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

Published online by Cambridge University Press:  13 July 2018

M. Matalytski  and
D. Kopats
M. Matalytski
Affiliation:
Czestochowa University of Technology, Institute of Mathematics, Czestochowa, Poland E-mail: m.matalytski@gmail.com
D. Kopats
Affiliation:
Faculty of Mathematics and Computer Science, Grodzenskij dzyarzhauny universitat imya Yanki Kupaly, Grodno, Belarus E-mail: dk80395@mail.ru
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Abstract

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.

Keywords

classescombined with the method of seriesG-networkmethod of successive approximationsnon-stationary regimepositive customers and signals of multiple
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 33 , Issue 3 , July 2019 , pp. 404 - 416
Copyright
Copyright © Cambridge University Press 2018 

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References

1.Gelenbe, E. (1991). Product form queueing networks with negative and positive customers. Journal of Applied Probability 28(1): 656–663.Google Scholar
2.Matalytski, M. & Naumenko, V. (2013). Non-stationary analysis of queueing network with positive and negative messages. Journal of Applied Mathematics and Computational Mechanics 12(2): 61–71.Google Scholar
3.Gelenbe, E. (1994). G-networks: a unifying model for neural and queueing networks. Annals of Operations Research 48: 433–461.Google Scholar
4.Kim, H. & Gelenbe, E. (2013). G-networks towards synthetic biology. A brief review. Conference of the IEEE Engineering in Medicine and Biology Society 1: 579–583.Google Scholar
5.Matalytski, M. & Naumenko, V. (2016). Stochastic network with non-standard moving customers: monogr. Grodno: GrSU, 346 p.Google Scholar
6.Gelenbe, E. (1993). G-networks with triggered customer movement. Journal of Applied Probability 30(1): 742–748.Google Scholar
7.Gelenbe, E. (1993). G-networks with signals and batch removal. Probability in the Engineering and Informational Sciences 7: 335–342.Google Scholar
8.Matalytski, M. & Naumenko, V. (2014). Investigation of G-network with signals at a transient behavior. Journal of Applied Mathematics and Computational Mechanics 13(1): 75–86.Google Scholar
9.Fourneau, J.M., Gelenbe, E. & Suros, R. (1996). G-networks with multiple classes of negative and positive customers. Theoretical Computer Science 155: 141–156.Google Scholar
10.Kopats, D. & Matalytski, M. (2017). Analysis of network with positive and negative customer multiple classes in non-stationary mode. Vesnik Hrodzenskaha Dziarzhaunaha Universiteta Imia Ianki Kupaly. Seryia 2. Matematyka. Fizika. Infarmatyka, Vylichal'naia Tekhnika i Kiravanne 3: 150–161.Google Scholar
11.Gelenbe, E. & Labed, A. (1998). G-networks with multiple classes of signals and positive customers. European Journal of Operational Research 108: 293–305.Google Scholar
12.Naumenko, V. (2014). Analysis of Markov G-network by the modified method of successive approximations, combined with the method of series. Vesnik of Yanka Kupala State University of Grodno. Series 2. Mathematics. Physics. Informatics, Computer Technology and its Control 4(1): 125–134.Google Scholar
13.Matalytski, M. & Naumenko, V. (2014). Application of a z-transforms method for investigation of Markov G-networks. Journal of Applied Mathematics and Computational Mechanics 13(1): 61–73.Google Scholar
14.Naumenko, V. & Matalytski, M. (2014). Analysis of Markov network with revenues and positive and negative customers. Informatics 1(41): 5–14.Google Scholar
15.Naumenko, V. & Matalytski, M. (2015). Simulation modeling of HM-networks with consideration of positive and negative messages. Journal of Applied Mathematics and Computational Mechanics 14(2): 49–60.Google Scholar
16.Naumenko, V. & Matalytski, M. (2014). Finding expected revenues in Markov G-network with signals. Vesnik of Yanka Kupala State University of Grodno. Series 2. Mathematics. Physics. Informatics, Computer Technology and its Control 4(2): 134–143.Google Scholar
17.Matalytski, M. & Naumenko, V. (2013). Analysis of G-network at a transient behavior and its application // Computer Data Analysis and Modeling: Theoretical and Applied Stochastics: Proc. of the Tenth Intern. Conf., Minsk, September. 10–14: 2: 82–85.Google Scholar
18.Matalytski, . & Kopat, D. (2017). Finding expected revenues in G-network with signals and customers batch removal. Probability in the Engineering and Informational Sciences 31(4): 561–575.Google Scholar
19.Gelenbe, E. (2007). Steady-state solution of probabilistic gene regulatory networks. Physical Review E 76, 031903.Google Scholar
20.Gelenbe, E. (1993). Learning in the recurrent random neural network. Neural Computation 5(1): 154–164.Google Scholar