Comparison of quantized transfers of a given volume of gas through a pipe under its temporal and velocity probability distributions

. The paper considers the quantization of gas volume transfers along a hypothetical pipe. The volume of gas is represented by two probability distributions in temporal and velocity form. The known model of optimal in the sense of filling information quantization is applied to quantization of material objects. There is a constant size gap between quanta. Technological aspects of formation of quanta are not considered. On the basis of the carried out minimum cost study the advantage of velocity quantization over time quantization for a number of selected quanta is determined. A normal probability distribution is taken as the initial distribution of the gas volume. The results of the article can be applied to any material and information processes. The main purpose in future research is recommended for any complex network processes, analysis and synthesis of their quality


Introduction
In the articles [1,2] the question about the existence along with the probability distributions of velocity probability distributions was considered earlier. The idea of the problem goes back to R. Fisher [3], where the velocity distributions were called fiducial. The author of this paper calls them dual, opposite to each other, and the very principle of their constructionmathematical dualism.
The question is raised that these types of distributions can be useful and applied to solve practical problems.
The purpose of this paper is to show their usefulness on the simplest example of transmission of a given volume of gas through a pipe using the information quantization, optimal in the sense of filling, proposed by the authors of the paper [4]. Brief meaning of this article in the mathematical form is presented in the form of following expression: where ( ) − probability density of quantized random amount of information, − quantum of information, − space between quanta, − maximal integer number of information quanta with deficiency. An additional unit in the integral is introduced for the quantum, which contains information not taken into account in the filled quanta. ( ) − minimum memory size for placing thus quantized information. The equation in order to find the optimal value of the quantum 0 is solved mathematically strictly [4] or numerically with a certain accuracy. We use expression (1) to solve the set problem of comparison of the process and quality of gas volume transfer through the pipe at two dual methods of its transfer, namely at time and velocity probability distributions.

The principle of duality of distributions and its main relations
Let the time of occurrence of an event be a random variable subject ̂ to the distribution ( ). Consider the quantity ̂= 1 . It has the meaning of the rate of occurrence of a single event ( ). Let us find its probability distribution Assuming that both random variables are positive and their probability densities exist ( ), ( ). Then according to [5] we have: where − is the Dirac delta function. The corresponding (2) distribution function is and the additional function (reliability) has the form where ̄( ) = 1 − ( ). The intensity of the event occurrence is determined by the formula Let us introduce a corresponding expression for the velocity intensity of the event occurrence: .
From formula (6) we obtain the expression known in the theory of random processes: Similarly, we can write the formula substituting into which (6) after a simple transformation we obtain (4), which should be expected. The Laplace transform of the distribution density (2) is equal to It is also true that the relation of the form where * , − image symbol and the Laplace variable. From expressions (7) and (8) differentiating by , taking into account that = 0, it is easy to see that there are no moments of a random variable în the domain [0, ∞).

Comparison of quantum transfers of gas through the pipe under temporal and velocity distributions of its volume probabilities
The same pipe is supposed to send gas from one point to another by quanta separated from each other by the same gaps. Two strategies for forwarding the same volume of gas are considered, related to the temporal and velocity representation of the volume. The gaps between the quanta are the same in both cases. It is required to compare quantitatively between each other the effects of the forwarding processes.
The mathematical expectation of the value of the quantized time volume of the gas is determined by the expression and the mathematical expectation of the value of the quantized velocity volume of the gas by the expression The value of the space between quanta is assumed to be equal to = 5.

Partial properties of quanta. Final estimation of the quantization effect
Quantum properties include: size of quantum, cost value of quantum, number of quanta at quantization, total cost of given quantization, total cost of gaps at given quantum. Total costs refer to the sum of costs for a particular quantum plus the sum of costs for gaps in that quantum. Partial estimation refers to the above four quanta. Each quantum was considered separately in time and velocity representations. The results of the estimation are summarized in Table 1. Table 1. Partial estimation of the quantum. The total volume of time slices and their intervals: 1. Type of quantum -time (M) or speed (N). 2. Quantum volume -time (nv) or velocity (n). 3. The total volume of quanta is temporal (∑ ) or high-speed (∑ ). 4. Total volume of gaps in time or speed quantization (∑ ). 5. Total volume of time quanta and gaps or total volume of velocity quanta and their gaps(∑ и ∑ and∑ и ∑ ).
Explanation of definitions using the example of a temporal distribution. Costs for the amount of gaps ⋅ ∫ ( ( ) + 1) ⋅ ( ) .

Costs for speed distribution
Similarly, the costs for speed distribution are determined ( ). Total final evaluation. Judging by the total costs of quantization (Table 1) quantization of the velocity distribution for all quanta except for the quantum of the last line with the result ∑ : = 101.658, has advantages in comparison with time quantization. The paper did not consider technological aspects of preparation of time and velocity quanta and other physical factors related to gas transfer through the tube.

Conclusion
The process of quantization of gas transfer in a hypothetical pipe is considered at two representations of gas volume in the form of probability distributions -temporal and velocity. The basis for the dual representation was the author's proposal to introduce two versions of probability distributions, obtained by him earlier.
The idea and mathematical form of optimal quantization in the sense of minimum cost is borrowed from the article [4], in which the authors considered quantization of information with gaps between quanta in order to determine the time costs for its storage. In this article, the author applied this idea not to information, but to material means, this is its first difference. The second difference is the consideration of quantization for dual distributions. For the first time the idea of dual distribution was proposed by R. Fisher [3], who called it fiducial, parametric.
The material of the article can be applied to any physical and informational processes. It can serve as a starting point for research of various networks, more precisely, network processes of application nature, informational and material. During the study of these networks direct tasks -analysis and inverse tasks -synthesis of ensuring of their required quality can be set and solved. For an additional acquaintance with velocity distribution, dualism in the theory of random processes one may refer to R. Fisher's monograph [3] and references to articles published by him. The article [6,7] is useful for investigating issues of network analysis and synthesis. In our opinion, velocity research should be performed on