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

Deadline for manuscript submissions: closed (30 November 2023) | Viewed by 17640

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Special Issue Editors

Prof. Dr. Vladimir Rykov

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Guest Editor

Department of Applied Probability and Informatics, Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, 117198 Moscow, Russia
Interests: reliability; stochastic models; applied probability
Special Issues, Collections and Topics in MDPI journals

Prof. Dr. Dmitry Efrosinin

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Guest Editor

Institute for Stochastics, Johannes Kepler Unversity Linz, Altenbergerstrasse 69, 4040 Linz, Austria
Interests: control of stochastic models; applied probability and statistics

Special Issue Information

Dear Colleagues,

This Special Issue focuses on new approaches, methods and applications in stochastic modelling and optimal control. The topics of interest in this Special Issue are the modelling and optimization of telecommunication and engineering systems, theoretical methods for the analysis of teletraffic and queueing systems, and mathematical and computer modelling in engineering, natural sciences, economics and other areas where there is a need for stochastic analysis of random processes. Various studies on the reliability of stochastic systems are also suitable for this Special Issue. Additionally, we would like to highlight a group of papers related to statistical data analysis. This includes topics such as sensor data processing and visualization, the simulation modelling of various models and their use in machine learning algorithms, simulation-based optimization, machine learning algorithms with dynamic programming and so on.

We also invite authors to support this Special Issue with their own new investigations in the field of stochastic modeling and applied probability.

Prof. Dr. Vladimir Rykov
Prof. Dr. Dmitry Efrosinin
Guest Editors

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Keywords

Stochastic models
Markovization methods
Sensitivity analysis
Applied probability
Reliability theory and risk analysis

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Published Papers (12 papers)

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Research

22 pages, 1267 KiB

Open AccessArticle

Investigation of the Fork–Join System with Markovian Arrival Process Arrivals and Phase-Type Service Time Distribution Using Machine Learning Methods

by Vladimir Mironovich Vishnevsky, Valentina Ivanovna Klimenok, Aleksandr Mikhailovich Sokolov and Andrey Alekseevich Larionov

Mathematics 2024, 12(5), 659; https://doi.org/10.3390/math12050659 - 23 Feb 2024

Viewed by 765

Abstract

This paper presents a study of fork–join systems. The fork–join system breaks down each customer into numerous tasks and processes them on separate servers. Once all tasks are finished, the customer is considered completed. This design enables the efficient handling of customers. The customers enter the system in a MAP flow. This helps create a more realistic and flexible representation of how customers arrive. It is important for modeling various real-life scenarios. Customers are divided into

K \geq 2

tasks and assigned to different subsystems. The number of tasks matches the number of subsystems. Each subsystem has a server that processes tasks, and a buffer that temporarily stores tasks waiting to be processed. The service time of a task by the k-th server follows a PH (phase-type) distribution with an irreducible representation (

β_{k}

S_{k}

1 \leq k \leq K

. An analytical solution was derived for the case of

K = 2

when the input MAP flow and service time follow a PH distribution. We have efficient algorithms to calculate the stationary distribution and performance characteristics of the fork–join system for this case. In general cases, this paper suggests using a combination of Monte Carlo and machine learning methods to study the performance of fork–join systems. In this paper, we present the results of our numerical experiments. Full article

(This article belongs to the Special Issue Stochastic Modeling and Applied Probability, 2nd Edition)

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11 pages, 284 KiB

Open AccessArticle

On the Positive Recurrence of Finite Regenerative Stochastic Models

by Evsey Morozov and Vladimir Rykov

Mathematics 2023, 11(23), 4754; https://doi.org/10.3390/math11234754 - 24 Nov 2023

Viewed by 874

Abstract

We consider a general approach to establish the positive recurrence (stability) of regenerative stochastic systems. The approach is based on the renewal theory and a characterization of the remaining renewal time of the embedded renewal process generated by regeneration. We discuss how this analysis is simplified for some classes of the stochastic systems. The general approach is then illustrated by the stability analysis of a k-out-of-n repairable system containing n unreliable components with exponential lifetimes. Then we extend the stability analysis to the system with non-exponential lifetimes. Full article

(This article belongs to the Special Issue Stochastic Modeling and Applied Probability, 2nd Edition)

23 pages, 767 KiB

Open AccessArticle

Randomized Threshold Strategy for Providing Flexible Priority in Multi-Server Queueing System with a Marked Markov Arrival Process and Phase-Type Distribution of Service Time

by A. N. Dudin, S. A. Dudin and O. S. Dudina

Mathematics 2023, 11(12), 2669; https://doi.org/10.3390/math11122669 - 12 Jun 2023

Cited by 1 | Viewed by 1122

Abstract

In this paper, we analyze a multi-server queueing system with a marked Markov arrival process of two types of customers and a phase-type distribution of service time depending on the type of customer. Customers of both types are assumed to be impatient and renege from the buffers after an exponentially distributed number of times. The strategy of flexible provisioning of priorities is analyzed. It assumes a randomized choice of the customers from the buffers, with probabilities dependent on the relation between the number of customers in a priority finite buffer and the fixed threshold value. To simplify the construction of the underlying Markov chain and the derivation of the explicit form of its generator, we use the so-called generalized phase-type distribution. It is shown that the created Markov chain fits the category of asymptotically quasi-Toeplitz Markov chains. Using this fact, we show that the considered Markov chain is ergodic for any value of the system parameters and compute its stationary distribution. Expressions for key performance measures are presented. Numerical results that show how the parameters of the control strategy affect the system’s performance measurements are given. It is shown that the results can be used for managerial purposes and that it is crucial to take correlation in the arrival process into account. Full article

(This article belongs to the Special Issue Stochastic Modeling and Applied Probability, 2nd Edition)

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11 pages, 274 KiB

Open AccessArticle

Mathematical and Statistical Aspects of Estimating Small Oscillations Parameters in a Conservative Mechanical System Using Inaccurate Observations

by Gurami Tsitsiashvili, Alexey Gudimenko and Marina Osipova

Mathematics 2023, 11(12), 2643; https://doi.org/10.3390/math11122643 - 9 Jun 2023

Cited by 2 | Viewed by 891

Abstract

This paper selects a set of reference points in the form of an arithmetic progression for planning an experiment to evaluate the parameters of systems of differential equations. This choice makes it possible to construct estimates of the parameters of a system of first-order differential equations based on the reversibility of the observation matrix, as well as estimates of the parameters of a system of second-order differential equations describing vibrations in a mechanical system by switching to a system of first-order differential equations. In turn, the reversibility of the observation matrix used in parameter estimation is established using the Vandermonde formula. A volumetric computational experiment has been carried out showing how, with an increase in the number of observations in the vicinity of reference points and with a decrease in the step of arithmetic progression, the accuracy of estimates of the parameters of the analyzed system increases. Among the estimated parameters, the most important are the oscillation frequencies of a conservative mechanical system, which establish its proximity to resonance, and therefore, determine the stability and reliability of the system. Full article

(This article belongs to the Special Issue Stochastic Modeling and Applied Probability, 2nd Edition)

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16 pages, 436 KiB

Open AccessArticle

On Queues with Working Vacation and Interdependence in Arrival and Service Processes

by S Sindhu, Achyutha Krishnamoorthy and Dmitry Kozyrev

Mathematics 2023, 11(10), 2280; https://doi.org/10.3390/math11102280 - 13 May 2023

Cited by 5 | Viewed by 1277

Abstract

In this paper, we consider two queuing models. Model 1 considers a single-server working vacation queuing system with interdependent arrival and service processes. The arrival and service processes evolve by transitions on the product space of two Markovian chains. The transitions in the two Markov chains in the product space are governed by a semi-Markov rule, with sojourn times in states governed by the exponential distribution. In contrast, in the second model, we consider independent arrival and service processes following phase-type distributions with representation

(α, T)

of order m and

(β, S)

of order n, respectively. The service time during normal working is the above indicated phase-type distribution whereas that during working vacation is a phase-type distribution with representation

(β, θ S)

0 < θ < 1

. The duration of the latter is exponentially distributed. The latter model is already present in the literature and will be briefly described. The main objective is to make a theoretical comparison between the two. Numerical illustrations for the first model are provided. Full article

(This article belongs to the Special Issue Stochastic Modeling and Applied Probability, 2nd Edition)

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18 pages, 1486 KiB

Open AccessArticle

On Importance of Sensitivity Analysis on an Example of a k-out-of-n System

by Nika Ivanova

Mathematics 2023, 11(5), 1100; https://doi.org/10.3390/math11051100 - 22 Feb 2023

Cited by 1 | Viewed by 1486

Abstract

Reliability and sensitivity issues are very close and important problems in any technical system. The system’s sensitivity is understood as the dependence of its behavior on changes in some internal parameters. To perform sensitivity analysis, a general procedure based on a theoretical and numerical study is proposed and applied to a repairable k-out-of-n model. The results show the asymptotic insensitivity of the non-stationary and stationary characteristics of the system reliability to the shape of the repair-time distribution, as well as to the value of its coefficient of variation at a fixed mean. The proposed methodology can be useful to researchers and engineers at the designing stage of real systems, as well as applied to other stochastic reliability models. Full article

(This article belongs to the Special Issue Stochastic Modeling and Applied Probability, 2nd Edition)

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29 pages, 853 KiB

Open AccessArticle

Multi-Class, Multi-Server Queueing Inventory System with Batch Service

by Khamis A. K. ALMaqbali, Varghese C. Joshua and Achyutha Krishnamoorthy

Mathematics 2023, 11(4), 830; https://doi.org/10.3390/math11040830 - 6 Feb 2023

Cited by 3 | Viewed by 1643

Abstract

In this paper, we consider a queueing inventory system with batch arrival and batch service processes. Customers arrive in batches of sizes

1, 2, \dots, k

, according to a marked compound Poisson process. We call a batch of customers [...] Read more.

In this paper, we consider a queueing inventory system with batch arrival and batch service processes. Customers arrive in batches of sizes

1, 2, \dots, k

, according to a marked compound Poisson process. We call a batch of customers as belonging to j when there are j individual customers in that batch. The service facility has waiting rooms for each category of customers and also a room to serve them. Except for category 1, all other customers have finite waiting rooms. The service room has only a limited number of seats. These seats are arranged in such a fashion that customers belonging to category j have groups of seats, each with j seats for

j = 1, 2, \dots, k

. Customers are taken for service according to the availability of seats designated to each category. A category j customer can be taken for service only if j items are available in the inventory. The service time of customers of category j is exponentially distributed with parameter depending on j for

j = 1, 2, \dots, k

. The number of seats available in the service room for each category of customers is restricted to a finite number. The replenishment for items follows the

(s, S)

policy: fill up to S at the time of replenishment. Lead time follows an exponential distribution. We analyze the system in the equilibrium state. Performance characteristics are evaluated and a number of numerical illustrations are provided. Full article

(This article belongs to the Special Issue Stochastic Modeling and Applied Probability, 2nd Edition)

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36 pages, 1322 KiB

Open AccessFeature PaperArticle

Queueing Inventory System in Transport Problem

by Khamis A. K. Al Maqbali, Varghese C. Joshua, Ambily P. Mathew and Achyutha Krishnamoorthy

Mathematics 2023, 11(1), 225; https://doi.org/10.3390/math11010225 - 2 Jan 2023

Cited by 2 | Viewed by 1608

Abstract

In this paper, we consider the batch arrival of customers to a transport station. Customers belonging to each category is considered as a single entity according to a BMMAP. An Erlang clock of order m starts ticking when the transport vessel reaches the station. When the Lth stage of the clock is reached, an order for the next vessel is placed. The lead time for arrival of the vessel follows exponential distribution. There are two types of rooms in this system: the waiting rooms and the service rooms for customers in the transport station and in the vessel, respectively. The waiting room capacity for customers of type 1 is infinite whereas those for customer of type

2, \dots, k

are of finite capacities. The service room capacity

C_{j}

for customer type

j, j = 1, 2, \dots, k

is finite. Upon arrival, customers of category j occupy seats designated for that category in the vessel, provided there is at least one vacancy belonging to that category. The total number of vessels with the operator is

h^{*}

. The service time of each vessel follows exponential distribution with parameter

μ .

Each group of customers belong to category j searches independently for customers of this category to mobilize passengers when the Erlang clock reaches

L_{1}

where

L_{1} < L

. The search time for customers of category j follows exponential distribution with parameter

λ_{j} .

The stability condition is derived. Some performance measures are estimated. Full article

(This article belongs to the Special Issue Stochastic Modeling and Applied Probability, 2nd Edition)

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16 pages, 1574 KiB

Open AccessArticle

Optimal Control of Degrading Units through Threshold-Based Control Policies

by Dmitry Efrosinin and Natalia Stepanova

Mathematics 2022, 10(21), 4098; https://doi.org/10.3390/math10214098 - 3 Nov 2022

Viewed by 1072

Abstract

Optimal control problems are applied to a variety of dynamical systems with a random law of motion. In this paper we show that the random degradation processes defined on a discrete set of intermediate degradation states are also suitable for formulating and solving optimization problems and finding an appropriate optimal control policy. Two degradation models are considered in this paper: with random time to an instantaneous failure and with random time to a preventive maintenance. In both cases, a threshold-based control policy with two thresholds levels defining the signal state, after which an instantaneous failure or preventive maintenance can occur after a random time, and a maximum number of intermediate degradation states is applied. The optimal control problem is mainly solved in a steady-state regime. The main loss functional is formulated as the average cost per unit of time for a given cost structure. The Markov degradation models are used for numerical calculations of the optimal threshold policy and reliability function of the studied degrading units. Full article

(This article belongs to the Special Issue Stochastic Modeling and Applied Probability, 2nd Edition)

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33 pages, 4057 KiB

Open AccessArticle

Fluctuation Analysis of a Soft-Extreme Shock Reliability Model

by Jewgeni H. Dshalalow and Ryan T. White

Mathematics 2022, 10(18), 3312; https://doi.org/10.3390/math10183312 - 13 Sep 2022

Cited by 4 | Viewed by 1887

Abstract

In this paper, we deal with a mixed reliability system decaying from natural wear, occasional soft and hard shocks that eventually lead the system to failure. The aging process alone is linear and it is escalated through soft shocks such that they lead to the system’s soft failure when the combined damage exceeds a threshold M. The other threat is that posed by occasional hard shocks. When the total number of them identified as critical (each critical shock exceeds a fixed threshold H) reaches N, the system becomes disabled. With

N = 1

, a critical shock is extreme. The arrival stream of shocks is a renewal process marked by soft and hard shocks. We establish a formula for a closed form functional containing system’s time-to-failure, the state of the system upon its failure, and other useful statistical characteristics of the system using and embellishing fluctuation analysis and operational calculus. Special cases provide tame expressions that are computed and validated by simulation. Full article

(This article belongs to the Special Issue Stochastic Modeling and Applied Probability, 2nd Edition)

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13 pages, 1236 KiB

Open AccessArticle

Reliability Analysis of a Load-Sharing k-out-of-n System Due to Its Components’ Failure

by Vladimir Rykov, Nika Ivanova and Irina Kochetkova

Mathematics 2022, 10(14), 2457; https://doi.org/10.3390/math10142457 - 14 Jul 2022

Cited by 5 | Viewed by 2129

Abstract

The reliability characteristics of a k-out-of-

n : F

The reliability characteristics of a k-out-of-

n : F

system are studied in the case when a failure of one of the system’s components leads to increasing the load on others, resulting in their residual lifetimes decreasing. The situation is modeled with the help of changing the components and the system hazard rate function. The reliability function and other reliability characteristics are calculated. Numerical examples are considered. Full article

(This article belongs to the Special Issue Stochastic Modeling and Applied Probability, 2nd Edition)

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10 pages, 323 KiB

Open AccessArticle

Safety Margin Prediction Algorithms Based on Linear Regression Analysis Estimates

by Gurami Tsitsiashvili and Alexandr Losev

Mathematics 2022, 10(12), 2008; https://doi.org/10.3390/math10122008 - 10 Jun 2022

Cited by 1 | Viewed by 1414

Abstract

In this paper, we consider the problem of approximating the safety margin of a single instance of a technical system based on inaccurate observations at specified time points. The solution to this problem is based on the selection of a certain set of reference points in time, in a small neighbourhood of which a sufficiently large number of inaccurate measurements are carried out. Analogously with the failure rate, it is assumed that the rate of decrease in the safety margin over time is represented by a polynomial of the fourth degree, and the safety margin itself is a polynomial of the fifth degree. A system of linear algebraic equations is constructed that determine the coefficients of this polynomial through its values and the values of its derivative at reference points in time. These values themselves are estimated by the method of linear regression analysis based on numerous observations in a small neighbourhood of reference points in time. A detailed computational experiment is carried out to verify the accuracy of the approximation of a fifth-degree polynomial and alternative ways of estimating it are constructed in the vicinity of points where the quality of approximation is insufficient. Full article

(This article belongs to the Special Issue Stochastic Modeling and Applied Probability, 2nd Edition)

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