# Vol 25, No 3 (2017)

**Year:**2017**Articles:**10**URL:**https://journals.rudn.ru/miph/issue/view/970**DOI:**https://doi.org/10.22363/2312-9735-2017-25-3

## Full Issue

## Mathematical Teletraffic Theory and Telecommunication Networks

### About Probability Characteristics Evaluation in Queuing System with Limited Resources and Random Requirements

#### Abstract

Mobile data traffic increases on everyday basis for the last decade and is going to keep this trend in the near future. Exponential growth of data traffic in wireless networks accelerates the development of new technologies and the transition to 5G networks. The ongoing improvements will allow to increase the channels throughput and will allow devices to simultaneously support both cellular and Wi-Fi networks, and even allow direct device-to-device (D2D) connection without any base station involved. Evolving heterogeneous networks promise more efficient radio resources usage by a macro cell traffic offloading to the small cells and uplink and downlink decoupling (DUDe). A resource-sharing model in heterogeneous networks is for the first time proposed to be analyzed in terms of queuing system with random requirements. We suggest a multiserver queuing network with limited resources where each class of customers requires a random vector of resources to be served. It has been proved that stationary probabilities of the system with aggregated flow of customers with mean-weighted requirement are equal to the stationary probabilities of the suggested system. The analytical method for the key probability characteristics evaluation requires calculating all k-fold convolutions for each set of vectors requirements. We propose a recurrent computation algorithm for normalization constant evaluation and efficient formulas for blocking probability, mean volume and variance of the occupied resources.

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(3):209-216

## Mathematical Modeling

### Application of the Harmonic Linearization Method to the Study a Control Systems with a Self-Oscillatory Regime

#### Abstract

In data transmission networks implemented as systems with control, the phenomenon of global synchronization can occur. Outwardly, this phenomenon manifests itself as a self-oscillating mode in the system. This mode negatively affects the characteristics of the entire system, such as throughput and transmission delays. Relevant is the problem of finding the areas of occurrence of self-oscillation. The authors investigated this problem for the system as a whole. Also, the problem of isolating the elements of the system responsible for the appearance of an autooscillatory regime is urgent. The complexity of this problem is caused by the essentially nonlinear character of the system and its elements. Often, the linearization method is used for the decomposition of the system. But with the linearization, the self-oscillatory regime disappears. There is a need to find a method of decomposition, non-destructive self-oscillating mode of the system. As such a method, the authors suggest using the method of harmonic linearization. This method is used in the control theory. However, we must admit that this mathematical apparatus is little known to researchers specializing in the study of networks. The authors tried to describe in as much detail the process of research using the method of harmonic linearization. The method is used to study the influence of the form of RED-type function on the occurrence of self-oscillation mode. Thus, this material is more methodical than exploratory one.

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(3):234-252

### High-Accuracy Finite Element Method for Solving Boundary-Value Problems for Elliptic Partial Differential Equations

#### Abstract

A new computational scheme of the finite element method of a high order of accuracy for solving boundary value problems for an elliptic partial differential equation that preserves the continuity of the derivatives of the approximate solution in a bounded domain of a multidimensional Euclidean space is proposed. A piecewise continuous basis of the finite element method is generated using interpolation Hermite polynomials of several variables and ensures the continuity of not only the approximate solution but also its derivatives up to a given order on the boundaries of finite elements, depending on the smoothness of the variable coefficients of the equation and the boundary of the domain. The efficiency and accuracy order of the computational scheme, algorithm and program are demonstrated by the example of an exactly solvable boundary-value problem for a triangular membrane depending on the number of finite elements of the partition of the domain and the dimension of the eigenvector of the algebraic problem. It was shown that, in order to achieve a given accuracy of the approximate solution, for schemes of the finite element method with Hermite interpolation polynomials the dimension of the eigenvector is approximately two times smaller than for schemes with Lagrange interpolation polynomials that preserve on the boundaries of finite elements only the continuity of the approximate solution. The high-accuracy computational scheme of the finite element method is oriented to calculations of the spectral and optical characteristics of quantum-mechanical systems.

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(3):217-233

### The Boundary Value Problem for Elliptic Equation in the Corner Domain in the Numerical Simulation of Magnetic Systems

#### Abstract

Modern accelerator systems and detectors contain magnetic systems of complex geometrical configuration. Design and optimization of the magnetic systems demands solving a nonlinear boundary-value problem of magnetostatic. The region in which the boundary-value problem is solved, consists of two sub-domains: a domain of vacuum and a domain of ferromagnetic. In view of the complex geometrical configuration of magnetic systems, the ferromagnetic/vacuum boundary can be nonsmooth, i.e. it contains a corner point near of which the boundary is formed by two smooth curves crossed in a corner point at some angle. Thereby, the solution of such a problem has to be found by numerical methods, a question arises about the behavior of the boundary value problem solution around the angular point of the ferromagnetic. This work shows that if the magnetic permeability function meets certain requirements, the corresponding solution of the boundary value problem will have a limited gradient. In this paper, an upper estimate of maximum possible growth of the magnetic field in the corner domain is given. In terms of this estimate, a method of condensing the differential mesh near the corner domain is proposed. This work represents an algorithm of constructing an adaptive mesh in the domain with a boundary corner point of ferromagnetic taking into account the character of behavior of the solution of the boundary value problem. An example of calculating a model problem in the domain containing a corner point is given.

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(3):253-265

## Physics

### Magnetic Excitations of Graphene in 8-Spinor Realization of Chiral Model

#### Abstract

The simplest scalar chiral model of graphene suggested earlier and based on the SU(2) order parameter is generalized by including 8-spinor field as an additional order parameter for the description of spin (magnetic) excitations in graphene. As an illustration we study the interaction of the graphene layer with the external magnetic field. In the case of the magnetic field parallel to the graphene plane the diamagnetic effect is predicted, that is the weakening of the magnetic intensity in the volume of the material. However, for the case of the magnetic field orthogonal to the graphene plane the strengthening of the magnetic intensity is revealed in the central domain (at small r). Thus, the magnetic properties of the graphene prove to be strongly anisotropic.

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(3):266-275

### Pressure Operator for the Pöeschl-Teller Oscillator

#### Abstract

The quantum-mechanical properties of the strongly non-linear quantum oscillator in the Pöeschl-Teller model are considered. In the first place, the energy spectrum and its dependence upon the confinement parameter (i.e., the width of the “box”) are studied. Moreover, on the grounds of the Hellman-Feynman theorem the pressure operator in this model is obtained and (along with the energy spectrum) is studied in two main approximations: the “particle in the box” and “linear (harmonic) oscillator” for large and low values of the main quantum number; the critical value is also evaluated. Semi-classical approximation as well as perturbation theory for the Pöeschl-Teller are also considered. The results obtained here are intended for future thermodynamic calculations: first of all, for the generalization of the well-known Bloch result for the linear harmonic oscillator in the thermostat. To this end, the density matrix for the Pöeschl-Teller oscillator will be calculated and the full Carnot cycle conducted.

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(3):276-282

### On the Evolution of Converging Wave Packet of an Inverted Quantum Oscillator Driven by Homogeneous Harmonic Field

#### Abstract

The problem investigated refers to periodically driven 1D quantum inverted harmonic oscillator (IHO) with the Hamiltonian of . The model is used widely in huge quantum applications concerned unstable molecular complexes and ions under laser light affection. Non-stationary Schrödinger equation (NSE) was solved analytically and numerically by means of Maple 17 with initial wave function (w.f.) of generalized Gaussian type. This one described the converging 1D probability flux and fitted well the quantum operator of initial conditions (IC). For the IC one can observe, first, the collapse of w.f. packet into extremely narrow 1D space interval of length and, second, its spreading back up to its starting half width, and all that - at dimensionless times. At certain phases j defined by W and s0 the wave packet center displayed nonharmonic oscillating behavior near some slowly drifting space position within this time interval and after that leaved onto infinity while the unlimited packet spreading. And the phases themselves served as bifurcation points separating the NSE solutions with the outgoing to from those with. In “resonant” case of the values obeyed an inverted Fermi-Dirac formula of; for differing the asymptotic of obeyed well classical law.

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(3):283-294

## Computer Science

### Modern Technologies of Information Integration from Independent Sources and their Application in the Construction of an Information System that Combines Transport Timetables

#### Abstract

The problem of constructing an integrating system combining bus timetables of different bus depots is considered. We suppose that these bus depots are independent and, possibly, located at different regions of the country. The purpose of this integrated system is, in particular, the easement of finding available bus routes between two given points. To solve this problem, it is proposed to use an approach combining the advantages of mediator and data repository technologies. The article considers a model of three data sources, which are bus depots that display information about their timetables. It is assumed that all sources have similar conceptual schemes, but they have their own specific characteristics. In particular, there may be different names of tables and attributes in different sources and different distribution of attributes throw tables. Also at some sources may be an absence of certain attributes. We construct correspondence tables and a mediator that translate user queries to the sources. To identify the necessary sources, a small auxiliary repository is maintained that contains information about the stop points served by each of the sources. We describe the technology for updating the repository and the executing strategy for the user query, using the information contained in the repository.

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(3):295-305

## Articles

### Information about the authors

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(3):306-307

### Guidelines for Authors

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(3):308-309