We will prove that when uniformly distributed random numbers are sorted by value, their successive differences are a exponentially distributed random variable Ex(λ). For a set of n random numbers, the parameters of mathematical expectation and standard deviation is λ =n−1. The theorem was verified on four series of 200 sets of 101 random numbers each. The first series was obtained on the basis of decimals of the constant e=2.718281…, the second on the decimals of the constant π =3.141592…, the third on a Pseudo Random Number generated from Excel function RAND, and the fourth series of True Random Number generated from atmospheric noise. The obtained results confirm the application of the derived theorem in practice.

The exponential pheromone signal of the AI hybrid of Markovian Ant Queuing System -MAQS, is divided into the spatial and deposit pheromone fractions which have the identical values. A new hybrid is formed. The convolution of two new exponential signals has the Erlang distribution. Introducing the interstate in the process of markovization the Erlang Queuing Ant System -EQAS, is solved. Comparison of the average distance between artificial ants in MAQS and EAQS gave particular numerical specificity. The average distances are in ϕ equilibrium. Constant ϕ is a famous constant of the Golden ratio.

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There is not a single country in the world that is so rich that it can remove all level crossings or provide their denivelation in order to absolutely avoid the possibility of accidents at the intersections of railways and road traffic. In the Republic of Serbia alone, the largest number of accidents occur at passive crossings, which make up three-quarters of the total number of crossings. Therefore, it is necessary to constantly find solutions to the problem of priorities when choosing level crossings where it is necessary to raise the level of security, primarily by analyzing the risk and reliability at all level crossings. This paper presents a model that enables this. The calculation of the maximal risk of a level crossing is achieved under the conditions of generating the maximum entropy in the virtual operating mode. The basis of the model is a heterogeneous queuing system. Maximum entropy is based on the mandatory application of an exponential distribution. The system is Markovian and is solved by a standard analytical concept. The basic input parameters for the calculation of the maximal risk are the geometric characteristics of the level crossing and the intensities and structure of the flows of road and railway vehicles. The real risk is based on statistical records of accidents and flow intensities. The exact reliability of the level crossing is calculated from the ratio of real and maximal risk, which enables their further comparison in order to raise the level of safety, and that is the basic idea of this paper.