Open AccessAbout applicability of semi-Markov models of operation
Author(s) -
R. Z. Khayrullin
Publication year - 2019
Publication title -
journal of physics. conference series
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.21
H-Index - 85
eISSN - 1742-6596
pISSN - 1742-6588
DOI - 10.1088/1742-6596/1425/1/012143
Subject(s) - eigenvalues and eigenvectors , mathematics , markov chain , matrix differential equation , euler method , differential equation , ordinary differential equation , stochastic matrix , euler's formula , invariant (physics) , matrix (chemical analysis) , markov model , markov process , range (aeronautics) , computer science , mathematical analysis , physics , statistics , materials science , composite material , quantum mechanics , mathematical physics
The methodology is proposed for assessing the applicability of stationary semi-Markov models for the problems of efficient operation of control and measuring equipment. The methodology is based on the Euler method of construction a fundamental system of solutions for a system of ordinary differential equations with constant coefficients. The methodology is founded on the calculation of all eigenvalues (spectrum) and eigenvectors of the matrix of the system of differential equations. It is shown that for the classical model of operation of measuring equipment for a typical range of variation of the main parameters of the model (probabilities of failure, false failure and undetected failure), the characteristic equation of the system has two invariant eigenvalues and four eigenvalues that can be considered us functions of the model parameters. A qualitative analysis of the dependence of the spectrum of the matrix on the parameters of the model is carried out. The dependence of the non-invariant real eigenvalue, which plays a key role in the convergence of the solution of the dynamic model to the solution of the stationary semi-Markov model, on the parameters of the model is investigated in detail. The results of mathematical simulation are presented.