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

Published online by Cambridge University Press:  10 June 2019

Mikhail Matalytski  and
Dmitry Kopats
Mikhail Matalytski
Affiliation:
Institute of Mathematics, Czestochowa, University of Technology, Czestochowa, Poland E-mail: [email protected]
Dmitry Kopats
Affiliation:
Faculty of Mathematics and Computer Science, Grodno State University, Grodno, Belarus E-mail: [email protected]
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Abstract

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.

Keywords

probabilistic networksqueueing theory
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 35 , Special Issue 1: Learning, Optimization, and Theory of G-Networks , January 2021 , pp. 158 - 179
Copyright
Copyright © Cambridge University Press 2019

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References

1.Jackson, J.R. (1957) Networks of waiting lines. Operations Research 5(4): 518–521.CrossRefGoogle Scholar
2.Basharin, G.P., Bocharov, P.P., & Kogan, Ya.A. (1989) Analysis of queues in computer networks, Moscow. Science.Google Scholar
3.Gelenbe, E. (1991) Product form queueing networks with negative and positive customers. Journal of Applied Probability 28: 656–663.CrossRefGoogle Scholar
4.Malinkovsky, Y.V. (1991) A criterion for the representability of a stationary distribution of states of an open Markov queueing network with several classes of claims in the product form. Automation and Remote Control 4: 75–83 (in Russian: “Kriteriy predstavimosti statsionarnogo raspredeleniya sostoyaniy otkrytoy markovskoy seti obsluzhivaniya s neskol'kimi klassami zayavok v forme proizvedeniya”).Google Scholar
5.Serfozo, R. (1999) Introduction to stochastic networks. New York: Springer-Verlag.CrossRefGoogle Scholar
6.Ivnitsky, V.A. (2004) Theory of queueing networks. Moscow: Fizmatlit.Google Scholar
7.Gelenbe, E. & Schassberger, R. (1992) Stability of G-networks. Probability in the Engineering and Information Sciences 6(1): 271–276.CrossRefGoogle Scholar
8.Gelenbe, E. (1994) G-networks: a unifying model for neural and queueing networks. Annals of Operations Research 48: 433–461.CrossRefGoogle Scholar
9.Bocharov, P.P. & Vishnevsky, V.M. (2003) G-networks: development of the theory of multiplicative networks. Automation and Remote Control 5: 46–74 (in Russian: ‘G-seti: razvitiye teorii mul'tiplikativnykh setey’).Google Scholar
10.Kim, H. & Gelenbe, E. (2013) G-networks towards synthetic biology. A brief review. Conference Proceedings of IEEE Engineering in Medicine and Biology Society 1: 579–583.Google Scholar
11.Matalytsky, M.A. &. Naumenko, V.V. (2016) Stochastic Networks with Non-Standard Customers Movement, monograph. Grodno: Grodno State University (in Russian: ‘Stokhasticheskiye seti s nestandartnymi peremeshcheniyami zayavok’).Google Scholar
12.Gelenbe, E. (1993) G-networks with triggered customer movement. Journal of Applied Probability 30(1): 742–748.CrossRefGoogle Scholar
13.Gelenbe, E. (1993) G-networks with signals and batch removal. Probability in the Engineering and Informational Sciences 7: 335–342.CrossRefGoogle Scholar
14.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.CrossRefGoogle Scholar
15.Matalytski, M. & Naumenko, V. (2014) Investigation of G-network with signals at transient behavior. Journal of Applied Mathematics and Computational Mechanics 13(1): 75–86.CrossRefGoogle Scholar
16.Fourneau, J.N., & Gelenbe, E., & Suros, R. (1996) G-networks with multiple classes of negative and positive customers. Theoretical Computer Science 155: 141–156.CrossRefGoogle Scholar
17.Gelenbe, E. & Labed, A. (1998) G-networks with multiple classes of signals and positive customers. European Journal of Operation Research 108(2): 293–305.CrossRefGoogle Scholar
18.Matalytski, M. (2019) Analysis of the network with multiple classes of positive and negative customers at a transient regime. Probability in the Engineering and Informational Sciences 33(2): 172–185.CrossRefGoogle Scholar
19.Matalytski, M. (2017) Finding non-stationary state probability of G-networks with signal and customers batch removal. Probability in the Engineering and Informational Sciences 31(4): 396–412.CrossRefGoogle Scholar
20.Matalytski, M. (2019) Analysis of the network with multiple classes of positive customers and signals at a non-stationary regime. Probability in the Engineering and Informational Sciences 33(3): 404–416.CrossRefGoogle Scholar
21.Kemeny, J.G. & Sneell, J.L. (1976) Denumerable Markov Chains. New York, Heidelberg, Berlin: Springer Verlag.CrossRefGoogle Scholar
22.Valeyev, K.G. & Zhautykov, O.A. (1974) Beskonechnyye sistemy differentsial'nykh uravneniy (Infinite systems of differential equations). Nauka: Alma-Ata.Google Scholar
23.Korobeynik, Yu.F. (1970) Differentsial'nyye uravneniya beskonechnogo poryadka i beskonechnyye sistemy differentsial'nykh uravneniy (Differential equations of infinite order and infinite systems of differential equations). Izvestiya AN SSSR. Seriya mat. 34(4): 881–922.Google Scholar
24.Kopats, D.Y. (2018) Imitatsionnoye modelirovaniye G-seti s gruppovym udaleniyem zayavok (Simulation modeling of G-network with customers batch removal), Vectnik GrGU Ser 2. 8(3): 133–140.Google Scholar
25.Gelenbe, E. (2002) G-network with resets. Performance Evaluation 49(1–4): 179–191.CrossRefGoogle Scholar
26.Fourneau, J.M. (2016) G-networks of unreliable nodes. Probability in the Engineering Information Sciences 30(3): 361–378.CrossRefGoogle Scholar
27.Malinkovsky, Yu. & Kravchenko, S. (2001) Queueing networks with negative requests, with bypassing of request nodes and bounded queueing times, Izvestia NAN RB. Seriya Fiziko-mathematicheskikh Nauk 4: 118–122.Google Scholar
28.Matalytski, M. & Naumenko, V. (2012) Analysis of the queuing network with messages bypass of systems in transient behavior. Czestochowa: Scientific Research of the Institute of Mathematics and Computer Science, University of Technology. 11(2): 71–83.Google Scholar
29.Matalytski, M. & Naumenko, V. (2013) Non-stationary analysis of the queueing network with bypass of systems of many-type and absolute priority messages. Journal of Applied Mathematics and Computational Mechanics 12(1): 93–105.CrossRefGoogle Scholar
30.Naumenko, V. (2017) Issledovanie v nestacionarnom rezhime setei obsluzhivania s obshodami sistem zayavkami I ogranichenom vremeni ozhidania ikh v ocherediash (Investigation in non-stationary regime network with bypass of system and bounded waiting time for claims). Vectnik GrGU Ser 2 7(2): 141–152.Google Scholar
31.Naumenko, V. (2018) Imitacionoe modelirovanie setei obsluzhivania s obshodami sistem zayavkami I ogranichenom vremeni ozhidania ikh v ocherediash (Simulation modeling of queueing networks with bounded waiting time for claims and bypasses). Vectnik GrGU Ser 2 8(2): 120–128.Google Scholar
32.Matalytski, M. & Naumenko, V. (2013) Analysis of the queuing network with messages bypass of systems in transient behavior. Scientific Research of the Institute of Mathematics and Computer Science 11(2): 71–83.Google Scholar
33.Gelenbe, E. (2017) G-networks with adders. Future Internet 9(3): 34–40.Google Scholar