# Vol 26, No 4 (2018)

## Mathematical Theory of Teletraffic

### Towards the Analysis of the Queuing System Operating in the Random Environment with Resource Allocation

Zaryadov I.S., Tsurlukov V.V., Carvalho C.V., Zaytseva A.A., Milovanova T.A.

#### Abstract

The mathematical model of the system, that consists of a storage device and several homogeneous servers and operates in a random environment, and provides incoming applications not only services, but also access to resources of the system, is being constructed. The random environment is represented by two independent Markov processes. The first of Markov processes controls the incoming flow of applications to the system and the size of resources required by each application. The incoming flow is a Poisson one, the rate of the flow and the amount of resources required for the application are determined by the state of the external Markov process. The service time for applications on servers is exponential distributed. The service rate and the maximum amount of system resources are determined by the state of the second external Markov process. When the application leaves the system, its resources are returned to the system. In the system under consideration, there may be failures in accepting incoming applications due to a lack of resources, as well as loss of the applications already accepted in the system, when the state of the external Markov process controlling the service and provision of resources changes. A random process describing the functioning of this system is constructed. The system of equations for the stationary probability distribution of the constructed random process is presented in scalar form. The main tasks for further research are formulated.

Discrete and Continuous Models and Applied Computational Science. 2018;26(4):303-320
303-320

## Mathematical Modeling

### On Normal Modes of the Closed Waveguide with Discontinuous Filling

Malykh M.D.

#### Abstract

We consider a waveguide of a constant cross-section S with ideally conducting walls. We assume that the filling of waveguide doesn’tchange along its axis and is described by the piecewise continuous functions ε and μ defined on waveguide cross-section. We show that it is possible to make substitutionwhich allows to work only with continuous functions.

Instead of noncontinuous cross-components of an electromagnetic field E and H we offer to use four potentials ue,uh and ve,vh. We can prove as the generalization of Tikhonov—Samarskii theorem that any field in the waveguide allows representation in such form if we consider the potentials ue,uh as elements of Sobolev space Wo21(S) and the potentials ve,vh as elements of Sobolev space W21(S).

If ϵ and m are the piecewise constant functions then Maxwell’s equations written in four potentialsreduce to a pair of independent systems. This statement give us new approach to theinvestigation of spectral properties of waveguides. First, we can prove the completenessof the system of the normal waves in closed waveguides using standard functionalspaces. Secondly, we can offer new technique for calculation of the normal waves usingstandard finite elements. FreeFem++ program for calculation of disperse lines ofwaveguides is presented. The question of calculation of modes at great values of k=ω/c is also considered.

Discrete and Continuous Models and Applied Computational Science. 2018;26(4):321-330
321-330

### Influence of Noise on the DTW Metric Value in Object Shape Recognition

Gostev I.M., Sevastianov L.A.

#### Abstract

The paper sets out one of the methodologies on image processing and recognition of the form of graphic objects. In it, at the first stage preliminary processing of the image with the purpose of extracting of characteristic attributes of the form of objects is made. Contours of objects are used as such attributes. For transformation of 2D contours of objects to one-dimensional contour function ArcHeight method has been used. The algorithm for identification contour functions based on metrics DTW is developed. Definition of the identification function based on this method is introduced. Features of application of metrics DTW are stated at identification of the form of objects. Matrices of distances of combinations the sample-sample and the sample-not sample are presented. Results of calculations of metrics DTW on a plenty of real data are analyzed. It is shown, that the developed algorithm allows to identify the form of objects independently of their position and an angle of turn on the image. Influence of the noise imposed on the image of object, on value of the metrics is investigated. Theoretical and practical results of such dependence are received; it shows that in a wide range (up to the ratio a signal/noise 10 dB) value of the metrics practically does not change. The positive parties and lacks of the offered algorithm are noted at identification of the form of object.

Discrete and Continuous Models and Applied Computational Science. 2018;26(4):331-342
331-342

### MAPLE program for modelling hydrogen-like atoms in quantum mechanics with non-negative distribution function

Zorin A.V., Tretyakov N.P.

#### Abstract

The program is proposed for a realization of the symbolic algorithm based on the quantum mechanics with non-negative probability distribution function (QDF) and for calculations of energy levels for hydrogen-like atoms. The program is written up in the language MAPLE. In the framework of the algorithm an original Maple package for calculations of necessary functions, such as hydrogen wave functions, Sturmian functions and their Fourier-transforms, Clebsch-Gordan coefficients, etc. is proposed. Operators of observables are calculated on the basis of the QDF quantization rule. According to the Ritz method, eigenvalues of Ritz matrices represent spectral values of the quantity under investigation, i.e. energy. As an example, energy levels of hydrogen-like atoms are calculated and compared with experimental data retrieved from the NIST Atomic Spectra Database Levels Data. It turns out that this theory seems to be equivalent to the traditional quantum mechanics in regard to predictions of experimental values. However, the existence of a phase-space probabilistic quantum theory may be an important advance towards the explanation and interpretation of quantum mechanics.

Discrete and Continuous Models and Applied Computational Science. 2018;26(4):343-356
343-356

## Computer Science

### Session-level control in heterogeneous mobile radio networks with device-to-device connections

Andreev S.D., Samouylov K.E., Tyurlikov A.M.

#### Abstract

In the emerging fifth-generation mobile networks, the challenge of system capacity and user connection quality boosting becomes increasingly important. To this aim, it is possible to apply a novel direct communication technique that is built upon device-to-device (D2D) connectivity. Such heterogeneous interactions allow to offload data flows from a cellular network into the D2D system, which may operate in unlicensed frequencies. However, there emerge several problems with interference coordination and radio resource allocation. This work considers a model of the direct communication system with cellular assistance, which serves user-initiated data flows (sessions), as well as proposes an algorithm to control traffic offloading from a cellular network onto the D2D connections. Analytical and simulation results are offered to investigate this heterogeneous system with D2D communication capabilities.

Discrete and Continuous Models and Applied Computational Science. 2018;26(4):357-370
357-370

### Computational and Simulation Models of the Control System on Modelica

Apreutesey A.Y., Zavozina A.V., Korolkova A.V., Kulyabov D.S.

#### Abstract

When modeling network protocols, the choice of a model approach and a software implementation tool is a problem. The specificity of this subject area is that for the description of protocols usually the discrete-event approach is used. However, the discrete model approach has several disadvantages. It is poorly scalable, not well suited for describing dynamic systems. As an alternative to the discrete approach, a continuous approach is usually considered. But when modeling discrete events, continuous description becomes unnecessarily complicated and heavy. Events take the form of some restrictions on the continuous system, which are often not explicitly included in the continuous model, but have the form of additional semantic descriptions. The authors propose to use a hybrid (continuous-discrete) approach when modeling such systems. In the framework of the hybrid approach, the discrete system is recorded in a continuous form, and the events take the form of discrete transitions inherent in the approach. In addition, if it is based on the description of events, a simulation model can be obtained on the basis of a hybrid approach. This paper demonstrates the use of a hybrid approach to describe systems with control by the example of the interaction of the TCP protocol and the RED algorithm. The simplicity of creating both computational and simulation models of the system is demonstrated. The Modelica language is used as the implementation language.

Discrete and Continuous Models and Applied Computational Science. 2018;26(4):371-382
371-382

### Semantics of Big Data in Corporate Management Systems

Novikova G.M., Azofeifa E.J.

#### Abstract

The modern development of engineering, telecommunications, information and computer technologies allows for collecting, processing and storing huge volumes of data today. Among the first applications of Big Data there was the creation of corporate repositories that use gathered information for analysis and strategic decision-making. However, an unsystematic collection of information leads to the storage and processing of a large amount of non-essential data, while important information falls out of the analysts’ view. An important point is the analysis of the semantics and purpose of data collection, which define both the collection technology and infrastructure and the direction of subsequent processing and use of Big Data with the help of metrics that reduce data volume, leaving only essential information to process. As a first step towards this goal, we present a formalization approach of corporate Big Data using a partially observable Markov decision process (POMDP), and we show that it naturally aligns itself with the corporate governance system.

Discrete and Continuous Models and Applied Computational Science. 2018;26(4):383-392
383-392

## Cosmological Models

### Bianchi Type VIII Cosmological Models Described with Caplygin Gas Equation of State Fluid Sources

Yanishevskiy D.M.

#### Abstract

Within the general theory of relativity the Bianchi type VIII cosmological models with rotation and expansion have been built. It’s known that dark energy can be simulated by different kinds of energy-stress tensor, therefore the sources of gravitation in present article are an anisotropic fluid, with a pressure component satisfying to Chaplygin gas equation of state and a perfect fluid in the first case and an anisotropic fluid, Chaplygin gas and cosmological constant in the second case. It has been proved that the model, when expanding from Plank scale to the modern size gives satisfactory value of the angular velocity value. The found solutions can be used for effects taking place nowadays and at the inflationary stage.

Discrete and Continuous Models and Applied Computational Science. 2018;26(4):393-398
393-398

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