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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: m.matalytski@gmail.com
Dmitry Kopats
Affiliation:
Faculty of Mathematics and Computer Science, Grodno State University, Grodno, Belarus E-mail: dk80395@mail.ru
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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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