Mathematics

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21 pages, 387 KiB

Open AccessArticle

Analysis of the Message Propagation Speed in VANET with Disconnected RSUs

by Dhari Ali Mahmood and Gábor Horváth

Mathematics 2020, 8(5), 782; https://doi.org/10.3390/math8050782 - 13 May 2020

Cited by 13 | Viewed by 1706

Vehicular ad-hoc networks (VANETs), which are networks of communicating vehicles, provide the essential infrastructure for intelligent transportation systems. Thanks to the significant research efforts to develop the technological background of VANETs, intelligent transportation systems are nowadays becoming a reality. The emergence of VANETs [...] Read more.

Vehicular ad-hoc networks (VANETs), which are networks of communicating vehicles, provide the essential infrastructure for intelligent transportation systems. Thanks to the significant research efforts to develop the technological background of VANETs, intelligent transportation systems are nowadays becoming a reality. The emergence of VANETs has triggered a lot of research aimed at developing mathematical models in order to gain insight into the dynamics of the communication and to support network planning. In this paper we consider the message propagation speed on the highway, where messages can be exchanged not only between the vehicles, but also between the road-side infrastructure and the vehicles as well. In our scenario, alert messages are generated by a static message source constantly. Relying on an appropriately defined Markov renewal process, we characterize the message passing process between the road-side units, derive the speed of the message propagation, and provide the transient distribution of the distance where the message is available. Our results make it possible to determine the optimal distance between road-side units (RSUs) and to calculate the effect of speed restrictions on message propagation. Full article

(This article belongs to the Special Issue Queueing Methods in Reliability Theory and Information Technology Systems)

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9 pages, 891 KiB

Open AccessArticle

Computing the Stationary Distribution of Queueing Systems with Random Resource Requirements via Fast Fourier Transform

by Valeriy A. Naumov, Yuliya V. Gaidamaka and Konstantin E. Samouylov

Mathematics 2020, 8(5), 772; https://doi.org/10.3390/math8050772 - 12 May 2020

Cited by 2 | Viewed by 1581

Abstract

Queueing systems with random resource requirements, in which an arriving customer, in addition to a server, demands a random amount of resources from a shared resource pool, have proved useful to analyze wireless communication networks. The stationary distributions of such queuing systems are [...] Read more.

Queueing systems with random resource requirements, in which an arriving customer, in addition to a server, demands a random amount of resources from a shared resource pool, have proved useful to analyze wireless communication networks. The stationary distributions of such queuing systems are expressed in terms of truncated convolution powers of the cumulative distribution function of the resource requirements. Discretization of the cumulative distribution function and the application of the fast Fourier transform are a traditional way of calculating convolutions. We suggest finding truncated convolution powers of the cumulative distribution functions by calculating the convolution powers of the truncated cumulative distribution functions via fast Fourier transform. This radically decreases computational complexity. We introduce the concept of resource load and investigate the accuracy of the proposed method at low and high resource loads. It is shown that the proposed method makes it possible to quickly and accurately calculate truncated convolution powers required for the analysis of queuing systems with random resource requirements. Full article

(This article belongs to the Special Issue Queueing Methods in Reliability Theory and Information Technology Systems)

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15 pages, 492 KiB

Open AccessArticle

On Probability Characteristics for a Class of Queueing Models with Impatient Customers

by Yacov Satin, Alexander Zeifman, Alexander Sipin, Sherif I. Ammar and Janos Sztrik

Mathematics 2020, 8(4), 594; https://doi.org/10.3390/math8040594 - 15 Apr 2020

Cited by 7 | Viewed by 1875

Abstract

In this paper, a class of queueing models with impatient customers is considered. It deals with the probability characteristics of an individual customer in a non-stationary Markovian queue with impatient customers, the stationary analogue of which was studied previously as a successful approximation [...] Read more.

In this paper, a class of queueing models with impatient customers is considered. It deals with the probability characteristics of an individual customer in a non-stationary Markovian queue with impatient customers, the stationary analogue of which was studied previously as a successful approximation of a more general non-Markov model. A new mathematical model of the process is considered that describes the behavior of an individual requirement in the queue of requirements. This can be applied both in the stationary and non-stationary cases. Based on the proposed model, a methodology has been developed for calculating the system characteristics both in the case of the existence of a stationary solution and in the case of the existence of a periodic solution for the corresponding forward Kolmogorov system. Some numerical examples are provided to illustrate the effect of input parameters on the probability characteristics of the system. Full article

(This article belongs to the Special Issue Queueing Methods in Reliability Theory and Information Technology Systems)

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16 pages, 372 KiB

Open AccessArticle

Asymptotic Diffusion Analysis of Multi-Server Retrial Queue with Hyper-Exponential Service

by Alexander Moiseev, Anatoly Nazarov and Svetlana Paul

Mathematics 2020, 8(4), 531; https://doi.org/10.3390/math8040531 - 3 Apr 2020

Cited by 12 | Viewed by 1698

Abstract

A multi-server retrial queue with a hyper-exponential service time is considered in this paper. The study is performed by the method of asymptotic diffusion analysis under the condition of long delay in orbit. On the basis of the constructed diffusion process, we obtain [...] Read more.

A multi-server retrial queue with a hyper-exponential service time is considered in this paper. The study is performed by the method of asymptotic diffusion analysis under the condition of long delay in orbit. On the basis of the constructed diffusion process, we obtain approximations of stationary probability distributions of the number of customers in orbit and the number of busy servers. Using simulations and numerical analysis, we estimate the accuracy and applicability area of the obtained approximations. Full article

(This article belongs to the Special Issue Queueing Methods in Reliability Theory and Information Technology Systems)

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12 pages, 256 KiB

Open AccessArticle

Renewal Redundant Systems Under the Marshall–Olkin Failure Model. A Probability Analysis

by Boyan Dimitrov, Vladimir Rykov and Tatiana Milovanova

Mathematics 2020, 8(3), 459; https://doi.org/10.3390/math8030459 - 24 Mar 2020

Viewed by 1969

Abstract

In this paper a two component redundant renewable system operating under the Marshall–Olkin failure model is considered. The purpose of the study is to find analytical expressions for the time dependent and the steady state characteristics of the system. The system cycle process [...] Read more.

In this paper a two component redundant renewable system operating under the Marshall–Olkin failure model is considered. The purpose of the study is to find analytical expressions for the time dependent and the steady state characteristics of the system. The system cycle process characteristics are analyzed by the use of probability interpretation of the Laplace–Stieltjes transformations (LSTs), and of probability generating functions (PGFs). In this way the long mathematical analytic derivations are avoid. As results of the investigations, the main reliability characteristics of the system—the reliability function and the steady state probabilities—have been found in analytical form. Our approach can be used in the studies of various applications of systems with dependent failures between their elements. Full article

(This article belongs to the Special Issue Queueing Methods in Reliability Theory and Information Technology Systems)

18 pages, 729 KiB

Open AccessFeature PaperArticle

On Reliability of a Double Redundant Renewable System with a Generally Distributed Life and Repair Times

by Vladimir Rykov, Dmitry Efrosinin, Natalia Stepanova and Janos Sztrik

Mathematics 2020, 8(2), 278; https://doi.org/10.3390/math8020278 - 19 Feb 2020

Cited by 6 | Viewed by 1949

Abstract

The paper provides reliability analysis of a cold double redundant renewable system assuming that both life-time and repair time distributions are arbitrary. The proposed approach is based on the theory of decomposable semi-regenerative processes. We derive the Laplace–Stieltjes transform of two main reliability [...] Read more.

The paper provides reliability analysis of a cold double redundant renewable system assuming that both life-time and repair time distributions are arbitrary. The proposed approach is based on the theory of decomposable semi-regenerative processes. We derive the Laplace–Stieltjes transform of two main reliability measures like the distribution of the time between failures and the time to the first failure. The transforms are used to calculate corresponding mean times. It is further derived in closed form the time-dependent and time stationary state probabilities in terms of the Laplace transforms. Numerical results illustrate the effect of the type of distributions as well as their parameters on the derived reliability and probabilistic measures. Full article

(This article belongs to the Special Issue Queueing Methods in Reliability Theory and Information Technology Systems)

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14 pages, 2126 KiB

Open AccessArticle

Analysis of Queueing System MMPP/M/K/K with Delayed Feedback

by Agassi Melikov, Sevinj Aliyeva and Janos Sztrik

Mathematics 2019, 7(11), 1128; https://doi.org/10.3390/math7111128 - 18 Nov 2019

Cited by 9 | Viewed by 4017

Abstract

The model of multi-channel queuing system with Markov modulated Poisson process (MMPP) flow and delayed feedback is considered. After the customer is served completely, they will decide either to join the retrial group again for another service (feedback) with some state-dependent probability or [...] Read more.

The model of multi-channel queuing system with Markov modulated Poisson process (MMPP) flow and delayed feedback is considered. After the customer is served completely, they will decide either to join the retrial group again for another service (feedback) with some state-dependent probability or to leave the system forever with complimentary probability. Feedback calls organize an orbit of repeated calls (r-calls). If upon arrival of an r-call all the channels of the system are busy, then it either leaves the system with some state-dependent probability or with a complementary probability returns to orbit. Methods to calculate the steady-state probabilities of the appropriate three-dimensional Markov chain as well as performance measures of investigated system are developed. Results of numerical experiments are demonstrated. Full article

(This article belongs to the Special Issue Queueing Methods in Reliability Theory and Information Technology Systems)

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17 pages, 316 KiB

Open AccessArticle

Queueing Network with Moving Servers as a Model of Car Sharing Systems

by Chesoong Kim, Sergei Dudin and Olga Dudina

Mathematics 2019, 7(9), 825; https://doi.org/10.3390/math7090825 - 6 Sep 2019

Cited by 2 | Viewed by 2470

Abstract

We consider a queueing network with a finite number of nodes and servers moving between the nodes as a model of car sharing. The arrival process of customers to various nodes is defined by a marked Markovian arrival process. The customer that arrives [...] Read more.

We consider a queueing network with a finite number of nodes and servers moving between the nodes as a model of car sharing. The arrival process of customers to various nodes is defined by a marked Markovian arrival process. The customer that arrives at a certain node when there is no idle server (car) is lost. Otherwise, he/she is able to start the service. With known probability, which depends on the node and the number of available cars, this customer can balk the service and leave the system. The service time of a customer has an exponential distribution. Location of the server in the network after service completion is random with the known probability distribution. The behaviour of the network is described by a multi-dimensional continuous-time Markov chain. The generator of this chain is derived which allows us to compute the stationary distribution of the network states. The formulas for computing the key performance indicators of the system are given. Numerical results are presented. They characterize the dependence of some performance measures of the network and the nodes on the total number of cars (fleet size of the car sharing system) and correlation in the arrival process. Full article

(This article belongs to the Special Issue Queueing Methods in Reliability Theory and Information Technology Systems)

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