2026 , Volume 31, № 3, p.57-70

Stochastic (probabilistic) models (such as models of radiation and particle transfer; geophysical stochastic computer models; queuing theory; models described by stochastic differential equations; problems of data processing and self-learning of neural networks, etc.) describe many practically significant processes and phenomena. These models correspond to algorithms of numerical statistical modelling (or Monte Carlo methods), which allow implementing a computer approximation of the necessary functionals associated with the objects under study. In turn, computer modelling (numerical generation of sample values) of random variables is the main element of these algorithms. Given the relatively slow convergence of Monte Carlo methods (with an increase in the number of sample values used), it is necessary to employ, whenever possible, economical computer algorithms for modelling of random variables.

The paper proposes efficient universal algorithms with probability equalization for modelling one-dimensional absolutely continuous random variables with monotone densities on finite distribution segments: the double-sided algorithm of the rejection method with piecewise constant majorant and minorant and the modified algorithm of the ziggurat method. The record-breaking efficiency of the algorithms is determined by the fact that when modelling an auxiliary random variable with respect to a distribution density proportional to the majorant, the number of the segment of the monotonicity interval partition, which has a uniform discrete distribution in the case of probability equalization, is selected using a record efficient “formula with an integer part”.

Algorithms with probability equalization are included in the library of the PrEMA (probability equalization modelling algorithms) research dialog system (see https://prema.andronix1.ru), which allows comparing the running time of algorithms both with each other and with the formulas of the inverse distribution function (for the so-called elementary densities) that were considered as the most economical until recently.

In this paper, the comparison of new algorithms with probability equalization in application to computer modelling of the widely used power-law distribution is carried out. It is shown that these algorithms significantly (the double-sided algorithm by 1.76 times, the ziggurat-algorithm by 2.04 times) exceed the known formula of the inverse distribution function method for the power distribution in terms of low costs.