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1

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.

2

Application of HM-networks with unreliable systems for finding the memory capacity in the information systems

Matalytski M., Zajac P., Kopats D.

Journal of Applied Mathematics and Computational Mechanics

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2018

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Vol. 17, nr 2

51--63

EN

To solve the problem of determining the memory capacity of the information systems (IS), it was proposed to use a stochastic model, based on the use of HM (Howard-Matalytski) - queueing networks with incomes. This model takes into account the servicing of requests along with their volumes, the ability to change the volumes of the requests over time and the possibility of damaging IS nodes and their repairs, so servicing of demands can be interrupted randomly. The expressions are generated for the mean (expected) values of total requests volumes in the IS nodes.

3

Finding of expected volumes of requests in systems of the queueing network with a limited amount of spaces of expectations

Zając P., Matalytski M.

Studia Informatica

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2017

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Vol. 38, nr 1B

89--104

EN

We present a method of finding the expected volume of requests in open HM-network with homogeneous requests, bypass of nodes the network service systems. Were considered a case where the changes in volumes associated with transitions between states of the network are deterministic functions dependent states of network and time, and service systems are single line. Assumed that the probability of state network systems, the parameters of entrance flow of messages and service depend on time.

PL

Opisano metodę znalezienia oczekiwanej objętości zgłoszeń w otwartej HM-sieci z jednorodnymi zgłoszeniami i obejściami węzłów sieci systemów obsługi. Rozpatrywano przypadek, gdy zmiany objętości związanych z przejściami między stanami sieci są deterministycznymi funkcjami, zależnymi od stanów sieci i czasu, a systemy obsługi są jednoliniowe. Zakłada się, że prawdopodobieństwo stanów systemów sieci, parametry strumienia wejściowego zgłoszeń i obsługi zależą od czasu.

4

Expected volumes of requests in systems of the queueing network with a limited number of waiting places

Zając P., Matalytski M.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

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2017

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Vol. 22

141--160

EN

We present a method of finding the expected volume of requests in HM-network with homogeneous requests and bypass of the queueing systems of requests. The case was considered when the volume changes associated with the transitions between the states of the network are deterministic functions, depending on the state of the network and time, and the systems are single line. It is assumed that the probability of the states of the network systems, the parameters of the entrance flow of the requests and the service depend on the time.

5

Application of HM-networks with inpatient claims in finding the memory capacity in information systems

Matalytski M., Zajac P.

Journal of Applied Mathematics and Computational Mechanics

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2017

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Vol. 16, nr 3

47--59

EN

To solve the problem of determining the memory capacity of the information systems (IS), the use of the stochastic model is proposed, based on the use of НМ (Howard-Matalytski) - queueing networks with revenues. This model allows one to take into account time dependencies of the message processing from their capacities, the possibility changes of the messages capacities over time and also the possibility of leaving messages from queues in nodes of IS, without getting into them appropriate processing. The expressions for the mean (expected) values of total message capacities in the IS nodes have been obtained.

6

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.

7

Finding the probabilistically-temporal characteristics of Markov G-network with batch removal of positive customers

Matalytski M., Naumenko V., Kopats D.

Journal of Applied Mathematics and Computational Mechanics

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2016

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Vol. 15, nr 4

125-136

EN

The investigation of a Markov queueing network with positive and negative customers and positive customers batch removal has been carried out in the article. The purpose of the research is analysis of such a network at the non-stationary regime, finding the time-dependent state probabilities and mean number of customers. In the first part of this article, a description of the G-network operation is provided with one-line queueing systems. When a negative customer arrives to the system, the count of positive customers is reduced by a random value, which is set by some probability distribution. Then for the non-stationary state probabilities a Kolmogorov system was derived of differencedifferential equations. A technique for finding the state probabilities and the mean number of customers of the investigated network, based on the use of an apparatus of multidimensional generating functions has been proposed. The theorem about the expression for the generating function has been given. A model example has been calculated.

8

Diffusion approximation of the network with limited number of same type customers and time dependent service parameters

Matalytski M., Kopats D.

Journal of Applied Mathematics and Computational Mechanics

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2016

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Vol. 15, nr 2

77--84

EN

The article presents research of an open queueing network (QN) with the same types of customers, in which the total number of customers is limited. Service parameters are dependent on time, and the route of customers is determined by an arbitrary stochastic transition probability matrix, which is also dependent on time. Service times of customers in each line of the system is exponentially distributed. Customers are selected on the service according to FIFO discipline. It is assumed that the number of customers in one of the systems is determined by the process of birth and death. It generates and destroys customers with certain service times of rates. The network state is described by the random vector, which is a Markov random process. The purpose of the research is an asymptotic analysis of its process with a big number of customers, obtaining a system of differential equations (DE) to find the mean relative number of customers in the network systems at any time. A specific model example was calculated using the computer. The results can be used for modelling processes of customer service in the insurance companies, banks, logistics companies and other organizations.

9

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.

10

About one method of finding expected incomes in HM-queueing network with positive customers and signals

Matalytski M.

Journal of Applied Mathematics and Computational Mechanics

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2016

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Vol. 15, nr 1

87--98

EN

In the paper an open Markov HM(Howard-Matalytski)-Queueing Network (QN) with incomes, positive customers and signals (G(Gelenbe)-QN with signals) is investigated. The case is researched, when incomes from the transitions between the states of the network are random variables (RV) with given mean values. In the main part of the paper a description is given of G-network with signals and incomes, all kinds of transition probabilities and incomes from the transitions between the states of the network. The method of finding expected incomes of the researched network was proposed, which is based on using of found approximate and exact expressions for the mean values of random incomes. The variances of incomes of queueing systems (QS) was also found. A calculation example, which illustrates the differences of expected incomes of HM-networks with negative customers and QN without them and also with signals, has been given. The practical significance of these results consist of that they can be used at forecasting incomes in computer systems and networks (CSN) taking into account virus penetration into it and also at load control in such networks.

11

Simulation modeling of HM-networks with consideration of positive and negative messages

Naumenko V., Matalytski M.

Journal of Applied Mathematics and Computational Mechanics

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2015

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Vol. 14, nr 2

49--60

EN

In the article an open Queueing Network (QN) with positive and negative messages and incomes, which can be used in modeling of the behavior of viruses at the Information Systems and Networks (ISN), and also at forecasting costs taking into account virus penetration, was analyzed. Some numerical and analytical results were described on the analysis of exponential QN of the above type. It was given an algorithm of simulation modeling (SM) of HM-networks with positive and negative messages, based on the 0-point, which allows you to find incomes in such networks with arbitrary distribution of service times of positive messages. The results of the SM have been compared with the proposed analytical and numerical results. Sufficiently high accuracy of these methods was shown.

12

Investigation of networks with positive and negative messages, many-lines queueing systems and incomes

Naumenko V., Matalytski M.

Journal of Applied Mathematics and Computational Mechanics

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2015

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Vol. 14, nr 1

79--90

EN

The analysis of an open Markov Queueing Network with positive and negative messages, many-lines queueing systems and incomes has been carried out. External arrivals to the network, service times of rates and probabilities of messages transition between queueing systems (QS) depend on time. A method for finding the expected incomes of the network systems, the expressions for the average number of messages at the systems has been proposed.

13

Analysis of Markov HM-Queueing Networks with FIFO discipline at the transient regime and its application

Matalytski M.

Journal of Applied Mathematics and Computational Mechanics

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2015

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Vol. 14, nr 3

65--77

EN

In the paper Markov Queueing Networks (QN) are considered with the same types and different types of customers and incomes, FIFO discipline, which are probabilistic models of different Information Nets and Systems (INS). The incomes from the state transition of the network depend on servicing times of customers in the Queueing Systems (QS). The purpose of the research are design and development of methods and techniques of finding the probability-cost characteristics in such QN as effective analysis tools of INS. A closed Markov HM-network with the same types of customers has been investigated. Approximate expressions for the expected incomes of the QS were obtained. The method of finding the mean number of the customers was proposed. The analysis of an open HM-network with different types of customers and many-server queues has been carried out in the second part of the paper. Customers during the transition between QS can change its type. Approximate expressions for the expected incomes of the QS for each type of customer have been also obtained. A method for finding the mean number of servicing lines was described.

14

Zastosowanie HM-sieci kolejkowych dla wyznaczenia objętości pamięci systemów informacyjnych

Matalytski M., Naumenko V.

Studia Informatica

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2014

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Vol. 35, nr 3

63--69

PL

Dla rozwiązywania zagadnienia wyznaczenia objętości pamięci stochastycznych systemów informacyjnych (SI) proponowano nowy model matematyczny oparty na zastosowaniu HM (Howard-Matalytski) sieci kolejkowych z dochodami. Wskazany model pozwala na uwzględnienie zależności czasu opracowania komunikatu od jego objętości, a także możliwość zmiany objętości komunikatu z ciągiem czasu. Otrzymano wyrażenia dla wartości oczekiwanych objętości komunikatów w węzłach SI.

EN

To solve the problem of determining of memory volume of stochastic information (IS) systems of various configurations is proposed to use a new model based on the use of HM-queueing network, which takes into account the relationship between the volume of messages and the time of processing nodes in the system, the possibility of changing with time the volume of messages. It is obtained an expressions for the expected volumes of messages in the nodes IS.

15

The analysis of the three-level HM-network and its application for the incomes forecasting of a logistic transport system

Matalytski M., Kiturko O.

Journal of Applied Mathematics and Computational Mechanics

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2014

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Vol. 13, nr 3

141--153

EN

Investigation of the closed three-level HM-network, which is the model of the production transportation in a logistic transport system, is conducted in this article. The approximated expression for the expected income of the central system and the system of the nonhomogeneous differential equations for an average number of messages in network systems, when the incomes from transitions between network’s states are random variables with the set moments of the first two orders, is received. The relation for variance of the income of the central system of network (system of the top level) is also received.

16

Investigation of G-network with signals at transient behavior

Matalytski M., Naumenko V.

Journal of Applied Mathematics and Computational Mechanics

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2014

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Vol. 13, nr 1

75--86

EN

G-queueing network with positive messages and signals at transient behavior is considered. A system of difference-differential equations for the state probabilities of the network is obtained. To find them and the average characteristics of the network a technique was applied based on the use of apparatus of multivariate generating functions. An expression for the generating function was obtained. An example is calculated.

17

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.

18

Application of a z-transforms method for investigation of Markov G-networks

Matalytski M., Naumenko V.

Journal of Applied Mathematics and Computational Mechanics

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2014

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Vol. 13, nr 1

61--73

EN

The purpose of this research paper is to find the expected incomes in open Markov queueing networks with incomes, positive and negative messages at any time by the multidimensional transformations. Investigations were carried out in cases when incomes from the network transitions between the states are deterministic functions not dependent on network states and time. It is assumed that all network systems are one-line. It was proved the theorem on the expression for the multidimensional z-transform. An algorithm was proposed for calculation of expected incomes. It is calculated an example on the PC.

19

Analiza asymptotyczna wykładniczej sieci kolejkowej z parametrami zależnymi od czasu

Matalytski M.

Studia Informatica

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2013

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Vol. 34, nr 3

177--184

PL

W artykule przeanalizowano zamkniętą wykładniczą sieć kolejkową z dużą liczbą zgłoszeń i parametrami zależnymi od czasu. Otrzymano układ równań różnicowo-różniczkowych dla prawdopodobieństw stanów i układ równań różniczkowych do wyznaczenia średniej liczby zgłoszeń w systemach sieci. Obliczono przykład numeryczny

EN

The closed exponentional queueing network with the large number of messages and the time-dependent parameters is investigated. We have received the system of difference-differential equations for the state probabilities and system of differential equations for the average number of messages of network systems. The modeling example for their calculation is presented.

20

The asymptotic analysis of the income in closed HM-structure with priority messages

Matalytski M., Kiturko O.

Journal of Applied Mathematics and Computational Mechanics

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2013

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Vol. 12, nr 2

51--59

EN

In the article the asymptotic analysis of closed exponential queueing HM-structure with priority messages is carried out with a large total number of messages, depending on time. The number of service lines in systems, the intensity of service messages in them, the probabilities of message transitions between systems also depends on time. It is proved that the density of the income distribution in the network systems in asymptotic satisfies differential equations in partial derivatives. This provided the inhomogeneous differential equation for the expected incomes system structure. An example of transport logistics shows how to solve such equations.