Analytical Solution of Kolmogorov Equations for Asymmetric Markov Chains with Four and Eight States | Chapter 10 | Mathematics and Computer Science: Contemporary Developments Vol. 10

Aims: Systems consisting of two and three independent subsystems are considered. First, a technical system consisting of two independent subsystems (e.g. hybrid car) is considered. The purpose is to demonstrate that the coefficients of the characteristic equation meet the demands of functional dependence matching the proposed visible analytical solution of the complete algebraic equation of the fourth order.

Then a system consisting of three independent subsystems is considered. The purpose is the development of an analytical method for solving octic Kolmogorov equations to control a random process while varying the intensity of the progress of renewal and degradation of failures.

Methodology: Differential Kolmogorov equations, describing homogenous Markovian processes with discrete states and continuous time, are listed in symmetric matrix form. The applied strategy relies upon a concept of mathematical description harmonization; representation of eight states of the considered Markov chain by an asymmetric graph; the ordered matrix of transitional probabilities of octic Kolmogorov equations; and characteristic specific octic equation in which roots are distributed symmetrically within a complex plane.

Results: Properties of symmetry of matrix of subsystem failure and recovery flow intensity are analyzed. Dependences of characteristic equation coefficients on the intensity of failure and recovery flows are obtained. Depending upon the intensity of failure and recovery flows, four roots of the characteristic equation are analytically found. Analytical temporal solutions have been obtained for octic probabilities of the Markov chain in the form of the ordered determinants for indices of eight roots as well as indices of eight states including column vector of the initial conditions. An example of structures consisting of three independently functioning processes, the random process of the transition of the structure through eight possible states with a known initial state is determined in time is considered.

Conclusion: Heuristically, formulas have been derived evaluating eight roots of the characteristic Kolmogorov equation; one of them is zero root due to the intensity of failures and renewals of three subsystems eight states of which compose generally asymmetric Markov chain. The proposed analytical solution of the octic algebraic equation is verified using Vieta’s formulas.

 

Author (s) Details

Victor V. Kravets
Dnipro University of Technology, Dnipro, Ukraine.

 

M.I. Kapitsaf
Ukrainian State University of Science and Technologies, Dnipro, Ukraine.

 

I.V. Domanskyi
Ukrainian State University of Science and Technologies, Dnipro, Ukraine.

 

Vladimir V. Kravets
Type Vasyl Stefanyk Precarpathian National University, Ivano-Frankivsk, Ukraine.

 

T.S. Hryshechkina
Ukrainian State University of Science and Technologies, Dnipro, Ukraine.

 

S. O. Zakurday
O. M. Beketov National University of Urban Economy, Ukraine.

 

Konstantin M. Bass
Dnipro University of Technology, Dnipro, Ukraine.

 

Larisa A. Tokar
Dnipro University of Technology, Dnipro, Ukraine.

 

Please see the book here:- https://doi.org/10.9734/bpi/mcscd/v10/3410