Development of a method for state estimation and optimisation of multifactor semi-Markov systems - Raskin, Lev; Sukhomlyn, Larysa; Karpenko, Viacheslav; Sokolov, Dmytro; Vlasenko, Vitalii (Technology audit and production reserves, 2025)
The object of this research is a method for solving problems of analysis and optimization of semi-Markov systems. The importance of this topic is determined by the following circumstances. First, traditional, standard theoretical and practical problems of stochastic system research are solved analytically only for Markov systems for which the laws of distribution of the duration of stay in each state before leaving are exponential. Clearly, this strict requirement is not met for real systems. Second, a general method of analytical study does not exist for many probabilistic systems. Third, only numerical methods for solving such problems are available and feasible. Moreover, in each case, a solution can only be obtained for the specific system under study, operating under specific conditions. Clearly, such a solution is uninformative and practically useless for optimization problems of multifactor systems. In this regard, the study aims to develop a universal method for solving analysis and optimization problems, suitable for any semi-Markov systems. The proposed method for solving the formulated problem solves it in two stages. In the first stage, a matrix of distribution densities is found by processing experimental data, representing the duration of the system's stay in each state before transitioning to another state. It is crucial that the densities be in the Erlang distribution class of some order. These densities are found using the least-squares method, using histograms obtained by processing the experimental data. In the second stage, the resulting distribution densities are used to construct a system of differential equations for the probabilities of the system's stay in each possible state. This constructively utilizes the unique property of Erlang distributions: any Erlang flow is a sifted simplest Poisson flow. Sequentially completing these two stages yields a solution to the problem of studying any probabilistic (semi-Markov) systems. Thus, the method proposed in this paper for solving problems of analysis and optimization of semi-Markov systems is universal.