Finding the probability distribution of states in the fuzzy markov systems

Abstract

A problem on finding the stationary distributions of probabilities of states for the Markov systems under conditions of uncertainty is solved. It is assumed that parameters of the analyzed Markov and semi-Markov systems (matrix of transition intensities, analytical description of distribution functions of the durations of being in states of the system before exiting, as well as a matrix of transition probabilities) are not clearly assigned. In order to describe the fuzziness, we employ the Gaussian membership functions, as well as functions of the type. The appropriate procedure of systems analysis is based on the developed technology for solving the systems of linear algebraic equations with fuzzy coefficients. In the problem on analysis of a semi-Markov system, the estimation of components of the stationary distribution of probabilities of states of the system is obtained by the minimization of a complex criterion. The criterion considers the measure of deviation of the desired distribution from the modal one, as well as the level of compactness of membership functions of the fuzzy result of solution. In this case, we apply the rule introduced for the calculation of expected value of fuzzy numbers. The criterion proposed is modified through the introduction of weight coefficients, which consider possible differences in the levels of requirements to different components of the criterion.Розв'язана задача відшукання стаціонарних розподілів ймовірностей станів для марковських систем в умовах невизначеності. Передбачається, що параметри аналізованих систем задані нечітко. У задачі аналізу напівмарковських системи оцінка компонентів стаціонарного розподілу ймовірностей станів системи отримана шляхом мінімізації комплексного критерію. Критерій враховує міру відхилення шуканого розподілу від модального, а також рівень компактності функції приналежності нечіткого результату рішення.