Mathematics

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18 pages, 622 KiB

Open AccessArticle

Modeling and Performance Evaluation of a Cellular Network with OMA and NOMA Users with Batch Arrivals by Means of an M^[X]/M/S/0 Model

by Luis Alberto Vásquez-Toledo, Carlos González-Flores, Miguel Lopez-Guerrero, Alfonso Prieto-Guerrero, José Alfredo Tirado-Méndez, Ricardo Marcelín-Jiménez, Enrique Rodriguez-Colina, Michael Pascoe-Chalke and Francisco R. Castillo-Soria

Mathematics 2024, 12(21), 3400; https://doi.org/10.3390/math12213400 - 30 Oct 2024

Viewed by 468

Abstract

Nowadays, efficient spectrum usage is one of the most important design principles to take into account in wireless communications due to the exponential growth of mobile devices. In that sense, solutions such as Non-Orthogonal Multiple Access (NOMA) and cognitive radio (CR) have been [...] Read more.

Nowadays, efficient spectrum usage is one of the most important design principles to take into account in wireless communications due to the exponential growth of mobile devices. In that sense, solutions such as Non-Orthogonal Multiple Access (NOMA) and cognitive radio (CR) have been proposed. In essence, NOMA allows some interference level by using non-orthogonal resource allocation with a tolerable increase in receiver complexity employing successive interference cancellation (SIC). In this work, a novel mathematical model of teletraffic for users performing accessment, simultaneously, by means of Orthogonal Multiple Access (OMA) and NOMA, is developed using a Markovian process that considers bursts of arrivals to model the access schemes. This novel procedure implies a closed-form solution of the proposed system compared to other works where these parameters are estimated assuming the moment generating function obtained with approximation models. The model is validated with a discrete event simulator, considering different scenarios and simulation conditions. The simulation results are in agreement with the mathematical solution proposed. Full article

(This article belongs to the Special Issue Stochastic Processes: Theory, Simulation and Applications)

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20 pages, 581 KiB

Open AccessFeature PaperArticle

Mittag–Leffler Fractional Stochastic Integrals and Processes with Applications

by Enrica Pirozzi

Mathematics 2024, 12(19), 3094; https://doi.org/10.3390/math12193094 - 2 Oct 2024

Viewed by 1560

Abstract

We study Mittag–Leffler (ML) fractional integrals involved in the solution processes of a system of coupled fractional stochastic differential equations. We introduce the ML fractional stochastic process as a ML fractional stochastic integral with respect to a standard Brownian motion. We provide some [...] Read more.

We study Mittag–Leffler (ML) fractional integrals involved in the solution processes of a system of coupled fractional stochastic differential equations. We introduce the ML fractional stochastic process as a ML fractional stochastic integral with respect to a standard Brownian motion. We provide some representation formulas of solution processes in terms of Mittag–Leffler fractional integrals and processes. Computable expressions of the mean functions and of the covariances of such processes are specifically given. The application in neuronal modeling is provided, and all involved functions and processes are specifically determined. Numerical evaluations are carried out and some results are shown and discussed. Full article

(This article belongs to the Special Issue Stochastic Processes: Theory, Simulation and Applications)

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17 pages, 731 KiB

Open AccessArticle

New Computer Experiment Designs with Area-Interaction Point Processes

by Ahmed Ait Ameur, Hichem Elmossaoui and Nadia Oukid

Mathematics 2024, 12(15), 2397; https://doi.org/10.3390/math12152397 - 31 Jul 2024

Viewed by 747

Abstract

This article presents a novel method for constructing computer experiment designs based on the theory of area-interaction point processes. This method is essential for capturing the interactions between different elements within a modeled system, offering a more flexible and adaptable approach compared with [...] Read more.

This article presents a novel method for constructing computer experiment designs based on the theory of area-interaction point processes. This method is essential for capturing the interactions between different elements within a modeled system, offering a more flexible and adaptable approach compared with traditional mathematical modeling. Unlike conventional rough models that rely on simplified equations, our method employs the Markov Chain Monte Carlo (MCMC) method and the Metropolis–Hastings algorithm combined with Voronoi tessellations. It uses a new dynamic called homogeneous birth and death dynamics of a set of points to generate the designs. This approach does not require the development of specific mathematical models for each system under study, making it universally applicable while achieving comparable results. Furthermore, we provide an in-depth analysis of the convergence properties of the Markov Chain to ensure the reliability of the generated designs. An expanded literature review situates our work within the context of existing research, highlighting its unique contributions and advancements. A comparison between our approach and other existing computer experiment designs has been performed. Full article

(This article belongs to the Special Issue Stochastic Processes: Theory, Simulation and Applications)

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15 pages, 312 KiB

Open AccessArticle

A Matrix-Multiplicative Solution for Multi-Dimensional QBD Processes

by Valeriy Naumov

Mathematics 2024, 12(3), 444; https://doi.org/10.3390/math12030444 - 30 Jan 2024

Viewed by 799

Abstract

We consider an irreducible positive-recurrent discrete-time Markov process on the state space

X = ℤ_{+}^{M} \times J

, where

ℤ_{+}

is the set of non-negative integers and

J = {1, 2, \dots, n}

. The [...] Read more.

We consider an irreducible positive-recurrent discrete-time Markov process on the state space

X = ℤ_{+}^{M} \times J

, where

ℤ_{+}

is the set of non-negative integers and

J = {1, 2, \dots, n}

. The number of states in

J

may be either finite or infinite. We assume that the process is a homogeneous quasi-birth-and-death process (QBD). It means that the one-step transition probability between non-boundary states

(k, i)

and

(n, j)

may depend on

i, j

, and

n - k

but not on the specific values of

k

and

n

. It is shown that the stationary probability vector of the process is expressed through square matrices of order

n

, which are the minimal non-negative solutions to nonlinear matrix equations. Full article

(This article belongs to the Special Issue Stochastic Processes: Theory, Simulation and Applications)

15 pages, 651 KiB

Open AccessArticle

Wasserstein Dissimilarity for Copula-Based Clustering of Time Series with Spatial Information

by Alessia Benevento and Fabrizio Durante

Mathematics 2024, 12(1), 67; https://doi.org/10.3390/math12010067 - 24 Dec 2023

Cited by 1 | Viewed by 1177

Abstract

The clustering of time series with geo-referenced data requires a suitable dissimilarity matrix interpreting the comovements of the time series and taking into account the spatial constraints. In this paper, we propose a new way to compute the dissimilarity matrix, merging both types [...] Read more.

The clustering of time series with geo-referenced data requires a suitable dissimilarity matrix interpreting the comovements of the time series and taking into account the spatial constraints. In this paper, we propose a new way to compute the dissimilarity matrix, merging both types of information, which leverages on the Wasserstein distance. We then make a quasi-Gaussian assumption that yields more convenient formulas in terms of the joint correlation matrix. The method is illustrated in a case study involving climatological data. Full article

(This article belongs to the Special Issue Stochastic Processes: Theory, Simulation and Applications)

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31 pages, 1302 KiB

Open AccessEditor’s ChoiceArticle

Time-Inhomogeneous Finite Birth Processes with Applications in Epidemic Models

by Virginia Giorno and Amelia G. Nobile

Mathematics 2023, 11(21), 4521; https://doi.org/10.3390/math11214521 - 2 Nov 2023

Cited by 2 | Viewed by 1264

Abstract

We consider the evolution of a finite population constituted by susceptible and infectious individuals and compare several time-inhomogeneous deterministic models with their stochastic counterpart based on finite birth processes. For these processes, we determine the explicit expressions of the transition probabilities and of [...] Read more.

We consider the evolution of a finite population constituted by susceptible and infectious individuals and compare several time-inhomogeneous deterministic models with their stochastic counterpart based on finite birth processes. For these processes, we determine the explicit expressions of the transition probabilities and of the first-passage time densities. For time-homogeneous finite birth processes, the behavior of the mean and the variance of the first-passage time density is also analyzed. Moreover, the approximate duration until the entire population is infected is obtained for a large population size. Full article

(This article belongs to the Special Issue Stochastic Processes: Theory, Simulation and Applications)

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12 pages, 1109 KiB

Open AccessArticle

Numerical Computation of Distributions in Finite-State Inhomogeneous Continuous Time Markov Chains, Based on Ergodicity Bounds and Piecewise Constant Approximation

by Yacov Satin, Rostislav Razumchik, Ilya Usov and Alexander Zeifman

Mathematics 2023, 11(20), 4265; https://doi.org/10.3390/math11204265 - 12 Oct 2023

Viewed by 815

Abstract

In this paper it is shown, that if a possibly inhomogeneous Markov chain with continuous time and finite state space is weakly ergodic and all the entries of its intensity matrix are locally integrable, then, using available results from the perturbation theory, its [...] Read more.

In this paper it is shown, that if a possibly inhomogeneous Markov chain with continuous time and finite state space is weakly ergodic and all the entries of its intensity matrix are locally integrable, then, using available results from the perturbation theory, its time-dependent probability characteristics can be approximately obtained from another Markov chain, having piecewise constant intensities and the same state space. The approximation error (the taxicab distance between the state probability distributions) is provided. It is shown how the Cauchy operator and the state probability distribution for an arbitrary initial condition can be calculated. The findings are illustrated with the numerical examples. Full article

(This article belongs to the Special Issue Stochastic Processes: Theory, Simulation and Applications)

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14 pages, 368 KiB

Open AccessArticle

Optimizing Air Pollution Modeling with a Highly-Convergent Quasi-Monte Carlo Method: A Case Study on the UNI-DEM Framework

by Venelin Todorov, Slavi Georgiev, Ivan Georgiev, Snezhinka Zaharieva and Ivan Dimov

Mathematics 2023, 11(13), 2919; https://doi.org/10.3390/math11132919 - 29 Jun 2023

Cited by 2 | Viewed by 1155

Abstract

In this study, we present the development of an advanced air pollution modeling approach, which incorporates cutting-edge stochastic techniques for large-scale simulations of long-range air pollutant transportation. The Unified Danish Eulerian Model (UNI-DEM) serves as a crucial mathematical framework with numerous applications in [...] Read more.

In this study, we present the development of an advanced air pollution modeling approach, which incorporates cutting-edge stochastic techniques for large-scale simulations of long-range air pollutant transportation. The Unified Danish Eulerian Model (UNI-DEM) serves as a crucial mathematical framework with numerous applications in studies concerning the detrimental effects of heightened air pollution levels. We employ the UNI-DEM model in our research to obtain trustworthy insights into critical questions pertaining to environmental preservation. Our proposed methodology is a highly convergent quasi-Monte Carlo technique that relies on a unique symmetrization lattice rule. By fusing the concepts of special functions and optimal generating vectors, we create a novel algorithm grounded in the component-by-component construction method, which has been recently introduced. This amalgamation yields particularly impressive outcomes for lower-dimensional cases, substantially enhancing the performance of the most advanced existing methods for calculating the Sobol sensitivity indices of the UNI-DEM model. This improvement is vital, as these indices form an essential component of the digital ecosystem for environmental analysis. Full article

(This article belongs to the Special Issue Stochastic Processes: Theory, Simulation and Applications)

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13 pages, 1383 KiB

Open AccessArticle

Solving Stochastic Nonlinear Poisson-Boltzmann Equations Using a Collocation Method Based on RBFs

by Samaneh Mokhtari, Ali Mesforush, Reza Mokhtari, Rahman Akbari and Clemens Heitzinger

Mathematics 2023, 11(9), 2118; https://doi.org/10.3390/math11092118 - 29 Apr 2023

Viewed by 1430

Abstract

In this paper, we present a numerical scheme based on a collocation method to solve stochastic non-linear Poisson–Boltzmann equations (PBE). This equation is a generalized version of the non-linear Poisson–Boltzmann equations arising from a form of biomolecular modeling to the stochastic case. Applying [...] Read more.

In this paper, we present a numerical scheme based on a collocation method to solve stochastic non-linear Poisson–Boltzmann equations (PBE). This equation is a generalized version of the non-linear Poisson–Boltzmann equations arising from a form of biomolecular modeling to the stochastic case. Applying the collocation method based on radial basis functions (RBFs) allows us to deal with the difficulties arising from the complexity of the domain. To indicate the accuracy of the RBF method, we present numerical results for two-dimensional models, we also study the stability of this method numerically. We examine our results with the RBF-reference value and the Chebyshev Spectral Collocation (CSC) method. Furthermore, we discuss finding the appropriate shape parameter to obtain an accurate numerical solution besides greatest stability. We have exerted the Newton–Raphson approach for solving the system of non-linear equations resulting from discretization by the RBF technique. Full article

(This article belongs to the Special Issue Stochastic Processes: Theory, Simulation and Applications)

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