Mathematical Model of the Delay in Communication Networks Based on QS with a Time Lag

In the mathematical modeling of modern computer networks, telecommunication networks, traffic flows, logistics and many others, the methods of queuing theory are widely used. In turn, in studies of queuing systems (QS) G/G/1 with arbitrary distribution laws of intervals between adjacent requirements of the incoming flow and their service time, the spectral decomposition method (MSD) of solving the Lindley integral equation is often used. This method is based on the search for zeros and poles of the constructed spectral decomposition in the form of some fractional-rational function using numerical methods to determine the roots of polynomials. In this case, the coefficients of the polynomial in the numerator of the expansion are expressed through the unknown parameters of the distribution laws used to describe the QS. In the case of teletraffic research, usually these unknown parameters of the distribution laws can be determined through the numerical characteristics of the intervals between traffic packets by the method of moments. The purpose of this article is to present a fundamentally new mathematical model of a system formed by two flows with distribution laws shifted to the right. This is possible only for those probability distribution laws whose density functions are Laplace transformable. The main advantages of such systems, let us call them time lag systems, are that they provide less queue latency compared to conventional systems, and that they extend the range of traffic parameters. The article presents the results obtained on the average delay of requests in the queue for a system with exponential and hyper-Erlang distributions, an algorithm for calculating the average delay and the results of computational experiments in the Mathcad package.

Simulation of Data Losses, Nonlinearity and Modulation Impact in RPAS/UAV Swarms

Intelligence of Remotely Piloted Air System (RPAS) swarms depends on reliable communications. The parallelism and distributed characteristics of swarm intelligence provide self-adapting and reliable capabilities. This article is devoted to the calculation of packet losses and the impact of traffic parameters on the data exchange with swarms. Original swarm models were created with the help of MATLAB and NetCracker packages. Dependences of data packet losses on the transaction size are calculated for different RPAS number in a swarm using NetCracker software. Data traffic with different parameters and statistical distribution laws was considered. The effect of different distances to drones on the base station workload has been simulated. Data transmission in a swarm was studied using MATLAB software depending on the signal-to-noise ratio, nonlinearity levels of base station amplifier, signal modulation types, base station antenna diameters, and signal phase offsets. The data obtained allows foresee the operation of RPAS communication channels in swarms.

Informative sign of determining the category of transmitted data in high-speed information transmission networks.

Difficulties arise with the recognition of the traffic category in the conditions of widespread use in telecommunication networks of many different non-standardized data transmission protocols. The article discusses the features of data transmission for the category of real-time data and data not critical to delays. As an informative feature for recognizing a data category, it is proposed to use an estimate of the bit rate of a single subscriber. The paper presents the results of an experiment to estimate the bit rate of a single subscriber of the assessment, substantiate the distribution laws of estimates, and determine the threshold value of a feature for recognizing a traffic category.

Application of bayesian scientific approach to constructing the statistical estimations for solving metrological and measurement problems

Nowadays, constructing effective statistical estimates with a limited amount of statistical information constitutes a significant practical problem. The article is devoted to applying the Bayesian scientific approach to the construction of statistical estimates of the parameters of the laws of distribution of random variables. Five distribution laws are considered: The Poisson law, the exponential law, the uniform law, the Pareto law, and the ordinary law. The concept of distribution laws that conjugate with the observed population was introduced and used. It is shown that for considered distribution laws, the parameters of the laws themselves are random variables and obey the typical law, gamma law, gamma - normal law, and Pareto law. Recalculation formulas are obtained to refine the parameters of these laws, taking into account posterior information. If we apply the recalculation formulas several times in a row, we will get some convergent process. Based on a converging process, it is possible to design a process for self-learning a system or self-tuning a system. The developed scientific approach was applied to solve the measuring problems for the testing measuring devices and technical systems. The results of constructing point estimates and constructing interval estimates for these laws' parameters are given. The results of comparison with the corresponding statistical estimates constructed by the classical maximum likelihood method are presented.

DEVELOPMENT OF A MODEL FOR THE DYNAMICS OF PROBABILITIES OF STATES OF SEMI-MARKOV SYSTEMS

The subject is the study of the dynamics of probability distribution of the states of the semi-Markov system during the transition process before establishing a stationary distribution. The goal is to develop a technology for finding analytical relationships that describe the dynamics of the probabilities of states of a semi-Markov system. The task is to develop a mathematical model that adequately describes the dynamics of the probabilities of the states of the system. The initial data for solving the problem is a matrix of conditional distribution laws of the random duration of the system's stay in each of its possible states before the transition to some other state. Method. The traditional method for analyzing semi-Markov systems is limited to obtaining a stationary distribution of the probabilities of its states, which does not solve the problem. A well-known approach to solving this problem is based on the formation and solution of a system of integral equations. However, in the general case, for arbitrary laws of distribution of the durations of the stay of the system in its possible states, this approach is not realizable. The desired result can only be obtained numerically, which does not satisfy the needs of practice. To obtain the required analytical relationships, the Erlang approximation of the original distribution laws is used. This technique significantly increases the adequacy of the resulting mathematical models of the functioning of the system, since it allows one to move away from overly obligatory exponential descriptions of the original distribution laws. The formal basis of the proposed method for constructing a model of the dynamics of state probabilities is the Kolmogorov system of differential equations for the desired probabilities. The solution of the system of equations is achieved using the Laplace transform, which is easily performed for Erlang distributions of arbitrary order. Results. Analytical relations are obtained that specify the desired distribution of the probabilities of the states of the system at any moment of time. The method is based on the approximation of the distribution laws for the durations of the stay of the system in each of its possible states by Erlang distributions of the proper order. A fundamental motivating factor for choosing distributions of this type for approximation is the ease of their use to obtain adequate models of the functioning of probabilistic systems. Conclusions. A solution is given to the problem of analyzing a semi-Markov system for a specific particular case, when the initial distribution laws for the duration of its sojourn in possible states are approximated by second-order Erlang distributions. Analytical relations are obtained for calculating the probability distribution at any time.

Analysis of distribution laws of transformer oil indicators in 110-330 kV transformers

Introduction. Ensuring the operational reliability of power transformers is an urgent task for the power industry in Ukraine and for most foreign countries. One of the ways to solve this problem is the correction of maximum permissible values of insulation parameters. However, such a correction is fundamentally impossible without an analysis of the laws of distribution of diagnostic indicators in the equipment with different states. The purpose of the research is to analyse the laws of distribution of the quality indicators of transformer oil with different states in 110 and 330 kV transformers. Novelty. It was found that both 330 kV autotransformers and 110 kV transformers have the displacements between the mathematical expectations of the distribution density of usable oil indicators. It caused by different service life of the analysed transformers and different values of load factors. This indicates the need to consider the influence of these factors when correcting the maximum permissible values of oil indicators. Also, the presence of displacement between the distribution densities of some indicators of usable oil in 110 kV transformers and 330 kV autotransformers has been revealed. It indicates a different intensity of oxidation reactions in transformers with different voltage class. In order to reduce the heterogeneity of initial data the procedure of statistical processing of in-service test results has been proposed as a method. This procedure combines the use of a priori information about the service life of equipment and values of load factors with the elements of statistical hypothesis testing. The results of the analysis of the distribution laws of transformer oil indicators with different states have shown that for both usable and unusable oil the values of oil indicators obey the Weibull distribution. Values of the shape and scale parameters for each of the obtained indices arrays have been obtained, as well as calculated and critical values of the goodness-of-fit criteria. Practical value. Obtained values of the distribution law parameters of the transformer oil indicators with different states, considering the service life and operating conditions allow to perform the correction of the maximum permissible values of the indicators using the statistical decision-making methods.

Automated system for identification of data distribution laws by analysis of histogram proximity with sample reduction

The error in the identification of the distribution law entails an incorrect assessment of other characteristics (standard deviation, kurtosis, antikurtosis, etc.). The article is devoted to the development of accessible and simple software products for solving problems of identifying distribution laws and determining the optimal size of a data sample. The paper describes a modified method for identifying the law of data distribution by visual analysis of the proximity of histograms with a reduction in the sample size with software implementation. The method allows choosing the most probable distribution law from a wide base of the set. The essence of the method consists in calculating the entropy coefficient and absolute entropy error for the initial and half data sample, determining the optimal method for processing the histogram using visual analysis of the proximity of histograms, and identifying the data distribution law. The experimental data processing model makes it possible to take into account the statistical properties of real data and can be applied to various arrays, and allows to reduce the sample size required for analysis. An automated system for identifying the laws of data distribution with a simple and intuitive interface has been developed. The results of the study on real data indicate an increase in the reliability of the identification of the data distribution law.

Composition and superposition failures probability distribution laws for assessing military electrical products reliability

An implementation of the developed computational structural method for assessing the reliability of complex electrical products. The basis of this method is the use of combinations of composition and superposition laws of distribution of probabilities of failures. The estimation of reliability of a rotating transformer type ВT-5. The calculated reliability indices are compared with results of standard test ВT-5 for reliability. Received high convergence of the results.

Evaluation of reliability of quickly repairable components of transshipment machines and mechanisms

When calculating the reliability indicators of transshipment machines and mechanisms, the required values are not always obtained explicitly due to systematic difficulties (complex distribution laws) or limited initial data. The optimal stock of spare parts for transshipment machines and mechanisms depends on the reliability indicators, and the nature and intensity of their use. Downtime due to the failure of transshipment machines and mechanisms used for ship handling leads to greater losses than downtime of these machines in the warehouse. Failures of transshipment machines and mechanisms can be due to the failure of non-repairable and quickly repairable components, and those with a long repair time. To calculate the main reliability indicators of quickly repairable components, the most common laws of distribution of random variables were employed. The paper considers the methodological aspects of the probabilistic reliability estimate of quickly repairable components of transshipment machines and mechanisms.