this paper, a class of queueing models with impatient customers is considered. It deals with the probability characteristics of an individual customer in a non-stationary Markovian queue with impatient customers, the stationary analogue of which was studied previously as a successful approximation [...] Read more.

In this paper, a class of queueing models with impatient customers is considered. It deals with the probability characteristics of an individual customer in a non-stationary Markovian queue with impatient customers, the stationary analogue of which was studied previously as a successful approximation of a more general non-Markov model. A new mathematical model of the process is considered that describes the behavior of an individual requirement in the queue of requirements. This can be applied both in the stationary and non-stationary cases. Based on the proposed model, a methodology has been developed for calculating the system characteristics both in the case of the existence of a stationary solution and in the case of the existence of a periodic solution for the corresponding forward Kolmogorov system. Some numerical examples are provided to illustrate the effect of input parameters on the probability characteristics of the system. Full article

(This article belongs to the Special Issue Queueing Methods in Reliability Theory and Information Technology Systems)

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25 pages, 588 KB

Open AccessFeature PaperReview

Two Approaches to the Construction of Perturbation Bounds for Continuous-Time Markov Chains

by Alexander Zeifman, Victor Korolev and Yacov Satin

Mathematics 2020, 8(2), 253; https://doi.org/10.3390/math8020253 - 14 Feb 2020

Cited by 25 | Viewed by 4478

Abstract

This paper is largely a review. It considers two main methods used to study stability and to obtain appropriate quantitative estimates of perturbations of (inhomogeneous) Markov chains with continuous time and a finite or countable state space. An approach is described to the [...] Read more.

This paper is largely a review. It considers two main methods used to study stability and to obtain appropriate quantitative estimates of perturbations of (inhomogeneous) Markov chains with continuous time and a finite or countable state space. An approach is described to the construction of perturbation estimates for the main five classes of such chains associated with queuing models. Several specific models are considered for which the limit characteristics and perturbation bounds for admissible “perturbed” processes are calculated. Full article

(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications)

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17 pages, 417 KB

Open AccessArticle

Mean Field Game with Delay: A Toy Model

by Jean-Pierre Fouque and Zhaoyu Zhang

Risks 2018, 6(3), 90; https://doi.org/10.3390/risks6030090 - 1 Sep 2018

Cited by 10 | Viewed by 4317

Abstract

We study a toy model of linear-quadratic mean field game with delay. We “lift” the delayed dynamic into an infinite dimensional space, and recast the mean field game system which is made of a forward Kolmogorov equation and a backward Hamilton-Jacobi-Bellman equation. We [...] Read more.

We study a toy model of linear-quadratic mean field game with delay. We “lift” the delayed dynamic into an infinite dimensional space, and recast the mean field game system which is made of a forward Kolmogorov equation and a backward Hamilton-Jacobi-Bellman equation. We identify the corresponding master equation. A solution to this master equation is computed, and we show that it provides an approximation to a Nash equilibrium of the finite player game. Full article

(This article belongs to the Special Issue Systemic Risk in Finance and Insurance)

18 pages, 330 KB

Open AccessArticle

Linear–Quadratic Mean-Field-Type Games: A Direct Method

by Tyrone E. Duncan and Hamidou Tembine

Games 2018, 9(1), 7; https://doi.org/10.3390/g9010007 - 12 Feb 2018

Cited by 42 | Viewed by 9647

Abstract

In this work, a multi-person mean-field-type game is formulated and solved that is described by a linear jump-diffusion system of mean-field type and a quadratic cost functional involving the second moments, the square of the expected value of the state, and the control [...] Read more.

In this work, a multi-person mean-field-type game is formulated and solved that is described by a linear jump-diffusion system of mean-field type and a quadratic cost functional involving the second moments, the square of the expected value of the state, and the control actions of all decision-makers. We propose a direct method to solve the game, team, and bargaining problems. This solution approach does not require solving the Bellman–Kolmogorov equations or backward–forward stochastic differential equations of Pontryagin’s type. The proposed method can be easily implemented by beginners and engineers who are new to the emerging field of mean-field-type game theory. The optimal strategies for decision-makers are shown to be in a state-and-mean-field feedback form. The optimal strategies are given explicitly as a sum of the well-known linear state-feedback strategy for the associated deterministic linear–quadratic game problem and a mean-field feedback term. The equilibrium cost of the decision-makers are explicitly derived using a simple direct method. Moreover, the equilibrium cost is a weighted sum of the initial variance and an integral of a weighted variance of the diffusion and the jump process. Finally, the method is used to compute global optimum strategies as well as saddle point strategies and Nash bargaining solution in state-and-mean-field feedback form. Full article

(This article belongs to the Special Issue Mean-Field-Type Game Theory)

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9 pages, 687 KB

Open AccessArticle

Fokker-Planck Equation and Thermodynamic System Analysis

by Umberto Lucia and Gianpiero Gervino

Entropy 2015, 17(2), 763-771; https://doi.org/10.3390/e17020763 - 9 Feb 2015

Cited by 10 | Viewed by 7591

Abstract

The non-linear Fokker-Planck equation or Kolmogorov forward equation is currently successfully applied for deep analysis of irreversibility and it gives an excellent approximation near the free energy minimum, just as Boltzmann’s definition of entropy follows from finding the maximum entropy state. A connection [...] Read more.

The non-linear Fokker-Planck equation or Kolmogorov forward equation is currently successfully applied for deep analysis of irreversibility and it gives an excellent approximation near the free energy minimum, just as Boltzmann’s definition of entropy follows from finding the maximum entropy state. A connection to Fokker-Planck dynamics and the free energy functional is presented and discussed—this approach has been particularly successful to deal with metastability. We focus our attention on investigating and discussing the fundamental role of dissipation analysis in metastable systems. The major novelty of our approach is that the obtained results enable us to reveal an appealing, and previously unexplored relationship between Fokker-Planck equation and the associated free energy functional. Namely, we point out that the dynamics may be regarded as a gradient flux, or a steepest descent, for the free energy. Full article

(This article belongs to the Special Issue Entropic Aspects in Statistical Physics of Complex Systems)

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