Mathematics

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23 pages, 1799 KiB

Open AccessArticle

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

Viewed by 883

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

Open AccessArticle

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 1567

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

Open AccessArticle

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 - 05 May 2023

Cited by 1 | Viewed by 1327

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

Open AccessArticle

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 - 04 May 2023

Cited by 1 | Viewed by 931

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

Open AccessFeature PaperArticle

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 766

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

Open AccessArticle

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 1173

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

Open AccessArticle

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 - 02 Jan 2023

Cited by 4 | Viewed by 1009

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

Open AccessArticle

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 - 05 Dec 2022

Viewed by 1047

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 (P_{0}, P_{t})

from the starting point

P_{0}

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

Open AccessArticle

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 2 | Viewed by 880

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

Open AccessFeature PaperArticle

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 1218

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

Open AccessArticle

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 1307

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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