Analytical Solution of Kolmogorov Equations for Four-Condition Homogenous, Symmetric and Ergodic System
Affiliation(s)
Department of Automobiles and Automobile Sector, National Mining University, Dnepropetrovsk, Ukraine. Foreign Languages Department National, Mining University, Dnepropetrovsk, Ukraine.
Department of Automobiles and Automobile Sector, National Mining University, Dnepropetrovsk, Ukraine. Foreign Languages Department National, Mining University, Dnepropetrovsk, Ukraine.
ABSTRACT
Technical system consisting of two independent subsystems (e.g. hybrid car) is considered. Graduated state graph being homogenous ergodic system of symmetric structure is constructed for the system. Differential Kolmogorov equations, describing homogenous Markovian processes with discrete states and continuous time, are listed in symmetric matrix form. Properties of symmetry of matrix of subsystem failure and recovery flow intensity are analyzed. Dependences of characteristic equation coefficients on intensity of failure and recovery flows are obtained. It is demonstrated that the coefficients of characteristic equation meet the demands of functional dependence matching proposed visible analytical solution of complete algebraic equation of fourth order. Depending upon intensity of failure and recovery flows, four roots of characteristic equation are analytically found out. Analytical formulae for state probability of interactive technical system depending upon the roots of characteristic equation are determined using structurally ordered symmetric determinants, involving proper column of set initial data as well as subsystem failure and recovery flow intensity are proposed.
Technical system consisting of two independent subsystems (e.g. hybrid car) is considered. Graduated state graph being homogenous ergodic system of symmetric structure is constructed for the system. Differential Kolmogorov equations, describing homogenous Markovian processes with discrete states and continuous time, are listed in symmetric matrix form. Properties of symmetry of matrix of subsystem failure and recovery flow intensity are analyzed. Dependences of characteristic equation coefficients on intensity of failure and recovery flows are obtained. It is demonstrated that the coefficients of characteristic equation meet the demands of functional dependence matching proposed visible analytical solution of complete algebraic equation of fourth order. Depending upon intensity of failure and recovery flows, four roots of characteristic equation are analytically found out. Analytical formulae for state probability of interactive technical system depending upon the roots of characteristic equation are determined using structurally ordered symmetric determinants, involving proper column of set initial data as well as subsystem failure and recovery flow intensity are proposed.
KEYWORDS
State Graph, Markovian Process, Kolmogorov Equations, Intensity Matrix, Characteristic Equations, State Probabilities
State Graph, Markovian Process, Kolmogorov Equations, Intensity Matrix, Characteristic Equations, State Probabilities
Cite this paper
Kravets, V. , Bass, K. , Kravets, V. and Tokar, L. (2014) Analytical Solution of Kolmogorov Equations for Four-Condition Homogenous, Symmetric and Ergodic System. Open Journal of Applied Sciences, 4, 497-500. doi: 10.4236/ojapps.2014.410048.
Kravets, V. , Bass, K. , Kravets, V. and Tokar, L. (2014) Analytical Solution of Kolmogorov Equations for Four-Condition Homogenous, Symmetric and Ergodic System. Open Journal of Applied Sciences, 4, 497-500. doi: 10.4236/ojapps.2014.410048.
References
[1] Venttsel, E.S. and Ovcharov L.A. (1991).Theory of Random Processes and Its Engineering Applications. Moscow, 384 p. [2] Smirnov, V.I. (1974) A Course on Higher Mathematics. Moscow, Vol. 2, 656p. [3] Kravets, V. and Chibushov, Y (1994) Method of Finding the Analytical Solution of the Algebraic Particular Aspect Equation, Poland, Rzeszow, Folia Sci. Univ. Tech. Resoviensis, Math. 16, 104-117. [4] Kravets, V.V. (1972) On the Solving Certain Engineering Problems, Amounting to Find Roots of Particular-Type Algebraic Equation, Selected works “Best Practices of Electrification of Mining-and-Processing Integrated Works in Dnepropetrovsk Region”, Dnepropetrovsk, 107-109 pp.
[1] Venttsel, E.S. and Ovcharov L.A. (1991).Theory of Random Processes and Its Engineering Applications. Moscow, 384 p. [2] Smirnov, V.I. (1974) A Course on Higher Mathematics. Moscow, Vol. 2, 656p. [3] Kravets, V. and Chibushov, Y (1994) Method of Finding the Analytical Solution of the Algebraic Particular Aspect Equation, Poland, Rzeszow, Folia Sci. Univ. Tech. Resoviensis, Math. 16, 104-117. [4] Kravets, V.V. (1972) On the Solving Certain Engineering Problems, Amounting to Find Roots of Particular-Type Algebraic Equation, Selected works “Best Practices of Electrification of Mining-and-Processing Integrated Works in Dnepropetrovsk Region”, Dnepropetrovsk, 107-109 pp.