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.

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.

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.

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