Evaluation of the complexity of modeling the dynamics of complex systems

Abstract

The article considers a new method for calculating the stationary values ​​of the probabilities of states of a complex system for processes with discrete states and continuous time. Markov models are adequate only for a very small class of real processes with an exponential probability distribution. Simulation methods in most cases lead to significant computational costs, as well as semi-Markov models. The possibility of an approach to modeling is shown taking into account the isomorphism of the structure of the set of states and the set of transitions of semi-Markov, Markov and simulation models for arbitrary distribution laws of random intervals in event flows. This approach is based on a set of theoretical provisions proved by the authors in previously published articles and monographs. It includes decomposition, simulation for individual states, the synthesis of an isomorphic Markov representation, and the final calculation of probabilities by solving systems of linear equations. The reduction in computational costs is achieved by equalizing the number of simulation implementations for different model states during decomposition, as well as by directly transferring the simplest flows to an isomorphic Markov representation. The upper O(n)-estimate of the complexity of the proposed algorithm approaches the lower Ω(n)-estimate for simulation modeling. At the same time, the lower Ω(n)-estimate is close to the complexity of solving systems of linear equations. The most significant gain is provided in studies related to the multiple estimation of probabilities on the model for various initial data in order to optimize the system parameters, since each subsequent experiment requires modification of the isomorphic representation for only one of the model states.