Open AccessSolution of the Kolmogorov system of equations for the generalized graph of states of a mobile feed hopper
Author(s) -
А. И. Купреенко,
Купреенко А. И,
Х М Исаев,
Исаев Х. М,
S.M. MIKHAYLICHENKO,
Михайличенко С. М
Publication year - 2017
Publication title -
traktory i selʹhozmašiny
Language(s) - English
Resource type - Journals
eISSN - 2782-425X
pISSN - 0321-4443
DOI - 10.17816/0321-4443-66353
Subject(s) - kolmogorov equations (markov jump process) , probabilistic logic , mathematics , fodder , probabilistic method , probability theory , work (physics) , probability distribution , statistics , differential equation , mathematical analysis , mechanical engineering , differential algebraic equation , ordinary differential equation , ecology , engineering , biology
To determine the probabilistic time for feeding animals with mobile feeders on farms of cattle, it is required to solve the system of Kolmogorov equations. Because of the massive calculations, this work can be done only with the use of computer technology. However, a specific Kolmogorov system of equations is suitable only for a certain number of components in the feed mix. Depending on the chosen feeding ration, this amount can vary significantly. Solving the Kolmogorov equations for feed mixtures with a different number of components is a very laborious process. Therefore, it is required to develop a mathematical model that allows solving the problem of determining the probabilistic feeding time for multicomponent fodder mixtures. In the course of this work, the positions of the theory of random processes, the theory of graphs, and the foundations of mathematical modeling were used. To accomplish the task, Kolmogorov's system of equations for 2-, 3-, and 4-component fodder mixtures was compiled and solved. The combinations of intensities «L» were replaced by the coefficients introduced for visual perception of formulas and the possibility to reveal the patterns of their development with a change in the number of components. The observed regularities are reflected in the algorithm. The final solution of the Kolmogorov equations is also presented, and a general formula is obtained for calculating the probability of finding a mobile feed hopper in the state of distribution of feed. The formula consists of the coefficients which are calculated according to the developed algorithm. Thus, using the proposed algorithm, there is no need to compile and solve Kolmogorov's systems of equations to determine the probability of finding a mobile feed mill in the state of distribution of food. The observed regularities are conveniently implemented in an electronic environment, for example, MS Excel, which will allow modeling of the technological process of preparation and distribution of feed mixes with a different number of components.