Вероятностные модели комбинаторных схем

Вестник Южно-Уральского государственного университета, серия «Математическое моделирование и программирование». 2020. Т. 13. № 3. С. 103–111.

Enatskaya N.

An enumerative method is proposed for the analysis of combinatorial schemes in the
pre-asymptotic region of variation of their parameters based on the construction of their
probabilistic mathematical model, which represents for each scheme an iterative
random process of sequential non-repeated formation of all its outcomes with a certain
discipline of their numbering by unitary addition of certain elements of the scheme to
a given value in it. Due to the importance for a number of studies of the scheme of
recurrence of listing its outcomes, if it does not lie in its nature, it can be achieved by
introducing into the scheme some restrictions that do not lead to a change in their set, do
not change their probability and should be taken into account. The design of the process
under the appropriate conditions of each scheme is graphically depicted by a graph with the
probabilities of iterative transitions specified in it, which determine the final distribution on
the set of its outcomes. On this basis, the problems of determining the number of outcomes
of a scheme, establishing a one-to-one correspondence between numbers and types of its
outcomes, called the numbering problem in direct and reverse statements, and finding the
probability distribution of all its final outcomes are solved, which makes it possible to model
them with the found distribution of playing out the outcome number and the subsequent
determination of its modeled form by the result of solving the direct numbering problem.
In the absence of an explicit formula for the number of outcomes of a scheme under certain
conditions, an estimate of it can be obtained from the results of their modeling, followed
by refinement of the numbering problem. The study of models of combinatorial schemes on
random processes with the introduction of probabilistic parameters expands the possibilities
of their use. The results of the analysis of schemes can be of a nature from numerical methods
and algorithms to analytical in the form of recurrence relations and explicit formulas.
 

Keywords: вероятностные модели, probabilistic models, метод графов, enumeration method, перечислительный метод, the graph method, numbering problem, rapid modelling, задача нумерации, быстрое моделирование