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Stability Problems for Stochastic Models: Analytics, Asymptotics, Estimation and Applications
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Stability Problems for Stochastic Models: Analytics, Asymptotics, Estimation and Applications

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

Deadline for manuscript submissions: 10 February 2025 | Viewed by 19593

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 main focus will be on stability problems related to these fields, including the 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

Manuscript Submission Information

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

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Published Papers (14 papers)

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14 pages, 299 KiB  
Article
Properties of the SURE Estimates When Using Continuous Thresholding Functions for Wavelet Shrinkage
by Alexey Kudryavtsev and Oleg Shestakov
Mathematics 2024, 12(23), 3646; https://doi.org/10.3390/math12233646 - 21 Nov 2024
Viewed by 320
Abstract
Wavelet analysis algorithms in combination with thresholding procedures are widely used in nonparametric regression problems when estimating a signal function from noisy data. The advantages of these methods lie in their computational efficiency and the ability to adapt to the local features of [...] Read more.
Wavelet analysis algorithms in combination with thresholding procedures are widely used in nonparametric regression problems when estimating a signal function from noisy data. The advantages of these methods lie in their computational efficiency and the ability to adapt to the local features of the estimated function. It is usually assumed that the signal function belongs to some special class. For example, it can be piecewise continuous or piecewise differentiable and have a compact support. These assumptions, as a rule, allow the signal function to be economically represented on some specially selected basis in such a way that the useful signal is concentrated in a relatively small number of large absolute value expansion coefficients. Then, thresholding is performed to remove the noise coefficients. Typically, the noise distribution is assumed to be additive and Gaussian. This model is well studied in the literature, and various types of thresholding and parameter selection strategies adapted for specific applications have been proposed. The risk analysis of thresholding methods is an important practical task, since it makes it possible to assess the quality of both the methods themselves and the equipment used for processing. Most of the studies in this area investigate the asymptotic order of the theoretical risk. In practical situations, the theoretical risk cannot be calculated because it depends explicitly on the unobserved, noise-free signal. However, a statistical risk estimate constructed on the basis of the observed data can also be used to assess the quality of noise reduction methods. In this paper, a model of a signal contaminated with additive Gaussian noise is considered, and the general formulation of the thresholding problem with threshold functions belonging to a special class is discussed. Lower bounds are obtained for the threshold values that minimize the unbiased risk estimate. Conditions are also given under which this risk estimate is asymptotically normal and strongly consistent. The results of these studies can provide the basis for further research in the field of constructing confidence intervals and obtaining estimates of the convergence rate, which, in turn, will make it possible to obtain specific values of errors in signal processing for a wide range of thresholding methods. Full article
(This article belongs to the Special Issue Stability Problems for Stochastic Models: Analytics, Asymptotics, Estimation and Applications)
12 pages, 323 KiB  
Article
On One Approach to Obtaining Estimates of the Rate of Convergence to the Limiting Regime of Markov Chains
by Yacov Satin, Rostislav Razumchik, Alexander Zeifman and Ilya Usov
Mathematics 2024, 12(17), 2763; https://doi.org/10.3390/math12172763 - 6 Sep 2024
Viewed by 651
Abstract
We revisit the problem of the computation of the limiting characteristics of (in)homogeneous continuous-time Markov chains with the finite state space. In general, it can be performed only numerically. The common rule of thumb is to interrupt calculations after quite some time, hoping [...] Read more.
We revisit the problem of the computation of the limiting characteristics of (in)homogeneous continuous-time Markov chains with the finite state space. In general, it can be performed only numerically. The common rule of thumb is to interrupt calculations after quite some time, hoping that the values at some distant time interval will represent the sought-after solution. Convergence or ergodicity bounds, when available, can be used to answer such questions more accurately; i.e., they can indicate how to choose the position and the length of that distant time interval. The logarithmic norm method is a general technique that may allow one to obtain such bounds. Although it can handle continuous-time Markov chains with both finite and countable state spaces, its downside is the need to guess the proper similarity transformations, which may not exist. In this paper, we introduce a new technique, which broadens the scope of the logarithmic norm method. This is achieved by firstly splitting the generator of a Markov chain and then merging the convergence bounds of each block into a single bound. The proof of concept is illustrated by simple examples of the queueing theory. Full article
(This article belongs to the Special Issue Stability Problems for Stochastic Models: Analytics, Asymptotics, Estimation and Applications)
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18 pages, 318 KiB  
Article
The Implicit Euler Scheme for FSDEs with Stochastic Forcing: Existence and Uniqueness of the Solution
by Kęstutis Kubilius
Mathematics 2024, 12(16), 2436; https://doi.org/10.3390/math12162436 - 6 Aug 2024
Viewed by 566
Abstract
In this paper, we focus on fractional stochastic differential equations (FSDEs) with a stochastic forcing term, i.e., to FSDE, we add a stochastic forcing term. Using the implicit scheme of Euler’s approximation, the conditions for the existence and uniqueness of the solution of [...] Read more.
In this paper, we focus on fractional stochastic differential equations (FSDEs) with a stochastic forcing term, i.e., to FSDE, we add a stochastic forcing term. Using the implicit scheme of Euler’s approximation, the conditions for the existence and uniqueness of the solution of FSDEs with a stochastic forcing term are established. Such equations can be applied to considering FSDEs with a permeable wall. Full article
(This article belongs to the Special Issue Stability Problems for Stochastic Models: Analytics, Asymptotics, Estimation and Applications)
23 pages, 1799 KiB  
Article
Stochastic Growth Models for the Spreading of Fake News
by Antonio Di Crescenzo, Paola Paraggio and Serena Spina
Mathematics 2023, 11(16), 3597; https://doi.org/10.3390/math11163597 - 19 Aug 2023
Cited by 1 | Viewed by 1425
Abstract
The propagation of fake news in online social networks nowadays is becoming a critical issue. Consequently, many mathematical models have been proposed to mimic the related time evolution. In this work, we first consider a deterministic model that describes rumor propagation and can [...] Read more.
The propagation of fake news in online social networks nowadays is becoming a critical issue. Consequently, many mathematical models have been proposed to mimic the related time evolution. In this work, we first consider a deterministic model that describes rumor propagation and can be viewed as an extended logistic model. In particular, we analyze the main features of the growth curve, such as the limit behavior, the inflection point, and the threshold-crossing-time, through fixed boundaries. Then, in order to study the stochastic counterparts of the model, we consider two different stochastic processes: a time non-homogeneous linear pure birth process and a lognormal diffusion process. The conditions under which the means of the processes are identical to the deterministic curve are discussed. The first-passage-time problem is also investigated both for the birth process and the lognormal diffusion process. Finally, in order to study the variability of the stochastic processes introduced so far, we perform a comparison between their variances. Full article
(This article belongs to the Special Issue Stability Problems for Stochastic Models: Analytics, Asymptotics, Estimation and Applications)
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7 pages, 230 KiB  
Article
Heavy-Tailed Probability Distributions: Some Examples of Their Appearance
by Lev B. Klebanov, Yulia V. Kuvaeva-Gudoshnikova and Svetlozar T. Rachev
Mathematics 2023, 11(14), 3094; https://doi.org/10.3390/math11143094 - 13 Jul 2023
Viewed by 2250
Abstract
We provide two examples of the appearance of heavy-tailed distributions in social sciences applications. Among these distributions are the laws of Pareto and Lotka and some new ones. The examples are illustrated through the construction of suitable toy models. Full article
(This article belongs to the Special Issue Stability Problems for Stochastic Models: Analytics, Asymptotics, Estimation and Applications)
7 pages, 247 KiB  
Article
On a New Characterization of Harris Recurrence for Markov Chains and Processes
by Peter Glynn and Yanlin Qu
Mathematics 2023, 11(9), 2165; https://doi.org/10.3390/math11092165 - 5 May 2023
Cited by 1 | Viewed by 1858
Abstract
This paper shows that Harris recurrent Markov chains and processes can be characterized as the class of Markov chains and processes for which there exists a random time T at which the distribution of the chain or process does not depend on its [...] Read more.
This paper shows that Harris recurrent Markov chains and processes can be characterized as the class of Markov chains and processes for which there exists a random time T at which the distribution of the chain or process does not depend on its initial condition. In particular, no independence assumptions concerning the post-T process or T play a role in the characterization. Since Harris chains and processes are known to contain infinite sequences of regeneration times exhibiting various independence properties, it follows that the existence of this single T implies the existence of infinitely many times at which regeneration occurs. Full article
(This article belongs to the Special Issue Stability Problems for Stochastic Models: Analytics, Asymptotics, Estimation and Applications)
21 pages, 484 KiB  
Article
Renewable k-Out-of-n System with the Component-Wise Strategy of Preventive System Maintenance
by Vladimir Rykov, Olga Kochueva and Elvira Zaripova
Mathematics 2023, 11(9), 2158; https://doi.org/10.3390/math11092158 - 4 May 2023
Cited by 2 | Viewed by 1300
Abstract
At the SMARTY-22 conference, a review of the regenerative methods development was presented, including its application to the study of a non-renewable k-out-of-n system. This paper develops the previous study for the renewable k-out-of-n system, including an investigation different [...] Read more.
At the SMARTY-22 conference, a review of the regenerative methods development was presented, including its application to the study of a non-renewable k-out-of-n system. This paper develops the previous study for the renewable k-out-of-n system, including an investigation different preventive maintenance strategies based on the system state observation. We also include the review of Smith’s regeneration idea development. Some new results are presented that form the basis for an algorithm for comparing preventing maintenance strategies with respect to the maximization of the availability factor. A numerical study was conducted for the 4-out-of-6 and 4-out-of-8 models. The study demonstrates the sensitivity of decision making to the shape of the repair time distribution. Full article
(This article belongs to the Special Issue Stability Problems for Stochastic Models: Analytics, Asymptotics, Estimation and Applications)
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15 pages, 448 KiB  
Article
Ergodicity and Related Bounds for One Particular Class of Markovian Time—Varying Queues with Heterogeneous Servers and Customer’s Impatience
by Yacov Satin, Rostislav Razumchik, Ivan Kovalev and Alexander Zeifman
Mathematics 2023, 11(9), 1979; https://doi.org/10.3390/math11091979 - 22 Apr 2023
Cited by 2 | Viewed by 1044
Abstract
We consider a non-standard class of Markovian time: varying infinite capacity queues with possibly heterogeneous servers and impatience. We assume that during service time, a customer may switch to the faster server (with no delay), when such a server becomes available and no [...] Read more.
We consider a non-standard class of Markovian time: varying infinite capacity queues with possibly heterogeneous servers and impatience. We assume that during service time, a customer may switch to the faster server (with no delay), when such a server becomes available and no other customers are waiting. As a result, customers in the queue may become impatient and leave it. Under this setting and with certain restrictions on the intensity functions, the quantity of interest, the total number of customers in the system, is the level-dependent birth-and-death process (BPD). In this paper, for the first time in the literature, explicit upper bounds for the distance between two probability distributions of this BDP are obtained. Using the obtained ergodicity bounds in combination with the sensitivity bounds, we assess the stability of BDP under perturbations. Truncation bounds are also given, which allow for numerical solutions with guaranteed truncation errors. Finally, we provide numerical results to support the findings. Full article
(This article belongs to the Special Issue Stability Problems for Stochastic Models: Analytics, Asymptotics, Estimation and Applications)
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25 pages, 1941 KiB  
Article
On the Control over the Distribution of Ticks Based on the Extensions of the KISS Model
by Vassili N. Kolokoltsov
Mathematics 2023, 11(2), 478; https://doi.org/10.3390/math11020478 - 16 Jan 2023
Viewed by 1533
Abstract
Ticks and tick-borne diseases present a well-known threat to the health of people in many parts of the globe. The scientific literature devoted both to field observations and to modeling the propagation of ticks continues to grow. To date, the majority of the [...] Read more.
Ticks and tick-borne diseases present a well-known threat to the health of people in many parts of the globe. The scientific literature devoted both to field observations and to modeling the propagation of ticks continues to grow. To date, the majority of the mathematical studies have been devoted to models based on ordinary differential equations, where spatial variability was taken into account by a discrete parameter. Only a few papers use spatially nontrivial diffusion models, and they are devoted mostly to spatially homogeneous equilibria. Here we develop diffusion models for the propagation of ticks stressing spatial heterogeneity. This allows us to assess the sizes of control zones that can be created (using various available techniques) to produce a patchy territory, on which ticks will be eventually eradicated. Using averaged parameters taken from various field observations we apply our theoretical results to the concrete cases of the lone star ticks of North America and of the taiga ticks of Russia. From the mathematical point of view, we give criteria for global stability of the vanishing solution to certain spatially heterogeneous birth and death processes with diffusion. Full article
(This article belongs to the Special Issue Stability Problems for Stochastic Models: Analytics, Asymptotics, Estimation and Applications)
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16 pages, 316 KiB  
Article
Double-Sources Queuing-Inventory Systems with Finite Waiting Room and Destructible Stocks
by Agassi Melikov, Ramil Mirzayev and Janos Sztrik
Mathematics 2023, 11(1), 226; https://doi.org/10.3390/math11010226 - 2 Jan 2023
Cited by 5 | Viewed by 1464
Abstract
Models of double-source queuing-inventory systems are studied in the presence of a finite buffer for waiting in the queue of consumer customers, where instant destruction of inventory is possible. It is assumed that the lead times of orders, as well as the cost [...] Read more.
Models of double-source queuing-inventory systems are studied in the presence of a finite buffer for waiting in the queue of consumer customers, where instant destruction of inventory is possible. It is assumed that the lead times of orders, as well as the cost of delivery from various sources, differ from each other. Replenishment of stocks from various sources is carried out according to the following scheme: if the inventory level drops to the reorder point s, then a regular order for the supply of inventory to a slow source is generated; if the inventory level falls below a certain threshold value r, where r < s, then the system instantly cancels the regular order and generates an emergency order to the fast source. Models of systems that use (s, S) or (s, Q) replenishment policies are studied. Exact and approximate methods for finding the performance measures of the models under study are proposed. The problems of minimizing the total cost are solved by choosing the appropriate values of the parameters s and r when using different replenishment policies. Numerical examples demonstrated the high accuracy of an approximate method as well as compared performance measures of the system under various replenishment policies. Full article
(This article belongs to the Special Issue Stability Problems for Stochastic Models: Analytics, Asymptotics, Estimation and Applications)
12 pages, 593 KiB  
Article
Random Motions at Finite Velocity on Non-Euclidean Spaces
by Francesco Cybo Ottone and Enzo Orsingher
Mathematics 2022, 10(23), 4609; https://doi.org/10.3390/math10234609 - 5 Dec 2022
Viewed by 1371
Abstract
In this paper, random motions at finite velocity on the Poincaré half-plane and on the unit-radius sphere are studied. The moving particle at each Poisson event chooses a uniformly distributed direction independent of the previous evolution. This implies that the current distance [...] Read more.
In this paper, random motions at finite velocity on the Poincaré half-plane and on the unit-radius sphere are studied. The moving particle at each Poisson event chooses a uniformly distributed direction independent of the previous evolution. This implies that the current distance d(P0,Pt) from the starting point P0 is obtained by applying the hyperbolic Carnot formula in the Poincaré half-plane and the spherical Carnot formula in the analysis of the motion on the sphere. We obtain explicit results of the conditional and unconditional mean distance in both cases. Some results for higher-order moments are also presented for a small number of changes of direction. Full article
(This article belongs to the Special Issue Stability Problems for Stochastic Models: Analytics, Asymptotics, Estimation and Applications)
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14 pages, 1066 KiB  
Article
Ergodicity Bounds and Limiting Characteristics for a Modified Prendiville Model
by Ilya Usov, Yacov Satin, Alexander Zeifman and Victor Korolev
Mathematics 2022, 10(23), 4401; https://doi.org/10.3390/math10234401 - 22 Nov 2022
Cited by 4 | Viewed by 1147
Abstract
We consider the time-inhomogeneous Prendiville model with failures and repairs. The property of weak ergodicity is considered, and estimates of the rate of convergence for the main probabilistic characteristics of the model are obtained. Several examples are considered showing how such estimates are [...] Read more.
We consider the time-inhomogeneous Prendiville model with failures and repairs. The property of weak ergodicity is considered, and estimates of the rate of convergence for the main probabilistic characteristics of the model are obtained. Several examples are considered showing how such estimates are obtained and how the limiting characteristics themselves are constructed. Full article
(This article belongs to the Special Issue Stability Problems for Stochastic Models: Analytics, Asymptotics, Estimation and Applications)
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16 pages, 346 KiB  
Article
Bounds for the Rate of Convergence in the Generalized Rényi Theorem
by Victor Korolev
Mathematics 2022, 10(22), 4252; https://doi.org/10.3390/math10224252 - 14 Nov 2022
Cited by 4 | Viewed by 1609
Abstract
In the paper, an overview is presented of the results on the convergence rate bounds in limit theorems concerning geometric random sums and their generalizations to mixed Poisson random sums, including the case where the mixing law is itself a mixed exponential distribution. [...] Read more.
In the paper, an overview is presented of the results on the convergence rate bounds in limit theorems concerning geometric random sums and their generalizations to mixed Poisson random sums, including the case where the mixing law is itself a mixed exponential distribution. The main focus is on the upper bounds for the Zolotarev ζ-metric as the distance between the pre-limit and limit laws. New results are presented that extend existing estimates of the rate of convergence of geometric random sums (in the well-known Rényi theorem) to a considerably more general class of random indices whose distributions are mixed Poisson, including generalized negative binomial (e.g., Weibull-mixed Poisson), Pareto-type (Lomax)-mixed Poisson, exponential power-mixed Poisson, Mittag-Leffler-mixed Poisson, and one-sided Linnik-mixed Poisson distributions. A transfer theorem is proven that makes it possible to obtain upper bounds for the rate of convergence in the law of large numbers for mixed Poisson random sums with mixed exponential mixing distribution from those for geometric random sums (that is, from the convergence rate estimates in the Rényi theorem). Simple explicit bounds are obtained for ζ-metrics of the first and second orders. An estimate is obtained for the stability of representation of the Mittag-Leffler distribution as a geometric convolution (that is, as the distribution of a geometric random sum). Full article
(This article belongs to the Special Issue Stability Problems for Stochastic Models: Analytics, Asymptotics, Estimation and Applications)
24 pages, 527 KiB  
Article
Multi-Server Queuing Production Inventory System with Emergency Replenishment
by Dhanya Shajin, Achyutha Krishnamoorthy, Agassi Z. Melikov and Janos Sztrik
Mathematics 2022, 10(20), 3839; https://doi.org/10.3390/math10203839 - 17 Oct 2022
Cited by 6 | Viewed by 1864
Abstract
We consider a multi-server production inventory system with an unlimited waiting line. Arrivals occur according to a non-homogeneous Poisson process and exponentially distributed service time. At the service completion epoch, one unit of an item in the on-hand inventory decreases with probability δ [...] Read more.
We consider a multi-server production inventory system with an unlimited waiting line. Arrivals occur according to a non-homogeneous Poisson process and exponentially distributed service time. At the service completion epoch, one unit of an item in the on-hand inventory decreases with probability δ, and the customer leaves the system without taking the item with probability (1−δ). The production inventory system adopts an (s,S) policy where the processing of inventory requires a positive random amount of time. The production time for a unit item is phase-type distributed. Furthermore, assume that an emergency replenishment of one item with zero lead time takes place when the on-hand inventory level decreases to zero. The emergency replenishment is incorporated in the system to ensure customer satisfaction. We derive the stationary distribution of the system and some main performance measures, such as the distribution of the production on/off time in a cycle and the mean emergency replenishment cycle time. Numerical experiments are conducted to illustrate the system performance. A cost function is constructed, and we examine the optimal number of servers to be employed. Furthermore, we numerically calculate the optimal values of the production starting level and maximum inventory level. Full article
(This article belongs to the Special Issue Stability Problems for Stochastic Models: Analytics, Asymptotics, Estimation and Applications)
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