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1

Asymptotic analysis of a closed G-network of unreliable nodes

Rusilko Tatiana

Journal of Applied Mathematics and Computational Mechanics

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2022

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Vol. 21, nr 2

91--102

EN

A closed exponential queueing G-network of unreliable multi-server nodes was studied under the asymptotic assumption of a large number of customers. The process of changing the number of functional servers in network nodes was considered as the birth-death process. The process of changing the number of customers at the nodes was considered as a continuous-state Markov process. It was proved that its probability density function satisfies the Fokker-Planck-Kolmogorov equation. The system of differential equations for the first-order and second-order moments of this process was derived. This allows us to predict the expectation, the variance and the pairwise correlation of the number of customers in the G-network nodes both in the transient and steady state.

2

Investigation of G-networks with restart at a non-stationary mode and their application

Matalytski Mikhail, Naumenko Victor, Kopats Dmitry

Journal of Applied Mathematics and Computational Mechanics

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2019

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Vol. 18, nr 2

41--51

EN

This article discusses the question of restarting the script, when restart is used by many users of the information network, which can be modelled as a G-network. Negative claims simulate the crash of the script and the re-sending of the request. Investigation of an open queuing network (QN) with several types of positive customers with the phase type of distribution of their service time and one type of negative customers have been carried out. Negative customers are signals whose effect is to restart one customers in a queue. A technique is proposed for finding the probability of states. It is based on the use of a modified method of successive approximations, combined with the method of a series. The successive approximations converge with time to a stationary distribution of state probabilities, the form of which is indicated in the article, and the sequence of approximations converges to the solution of the difference-differential equations (DDE) system. The uniqueness of this solution is proved. Any successive approximation is representable in the form of a convergent power series with an infinite radius of convergence, the coefficients of which satisfy recurrence relations, which is convenient for computer calculations. A model example illustrating the finding of time-dependent probabilities of network states using the proposed technique is also presented.

3

Finding the expected revenues in Markov networks with positive and negative customers at a stationary regime

Matalytski M., Kopats D.

Journal of Applied Mathematics and Computational Mechanics

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2018

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Vol. 17, nr 1

49--60

EN

Finding the expected revenues in the queueing systems (QS) of open Markov G-networks of two types, with positive and negative customers and with positive customers and signals, has been described in the paper. A negative customer arriving to the system destroys one positive customer if at least one is available in the system, thus reducing the number of positive customers in the system by one. The signal, coming into an empty system (where there are no positive customers), does not have any impact on the network and immediately disappears from it. Otherwise, if the system is not empty, when it receives a signal, the following events can occur: the incoming signal instantly moves the positive customer from one QS into another with a certain probability, or with the other probability, the signal is triggered as a negative customer.

4

Analysis of the queueing network with a random bounded waiting time of positive and negative customers at a non-stationary regime

Matalytski M., Naumenko V.

Journal of Applied Mathematics and Computational Mechanics

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2017

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Vol. 16, nr 1

97--108

EN

In the first part of the article, an investigation of an open Markov queueing network with positive and negative customers (G-networks) has been carried out. The network receives two exponential arrivals of positive and negative customers. Negative customers do not receive service. The waiting time of customers of both types in each system is bounded by a random variable having an exponential distribution with different parameters. When the waiting time of a negative customer in the queue is over it reduces the number of positive customers per unit if the system has positive customers. The Kolmogorov system of difference-differential equations for non-stationary state probabilities has been derived. The method for finding state probabilities of an investigated network, based on the use of apparatus of multidimensional generating functions has been proposed. Expressions for finding the mean number of positive and negative customers in the network systems have also been found. In the second part the same network has been investigated, but with revenues. The case when revenues from the network transitions between states are random variables with given mean values has been considered. A method for finding expected revenues of the network systems has been proposed. Obtained results may be used for modeling of computer viruses in information systems and networks and also for forecasting of costs, considering the viruses penetration.

5

Analysis of the queueing network with a random waiting time of negative customers at a non-stationary regime

Naumenko V., Matalytski M., Kopats D.

Journal of Applied Mathematics and Computational Mechanics

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2016

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Vol. 15, nr 3

111--122

EN

In the article a queueing network (QN) with positive customers and a random waiting time of negative customers has been investigated. Negative customers destroy positive customers on the expiration of a random time. Queueing systems (QS) operate under a heavy-traffic regime. The system of difference-differential equations (DDE) for state probabilities of such a network was obtained. The technique of solving this system and finding mean characteristics of the network, which is based on the use of multivariate generating functions was proposed.

6

Investigation of G-network with random delay of signals at nonstationary behaviour

Matalytski M., Naumenko V.

Journal of Applied Mathematics and Computational Mechanics

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2014

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Vol. 13, nr 3

155--166

EN

The object of this research is an open queueing G-network with signals with random delay. The purpose of the research is investigation of such a network at the transient behavior. It is considered the case when the intensity of the incoming flow of positive and negative messages and service intensity of messages do not depend on time. All the systems in the network are one-line. It is described the model of computer system DDoS-attacks, the effect of penetration of the virus in a computer network in the form of G-network with random delay of signals. Approximate expressions are obtained for the time-dependent probabilities of states and the average characteristics of the network. Examples are calculated.

7

Stochastic bounds on the transient behaviors of the G-networks

Pekergin N., Taleb-Castel H.

Studia Informatica

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2002

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Vol. 23, nr 4

255-275

EN

In this work, we study the transient behaviors of the G-networks which are the extension of the Jackson networks. In fact, the steady-state solution of these networks has a product-form solution, however any analytical solution for their transient behaviors is not known. Following the studies on the Jackson networks, we propose to study the transient behaviors of the G-networks by applying the stochastic comparison approach.

PL

W artykule badane są stany nieustalone sieci G, stanowiących rozszerzenie sieci Jacksona. Zarówno sieci G, jak i sieci Jacksona posiadają rozwiązanie produktowe w stanie ustalonym, natomiast nieznane jest ich zachowanie w stanie nieustalonym. Zaproponowano analizę stanów nieustalonych w sieciach G poprzez porządkowanie i stochastyczne porównywanie wielowymiarowych łańcuchów Markowa.