ANALYSIS OF THE NETWORK WITH MULTIPLE CLASSES OF POSITIVE AND NEGATIVE CUSTOMERS AT A TRANSIENT REGIME

2018 ◽  
Vol 33 (2) ◽  
pp. 172-185 ◽  
Author(s):  
Mikhail Matalytski
Keyword(s):  
Transient Regime ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Stationary Probability Distribution ◽  
Negative Customers ◽  
Network Systems ◽  
Modified Method ◽  
Computer Calculations ◽  
Network States ◽  
The Impact

This paper is devoted to the investigation of the G-network with multiple classes of positive and negative customers. The purpose of the investigation is to analyze such a network at a transient regime, to find the state probabilities of the network that depend on time. In the first part, a description of the functioning of G-networks with positive and negative customers is provided, when a negative customer when arriving to the system destroys a positive customer of its class. Streams of positive and negative customers arriving at each of the network systems are independent. Services of positive customers of all types occur in accordance with a random selection of them for service. For nonstationary probabilities of network states, a system of Kolmogorov's difference-differential equations (DDE) has been derived. A method for their finding is proposed. It is based on the use of a modified method of successive approximations, combined with the method of series. It is proved that successive approximations converge with time to a stationary probability distribution, the form of which is indicated in this paper, and the sequence of approximations converges to the unique solution of the DDE system. 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 determination of the time-dependent probabilities of network states using this technique has been calculated. The obtained results can be applied in modeling the behavior of computer viruses and attacks in information and telecommunication systems and networks, for example, as a model of the impact of some file viruses on server resources. variable.

FINDING NONSTATIONARY STATE PROBABILITIES OF OPEN MARKOV NETWORKS WITH MULTIPLE CLASSES OF CUSTOMERS AND VARIOUS FEATURES

2019 ◽  
pp. 1-22
Author(s):  
Mikhail Matalytski ◽  
Dmitry Kopats
Keyword(s):  
Queueing Networks ◽  
Recurrence Relations ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Stationary Probability Distribution ◽  
Modified Method ◽  
Nonstationary State ◽  
Customer Services ◽  
Network States ◽  
Dependent State

This paper discusses a system of difference-differential equations (DDE) that is satisfied by the time-dependent state probabilities of open Markov queueing networks with various features. The number of network states in this case and the number of equations in this system is infinite. Flows of customers arriving at the network are a simple and independent, the time of customer services is exponentially distributed. The intensities of transitions between the network states are deterministic functions depending on its states.To solve the system of DDE, we propose a modified method of successive approximations, combined with the method of series. The convergence of successive approximations with time to a stationary probability distribution, the form of which is indicated in the paper has been proved. The sequence of approximations converges to a unique solution of the system of equations. Any successive approximation can be represented as a convergent power series with an infinite radius of convergence, the coefficients of which satisfy recurrence relations, which is convenient for calculations on a computer. Examples of the analysis of Markov G-networks with various features have been presented.

ANALYSIS OF THE NETWORK WITH MULTIPLE CLASSES OF POSITIVE CUSTOMERS AND SIGNALS AT A NON-STATIONARY REGIME

2018 ◽  
Vol 33 (3) ◽  
pp. 404-416 ◽  
Author(s):  
M. Matalytski ◽  
D. Kopats
Keyword(s):  
Stationary State ◽  
Recurrence Relations ◽  
Stationary Regime ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Network Systems ◽  
Modified Method ◽  
Computer Calculations ◽  
Modeling Behavior ◽  
Selection Of

The object of research is G-network with positive customers and signals of multiple classes. The present paper describes an analysis of this network at a non-stationary regime, also provided a description of method for finding non-stationary state probabilities.At the beginning of the article, a description of the network with positive customers and signals is given. A signal when entering the system destroys a positive customer of its type or moves the customer of its type to another system. Streams of positive customers and signals arriving to each of the network systems are independent. Selection of positive customers of all classes for service – randomly. For non-stationary state probabilities of the network, the system of Kolmogorov difference-differential equations (DDE) has been derived. It is solved by a modified method of successive approximations, combined with the method of series. The convergence of successive approximations with time has been proved to the stationary distribution of probabilities, the form of which is indicated in the article, and the sequence of approximations converges to the unique solution of the DDE system. 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.The obtained results can be applied for modeling behavior of computer viruses and attack in computer systems and networks, for example, as model impact of some file viruses on server resources.

FINDING EXPECTED REVENUES IN G-NETWORK WITH MULTIPLE CLASSES OF POSITIVE AND NEGATIVE CUSTOMERS

2018 ◽  
Vol 33 (1) ◽  
pp. 105-120 ◽  
Author(s):  
Mikhail Matalytski
Keyword(s):  
Time Dependent ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Negative Customers ◽  
Network Systems ◽  
Computer Viruses ◽  
Modified Method ◽  
Telecommunication Systems ◽  
Computer Attacks ◽  
Selection Of

Investigation of the G-network with multiple classes of positive and negative customers has been carried out in the article. The purpose of the investigation is to analyze such a network at a transient regime, finding expected revenues in the network systems depending on time. A negative customer arriving to the system and destroys a positive customer of its class. Streams of positive and negative customers arriving to each of the network systems are independent. Services of positive customers of all types occur in accordance with a random selection of them for service. For the expected revenues, a system of Kolmogorov's difference-differential equations has been derived. A method for their finding is proposed. It is based on the use of a modified method of successive approximations, combined with the method of series. A model example illustrating the finding of time-dependent expected revenues of network systems has been calculated, which shows that the expected revenues of network systems can be either increasing or decreasing time functions. The obtained results can be applied in forecasting losses in information and telecommunication systems and networks from the penetration of computer viruses into it and conducting computer attacks.

scholarly journals Analysis of G-network with unreliable service systems

2019 ◽  
Vol 54 (4) ◽  
pp. 441-453
Author(s):  
D. Ya. Kopats ◽  
M. A. Matalytski
Keyword(s):  
Exponential Distribution ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Queuing Systems ◽  
Negative Customers ◽  
Correct Operation ◽  
Service Line ◽  
Modified Method ◽  
Network States ◽  
Stationary Probabilities

In this paper, the object of research is Markov’s network with positive and negative customers and unreliable service lines with single-line queuing systems (QS). The discipline of service of customers in the systems – FIFO (“first come first served”) and the service time of customers in each line of the QS network are distributed according to the exponential law with their parameters for each QS. The service lines in each QS are defeated by accidental breakdowns, and the time of correct operation of the service line in each SMO has an exponential distribution, with different parameters for each QS. After the breakdown, the line immediately begins to recover, and the recovery time also has an exponential distribution, the parameters of which are different for each QS. The aim of the study is to find the non-stationary probabilities of network states. To find them, a modified method of successive approximations combined with the method of series is proposed. This method allows one to remove the condition of high load. The properties of successive approximations are proved. On the basis of the obtained data, using a computer, a model example illustrating the finding of the time-dependent probabilities of network states is calculated. The results of this work can be applied to the modeling of various information systems and networks.

scholarly journals FINDING EXPECTED REVENUES IN G-NETWORK WITH SIGNALS AND CUSTOMERS BATCH REMOVAL

2017 ◽  
Vol 31 (4) ◽  
pp. 561-575 ◽  
Author(s):  
Mikhail Matalytski ◽  
Dmitry Kopats
Keyword(s):  
Recurrence Relations ◽  
Transition Probabilities ◽  
Queueing Network ◽  
Successive Approximations ◽  
State Transitions ◽  
Method Of Successive Approximations ◽  
Network Systems ◽  
Telecommunication Systems ◽  
Computer Attacks ◽  
Network States

The paper provides an analysis of G-network with positive customers and signals when signals arriving to the system move customer to another system or destroy in it a group of customers, reducing their number to a random value that is given by a probability distribution. The signal arriving to the system, in which there are no positive customers, does not exert any influence on the queueing network and immediately disappears from it. Streams of positive customers and signals arriving to each of the network systems are independent. Customer in the transition from one system to another brings the latest some revenue, and the revenue of the first system is reduced by this amount. A method of finding the expected revenues of the systems of such a network has been proposed. The case when the revenues from transitions between network states are deterministic functions depending on its states has been considered. A description of the network is given, all possible transitions between network states, transition probabilities, and revenues from state transitions are indicated. A system of difference-differential equations for the expected revenues of network systems has been obtained. To solve it, we propose a method of successive approximations, combined with the method of series. It is proved that successive approximations converge to the stationary solution of such a system of equations, and the sequence of approximations converges to a unique solution of the system. Each approximation can be represented as a convergent power series with an infinite radius of convergence, the coefficients of which are related by recurrence relations. Therefore, it is convenient to use them for calculations on a PC. The obtained results can be applied in forecasting losses in information and telecommunication systems and networks from the penetration of computer viruses into it and conducting computer attacks.

FINDING NON-STATIONARY STATE PROBABILITIES OF G-NETWORK WITH SIGNALS AND CUSTOMERS BATCH REMOVAL

2017 ◽  
Vol 31 (4) ◽  
pp. 396-412 ◽  
Author(s):  
Mikhail Matalytski
Keyword(s):  
Stationary State ◽  
Recurrence Relations ◽  
Queueing Network ◽  
Stationary Regime ◽  
Time Dependent ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Kolmogorov Equations ◽  
Modified Method ◽  
Dependent State

This paper is devoted to the research of an open Markov queueing network with positive customers and signals, and positive customers batch removal. A way of finding in a non-stationary regime time-dependent state probabilities has been proposed. The Kolmogorov system of difference-differential equations for state probabilities of such network was derived. The technique of its building, based on the use of the modified method of successive approximations combined with a series method, has been proposed. It is proved that the successive approximations converge over time to the stationary state probabilities, and the sequence of approximations converges to the unique solution of the Kolmogorov equations. Any successive approximation can be represented as a convergent power series with infinite radius of convergence, the coefficients of which satisfy the recurrence relations; that is useful for estimations. Model example illustrating the finding of time-dependent state probabilities of the network has been provided.

scholarly journals Impact of liquids with different densities

2015 ◽  
Vol 766 ◽  
pp. 5-27 ◽  
Author(s):  
Y. A. Semenov ◽  
G. X. Wu ◽  
A. A. Korobkin
Keyword(s):  
Velocity Potential ◽  
Normal Component ◽  
Mapping Function ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Complex Conjugate ◽  
Hodograph Method ◽  
Velocity Magnitude ◽  
Complex Velocity Potential ◽  
The Impact

AbstractThe collision of liquids of different densities is studied theoretically for the case of liquids having wedge-shaped configuration before the impact. Both liquids are assumed to be ideal and incompressible, and the velocity potential theory is used for the flow of each liquid. Surface tension and gravity effects are neglected. The problem is decomposed into two self-similar problems, one for each liquid. Across the interface between the liquids, continuity of the pressure and the normal component of the velocity is enforced through iteration. This determines the shape of the interface and other flow parameters. The integral hodograph method is employed to derive the solution consisting of analytical expressions for the complex-velocity potential, the complex-conjugate velocity, and the mapping function. They are all defined in the first quadrant of a parameter plane, in which the original boundary-value problem is reduced to a system of integro-differential equations in terms of the velocity magnitude and the velocity angle relative to the flow boundary. They are solved numerically using the method of successive approximations. The results are presented through streamlines, interface and free-surface shapes, the pressure and velocity distributions. Special attention is given to the structure of the splash jet rising as a result of the impact.

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