Abstract
The problem of probabilistic analysis of a complex dynamic system, which in the process of functioning passes from one state to another at random times, is considered. The methodology for calculating the conditional probabilities of the system getting into a given state at a given time t, provided that at the initial time the system was in any of the possible states is proposed. The initial data for analysis are a set of experimentally obtained values of the duration of the system stay in each of the states before transition to another state. Approximation of the resulting histograms using the Erlang distribution gives a set of distribution densities of the duration of the system stay in possible states before transition to other states. At the same time, the choice of the proper Erlang distribution order provides an adequate description of the semi-Markov processes occurring in the system. The mathematical model that relates the obtained distribution densities to the functions determining the probabilistic dynamics of the system is proposed. The model describes a random process of system transitions from any possible initial state to any other state during a given time interval. Using the model, a system of integral equations for the desired functions describing the probabilistic transition process is obtained. To solve these equations, the Laplace transform is used. As a result of solving the system of integral equations, functions are obtained that specify the probability distribution of the system states at any time t. The same functions also describe the asymptotic probability distribution of states. An illustrative example of solving the problem for the case when the distribution densities of the lengths of the system stay in possible states are described by the second-order Erlang distributions is given. The solution procedure is described in detail for the most natural special case, when the initial state is H 0
Highlights
A significant part of mathematical models of dynamic systems functioning processes are constructed and described in terms of general graph theory [1, 2]
The theoretical and practical interest is to solve a more complex problem – finding the probability distribution of the system stay in its possible states at an arbitrary point of time after the start of functioning from a given initial state
Comprehensive results are obtained for the case when the random process determining transitions from one state to another is discrete in the phase state space, and the distribution law of intervals between transitions is exponential [5, 6]
Summary
A significant part of mathematical models of dynamic systems functioning processes are constructed and described in terms of general graph theory [1, 2]. The theoretical and practical interest is to solve a more complex problem – finding the probability distribution of the system stay in its possible states at an arbitrary point of time after the start of functioning from a given initial state. The solution to this problem under the most general assumptions regarding the nature of the influencing random process is practically impossible. The problem of obtaining simple relations for calculating the probabilities of stay in each of the states at an Mathematics and cybernetics – applied aspects arbitrary point in time has not been studied enough.
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