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Stability Problems for Stochastic Models: Theory and Applications II
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Stability Problems for Stochastic Models: Theory and Applications II

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematics and Computer Science".

Deadline for manuscript submissions: closed (31 December 2021) | Viewed by 25091

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editors

Prof. Dr. Alexander Zeifman

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Guest Editor
1. Department of Applied Mathematics, Vologda State University, 160000 Vologda, Russia
2. Institute of Informatics Problems of the Federal Research Center “Computer Science and Control”, Russian Academy of Sciences, 119333 Moscow, Russia
3. Vologda Research Center of the Russian Academy of Sciences, 160014 Vologda, Russia
Interests: stochastic models; continuous-time Markov chains; queueing models; biological models
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Victor Korolev

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Guest Editor
1. Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, Russia
2. Moscow Center for Fundamental and Applied Mathematics, 119991 Moscow, Russia
3. Federal Research Center “Informatics and Control” of the Russian Academy of Sciences, 119333 Moscow, Russia
Interests: stochastic models; risk processes; queueing theory; limit theorems
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Alexander Sipin

E-Mail Website
Guest Editor
Department of Mathematics, Vologda State University, 160000 Vologda, Russia
Interests: stochastic models; monte carlo methods
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The aim of this Special Issue is to publish original research articles that cover recent advances in probability theory, stochastic processes, mathematical statistics, and their applications. The focus will especially be on stability problems related to this field, including treatment of limit theorems as the source of practical approximations, applications of limit distributions as probability models of statistical regularities in observed data, and stochastic processes as models of dynamic phenomena in various research areas, such as queuing theory, physics, biology, economics, medicine, reliability theory, and financial mathematics.

Potential topics include but are not limited to the following:

  • Limit theorems and stability problems;
  • Asymptotic theory of stochastic processes;
  • Stable distributions and processes;
  • Asymptotic statistics;
  • Discrete probability models;
  • Characterizations of probability distributions;
  • Insurance and financial mathematics;
  • Applied statistics;
  • Queueing theory including queueing network models;
  • Markov chains and processes;
  • Large deviations and limit theorems;
  • Random motions;
  • Stochastic biological models;
  • Reliability, availability, maintenance, and inspection;
  • Computational methods for stochastic models.

Prof. Dr. Alexander Zeifman
Prof. Dr. Victor Korolev
Prof. Dr. Alexander Sipin
Guest Editors

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Published Papers (12 papers)

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Research

17 pages, 356 KiB  
Article
The Estimators of the Bent, Shape and Scale Parameters of the Gamma-Exponential Distribution and Their Asymptotic Normality
by Alexey Kudryavtsev and Oleg Shestakov
Mathematics 2022, 10(4), 619; https://doi.org/10.3390/math10040619 - 17 Feb 2022
Cited by 4 | Viewed by 1399
Abstract
When modeling real phenomena, special cases of the generalized gamma distribution and the generalized beta distribution of the second kind play an important role. The paper discusses the gamma-exponential distribution, which is closely related to the listed ones. The asymptotic normality of the [...] Read more.
When modeling real phenomena, special cases of the generalized gamma distribution and the generalized beta distribution of the second kind play an important role. The paper discusses the gamma-exponential distribution, which is closely related to the listed ones. The asymptotic normality of the previously obtained strongly consistent estimators for the bent, shape, and scale parameters of the gamma-exponential distribution at fixed concentration parameters is proved. Based on these results, asymptotic confidence intervals for the estimated parameters are constructed. The statements are based on the method of logarithmic cumulants obtained using the Mellin transform of the considered distribution. An algorithm for filtering out unnecessary solutions of the system of equations for logarithmic cumulants and a number of examples illustrating the results obtained using simulated samples are presented. The difficulties arising from the theoretical study of the estimates of concentration parameters associated with the inversion of polygamma functions are also discussed. The results of the paper can be used in the study of probabilistic models based on continuous distributions with unbounded non-negative support. Full article
(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications II)
15 pages, 285 KiB  
Article
Comparing Distributions of Sums of Random Variables by Deficiency: Discrete Case
by Vladimir E. Bening and Victor Y. Korolev
Mathematics 2022, 10(3), 454; https://doi.org/10.3390/math10030454 - 30 Jan 2022
Cited by 1 | Viewed by 1667
Abstract
In the paper, we consider a new approach to the comparison of the distributions of sums of random variables. Unlike preceding works, for this purpose we use the notion of deficiency that is well known in mathematical statistics. This approach is used, first, [...] Read more.
In the paper, we consider a new approach to the comparison of the distributions of sums of random variables. Unlike preceding works, for this purpose we use the notion of deficiency that is well known in mathematical statistics. This approach is used, first, to determine the distribution of a separate random variable in the sum that provides the least possible number of summands guaranteeing the prescribed value of the (1−α)-quantile of the normalized sum for a given α∈(0,1), and second, to determine the distribution of a separate random variable in the sum that provides the least possible number of summands guaranteeing the prescribed value of the probability for the normalized sum to fall into a given interval. Both problems are solved under the condition that possible distributions of random summands possess coinciding three first moments. In both settings the best distribution delivers the smallest number of summands. Along with distributions of a non-random number of summands, we consider the case of random summation and introduce an analog of deficiency which can be used to compare the distributions of sums with random and non-random number of summands. The main mathematical tools used in the paper are asymptotic expansions for the distributions of R-valued functions of random vectors, in particular, normalized sums of independent identically distributed r.v.s and their quantiles. Along with the general case, main attention is paid to the situation where the summarized random variables are independent and identically distributed. The approach under consideration is applied to determination of the distribution of insurance payments providing the least insurance portfolio size under prescribed Value-at-Risk or non-ruin probability. Full article
(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications II)
15 pages, 822 KiB  
Article
Equilibrium in a Queueing System with Retrials
by Julia Chirkova, Vladimir Mazalov and Evsey Morozov
Mathematics 2022, 10(3), 428; https://doi.org/10.3390/math10030428 - 28 Jan 2022
Cited by 7 | Viewed by 2466
Abstract
We find an equilibrium in a single-server queueing system with retrials and strategic timing of the customers. We consider a set of customers, each of which must decide when to arrive to a queueing system during a fixed period of time. In this [...] Read more.
We find an equilibrium in a single-server queueing system with retrials and strategic timing of the customers. We consider a set of customers, each of which must decide when to arrive to a queueing system during a fixed period of time. In this system, after completion of service, the server seeks a customer blocked in a virtual orbit (orbital customer) to be served next, unless a new customer captures the server. We develop, in detail, a setting with two and three customers in the set, and formulate and discuss the problem for the general case with an arbitrary number of customers. The numerical examples for the system with two and three customers included as well. Full article
(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications II)
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12 pages, 2042 KiB  
Article
Conditions for Existence of Second-Order and Third-Order Filters for Discrete Systems with Additive Noises
by Mikhail Kamenshchikov
Mathematics 2022, 10(3), 370; https://doi.org/10.3390/math10030370 - 25 Jan 2022
Cited by 2 | Viewed by 2145
Abstract
The problem of constructing functional optimal observers (filters) for stochastic control systems with additive noises in discrete time are studied in this work. Under the assumption that there is no filter of the first order, necessary and sufficient conditions for the existence of [...] Read more.
The problem of constructing functional optimal observers (filters) for stochastic control systems with additive noises in discrete time are studied in this work. Under the assumption that there is no filter of the first order, necessary and sufficient conditions for the existence of filters of the second and third order are obtained in the canonical basis. Analytical expressions of the transfer function matrix from the input noise to the estimation error are presented. A numerical example is given to compare the performance of filters by the quadratic criterion in the steady state. Full article
(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications II)
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19 pages, 838 KiB  
Article
Resource Retrial Queue with Two Orbits and Negative Customers
by Ekaterina Lisovskaya, Ekaterina Fedorova, Radmir Salimzyanov and Svetlana Moiseeva
Mathematics 2022, 10(3), 321; https://doi.org/10.3390/math10030321 - 20 Jan 2022
Cited by 3 | Viewed by 2071
Abstract
In this paper, a multi-server retrial queue with two orbits is considered. There are two arrival processes of positive customers (with two types of customers) and one process of negative customers. Every positive customer requires some amount of resource whose total capacity is [...] Read more.
In this paper, a multi-server retrial queue with two orbits is considered. There are two arrival processes of positive customers (with two types of customers) and one process of negative customers. Every positive customer requires some amount of resource whose total capacity is limited in the system. The service time does not depend on the customer’s resource requirement and is exponentially distributed with parameters depending on the customer’s type. If there is not enough amount of resource for the arriving customer, the customer goes to one of the two orbits, according to his type. The duration of the customer delay in the orbit is exponentially distributed. A negative customer removes all the customers that are served during his arrival and leaves the system. The objects of the study are the number of customers in each orbit and the number of customers of each type being served in the stationary regime. The method of asymptotic analysis under the long delay of the customers in the orbits is applied for the study. Numerical analysis of the obtained results is performed to show the influence of the system parameters on its performance measures. Full article
(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications II)
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21 pages, 666 KiB  
Article
Self-Service System with Rating Dependent Arrivals
by Alexander Dudin, Olga Dudina, Sergei Dudin and Yulia Gaidamaka
Mathematics 2022, 10(3), 297; https://doi.org/10.3390/math10030297 - 19 Jan 2022
Cited by 12 | Viewed by 1825
Abstract
A multi-server infinite buffer queueing system with additional servers (assistants) providing help to the main servers when they encounter problems is considered as the model of real-world systems with customers’ self-service. Such systems are widely used in many areas of human activity. An [...] Read more.
A multi-server infinite buffer queueing system with additional servers (assistants) providing help to the main servers when they encounter problems is considered as the model of real-world systems with customers’ self-service. Such systems are widely used in many areas of human activity. An arrival flow is assumed to be the novel essential generalization of the known Markov Arrival Process (MAP) to the case of the dynamic dependence of the parameters of the MAP on the rating of the system. The rating is the process defined at any moment by the quality of service of previously arrived customers. The possibilities of a customers immediate departure from the system at the entrance to the system and the buffer due to impatience are taken into account. The system is analyzed via the use of the results for multi-dimensional Markov chains with level-dependent behavior. The transparent stability condition is derived, as well as the expressions for the key performance indicators of the system in terms of the stationary probabilities of the Markov chain. Numerical results are provided. Full article
(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications II)
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20 pages, 568 KiB  
Article
A Time-Inhomogeneous Prendiville Model with Failures and Repairs
by Virginia Giorno and Amelia G. Nobile
Mathematics 2022, 10(2), 251; https://doi.org/10.3390/math10020251 - 14 Jan 2022
Cited by 3 | Viewed by 1543
Abstract
We consider a time-inhomogeneous Markov chain with a finite state-space which models a system in which failures and repairs can occur at random time instants. The system starts from any state j (operating, F, R). Due to a failure, a transition [...] Read more.
We consider a time-inhomogeneous Markov chain with a finite state-space which models a system in which failures and repairs can occur at random time instants. The system starts from any state j (operating, F, R). Due to a failure, a transition from an operating state to F occurs after which a repair is required, so that a transition leads to the state R. Subsequently, there is a restore phase, after which the system restarts from one of the operating states. In particular, we assume that the intensity functions of failures, repairs and restores are proportional and that the birth-death process that models the system is a time-inhomogeneous Prendiville process. Full article
(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications II)
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16 pages, 668 KiB  
Article
Convergence Bounds for Limited Processor Sharing Queue with Impatience for Analyzing Non-Stationary File Transfer in Wireless Network
by Irina Kochetkova, Yacov Satin, Ivan Kovalev, Elena Makeeva, Alexander Chursin and Alexander Zeifman
Mathematics 2022, 10(1), 30; https://doi.org/10.3390/math10010030 - 22 Dec 2021
Cited by 2 | Viewed by 2243
Abstract
The data transmission in wireless networks is usually analyzed under the assumption of non-stationary rates. Nevertheless, they strictly depend on the time of day, that is, the intensity of arrival and daily workload profiles confirm this fact. In this article, we consider the [...] Read more.
The data transmission in wireless networks is usually analyzed under the assumption of non-stationary rates. Nevertheless, they strictly depend on the time of day, that is, the intensity of arrival and daily workload profiles confirm this fact. In this article, we consider the process of downloading a file within a single network segment and unsteady speeds—for arrivals, file sizes, and losses due to impatience. To simulate the scenario, a queuing system with elastic traffic with non-stationary intensity is used. Formulas are given for the main characteristics of the model: the probability of blocking a new user, the average number of users in service, and the queue. A method for calculating the boundaries of convergence of the model is proposed, which is based on the logarithmic norm of linear operators. The boundaries of the rate of convergence of the main limiting characteristics of the queue length process were also established. For clarity of the influence of the parameters, a numerical analysis was carried out and presented. Full article
(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications II)
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26 pages, 651 KiB  
Article
A Continuous-Time Network Evolution Model Describing 2- and 3-Interactions
by István Fazekas and Attila Barta
Mathematics 2021, 9(23), 3143; https://doi.org/10.3390/math9233143 - 6 Dec 2021
Cited by 1 | Viewed by 2315
Abstract
A continuous-time network evolution model is considered. The evolution of the network is based on 2- and 3-interactions. 2-interactions are described by edges, and 3-interactions are described by triangles. The evolution of the edges and triangles is governed by a multi-type continuous-time branching [...] Read more.
A continuous-time network evolution model is considered. The evolution of the network is based on 2- and 3-interactions. 2-interactions are described by edges, and 3-interactions are described by triangles. The evolution of the edges and triangles is governed by a multi-type continuous-time branching process. The limiting behaviour of the network is studied by mathematical methods. We prove that the number of triangles and edges have the same magnitude on the event of non-extinction, and it is eαt, where α is the Malthusian parameter. The probability of the extinction and the degree process of a fixed vertex are also studied. The results are illustrated by simulations. Full article
(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications II)
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19 pages, 663 KiB  
Article
Monte Carlo Algorithms for the Extracting of Electrical Capacitance
by Andrei Kuznetsov and Alexander Sipin
Mathematics 2021, 9(22), 2922; https://doi.org/10.3390/math9222922 - 17 Nov 2021
Cited by 5 | Viewed by 2106
Abstract
We present new Monte Carlo algorithms for extracting mutual capacitances for a system of conductors embedded in inhomogeneous isotropic dielectrics. We represent capacitances as functionals of the solution of the external Dirichlet problem for the Laplace equation. Unbiased and low-biased estimators for the [...] Read more.
We present new Monte Carlo algorithms for extracting mutual capacitances for a system of conductors embedded in inhomogeneous isotropic dielectrics. We represent capacitances as functionals of the solution of the external Dirichlet problem for the Laplace equation. Unbiased and low-biased estimators for the capacitances are constructed on the trajectories of the Random Walk on Spheres or the Random Walk on Hemispheres. The calculation results show that the accuracy of these new algorithms does not exceed the statistical error of estimators, which is easily determined in the course of calculations. The algorithms are based on mean value formulas for harmonic functions in different domains and do not involve a transition to a difference problem. Hence, they do not need a lot of storage space. Full article
(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications II)
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26 pages, 428 KiB  
Article
Optimal Prefetching in Random Trees
by Kausthub Keshava, Alain Jean-Marie and Sara Alouf
Mathematics 2021, 9(19), 2437; https://doi.org/10.3390/math9192437 - 1 Oct 2021
Viewed by 1847
Abstract
We propose and analyze a model for optimizing the prefetching of documents, in the situation where the connection between documents is discovered progressively. A random surfer moves along the edges of a random tree representing possible sequences of documents, which is known to [...] Read more.
We propose and analyze a model for optimizing the prefetching of documents, in the situation where the connection between documents is discovered progressively. A random surfer moves along the edges of a random tree representing possible sequences of documents, which is known to a controller only up to depth d. A quantity k of documents can be prefetched between two movements. The question is to determine which nodes of the known tree should be prefetched so as to minimize the probability of the surfer moving to a node not prefetched. We analyzed the model with the tools of Markov decision process theory. We formally identified the optimal policy in several situations, and we identified it numerically in others. Full article
(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications II)
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11 pages, 327 KiB  
Article
Bounds on the Rate of Convergence for MtX/MtX/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 2 | Viewed by 1633
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 l1. Full article
(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications II)
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