# Vol 25, No 1 (2017)

**Year:**2017**Articles:**9**URL:**https://journals.rudn.ru/miph/issue/view/917**DOI:**https://doi.org/10.22363/2312-9735-2017-25-1

## Full Issue

## Mathematical Teletraffic Theory and Telecommunication Networks

### The Analysis of Queuing System with General Service Distribution and Renovation

#### Abstract

We investigate the queueing system in which the losses of incoming orders due to the introduction of a special renovation mechanism are possible. The introduced queueing system consists of server with a general distribution of service time and a buffer of unlimited capacity. The incoming flow of tasks is a Poisson one. The renovation mechanism is that at the end of its service the task on the server may with some probability empty the buffer and leave the system, or with an additional probability may just leave the system. In order to study the characteristics of the system the Markov chain embedded upon the end of service times is introduced. Under the assumption of the existence of a stationary regime for the embedded Markov chain the formula for the probability generation function is obtained. With the help of the probability generation function such system characteristics as the probability of the system being empty, the average number of customers in the system, the probability of a task not to be dropped, the distribution of the service waiting time for non-dropped tasks, the average service waiting time for non-dropped requests are derived.

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(1):3-8

3-8

### Performance and Modeling of Session Setup Procedure Using Random Access Channel

#### Abstract

We observe nowadays increasing the number of devices using Internet for data transmission in LTE network. Researchers offer their algorithms and methods for base station load reduction. Our approach allows minimizing network resources using Random Access Channel for session setup procedure. We obtain all performance measures for this procedure between user equipment and LTE base station. Procedure includes four messages: preamble transmission on the physical random access channel (Msg1), random access response of existing RACH procedure (Msg2), connection request, a.k.a. Hybrid automatic repeat request on the physical uplink shared channel (Msg3) and connection resolution (Msg4). This approach is used for small data transmission. If two or more devices select the same preamble, then collision occurs, and it is necessary to retransmit another preamble. Similarly, Msg3 may be blocked due to lack of resources. Our analytical model includes the dependence on collision probability. This solution has been proposed by 3GPP technical report 37.868. We obtain such performance measures as collision probability, dependent upon the number of attempts and the overall number of attempts in the period; access success probability; statistics of number of preamble transmissions; statistics of access delay. Numerical experiment demonstrate the accuracy of the proposed analytical model in the form of Markov chains.

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(1):9-18

9-18

## Mathematical Modeling

### Propagation of Nonlinear Waves in Coaxial Physically Nonlinear Cylindrical Shells Filled with a Viscous Fluid

#### Abstract

Investigation of deformation waves behavior in elastic shells is one of the important trends in contemporary wave dynamics. There exist mathematical models of wave motions in infinitely long geometrically non-linear shells, containing viscous incompressible liquid, based on the related hydroelasticity problems, which are derived by the shells dynamics and viscous incompressible liquid equations in the form of generalized KdV equations. Also, mathematical models of the wave process in infinitely long geometrically non-linear coaxial cylindrical elastic shells are obtained by means of disturbances method. These models differ from the known ones by the consideration of incompressible liquid presence between the shells, based on the related hydroelasticity problems. These problems are described by shells dynamics and viscous incompressible liquid equations with corresponding edge conditions in the form of generalized KdV equations system. The paper presents the investigation of wave occurrences of two geometrically non-linear elastic coaxial cylindrical shells model of Kirchhoff-Love type, containing viscous incompressible liquid between them, as well as inside. The difference schemes of Crank-Nicholson type are obtained for the considered equations system by taking into account liquid impact and with the help of Gro¨bner bases construction. To generate these difference schemes, the basic integral difference correlations, approximating initial equations system, were used. The usage of Gro¨bner bases technology provides generating the schemes, for which it becomes possible to obtain discrete analogs of the laws of preserving initial equations system. To do this, equivalent transformations were made. On the basis of computation algorithm the complex of programs, permitting to construct graphs and obtain numerical solutions under exact solutions of coaxial shell dynamics equations system, was made.

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(1):19-35

19-35

### Algorithms for Solving the Parametric Self-Adjoint 2D Elliptic Boundary-Value Problem Using High-Accuracy Finite Element Method

#### Abstract

We consider the calculation schemes for solving elliptic boundary-value problems (BVPs) within the framework of the Kantorovich method that provides the reduction of an elliptic BVP to a system of coupled second-order ordinary differential equations (ODEs). The surface basis functions of the expansion depend on the independent variable of the ODEs parametrically. Here we use the basis functions calculated by means of the finite element method(FEM), as well as the probe parametric surface basis functions calculated in the analytical form. We propose new calculation schemes and algorithms for solving the parametric self-adjoint elliptic boundary-value problem (BVP) in a 2D finite domain, using high-accuracy finite element method (FEM) with rectangular and triangular elements. The algorithm and the programs calculate with the given accuracy the eigenvalues, the surface eigenfunctions and their first derivatives with respect to the parameter of the BVP for parametric self-adjoint elliptic differential equation with the Dirichlet and/or Neumann type boundary conditions on the 2D finite domain, and the potential matrix elements, expressed as integrals of the products of surface eigenfunctions and/or their first derivatives with respect to the parameter. The parametric eigenvalues (potential curves) and the potential matrix elements computed by the program can be used for solving bound-state and multi-channel scattering problems for systems of coupled second-order ODEs by means of the Kantorovich method. We demonstrate the efficiency of the proposed calculation schemes and algorithms in benchmark calculations of 2D elliptic BVPs describing quadrupole vibrations of a collective nuclear model.

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(1):36-55

36-55

### Simulation of Polarized Light Propagation in the Thin-Film Waveguide Lens

#### Abstract

The paper deals with the problem of electromagnetic TE-polarized monochromatic light diffraction on three-dimensional thickening of the waveguide layer of regular three-layered open planar dielectric waveguide, which forms thin-film waveguide lens. The authors propose an approximate mathematical model in which open waveguide is placed inside the auxiliary closed waveguide, that leads to well-posed diffraction problem. It is shown, that properties of guided modes of the open waveguide are stable with respect to shifts of the closed waveguide boundaries. So, the proposed approach describes the propagation of polarized light in the open smoothly irregular waveguide adequately. The three-dimensional thickening of the waveguide layer forces us to deal with electromagnetic field in vector form due to depolarization effect. The diffraction problem, presented in the work, is solved in adiabatic approximation by the small parameter of irregularity of the waveguide layer. The numerical experiments show that decreasing of the small parameter tends the reflection coefficient matrix to zero-matrix, tends the transmittance coefficient matrix to identity matrix, and besides the non-diagonal matrix elements, corresponding to modes interaction, tend to zero by an order faster than diagonal matrix elements, which shows that depolarization effects in the given configuration can be neglected.

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(1):56-68

56-68

### The Wave Model of the Propagation and Changes of Linguistic Information in the Indo-European Model Language Community

#### Abstract

Paper considers the wave mathematical model describing the distribution and change of linguistic information in Indo-European model linguistic community. Brief information about the Anatolian and Kurgan hypothesis of formation of Proto-Indo-Europeans (PIE) is given. The mathematical model of wave propagation and changing of information is described by system of integral-differential equations. Results of the preliminary theoretical analysis and computer modeling are given. In particular, an estimate of the maximum possible time (24000 years) of development/emergence of considered Indo-European “family” is obtained in the framework of this model. Results of computer modeling show, that from two main hypotheses of formation of the PIE - Anatolian and Kurgan, the latter better matches obtained by us temporary estimates. Finally, data on a hypothetical PIE alphabet - namely, the number of possible “characters/symbols” (or “letters”) alphabet of PIE on the basis of our data were obtained. Based on the results, it was found that the range of possible values for alphabetic characters is limited to about values: 3 32. In addition, there is a good correlation of our data with the findings of geneticists on a possible time of occurrence (14000-20000 years ago) of the haplogroup Y-DNA R1a (it is believed that PIE language speakers were primarily men with this haplogroup).

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(1):69-80

69-80

### Geometrization of Maxwell's Equations in the Construction of Optical Devices

#### Abstract

The development of physics in the XX-th century was closely linked to the development of the mathematical apparatus. The General Relativity demonstrated the power of the geometric approach. Unfortunately, the infiltration of this apparatus in other domains of physics is rather slow. For example, there were some attempts of integration of the geometric methods in electrodynamics, but until recently they remained only as a theoretical exercise. Interest to the geometric methods in electrodynamics is summoned by practical necessity. The following algorithm of designing of the electromagnetic device is possible. We construct the estimated trajectories of propagation of electromagnetic waves. Then we calculate the parameters of the medium along these trajectories. The inverse problem is also interesting. The paper considers the techniques of construction of optical devices based on the method of geometrization of Maxwell’s equations. The method is based on representation of material equations in the form of an effective space-time geometry. Thus we get a problem similar to that of some bimetric theory of gravity. That allows to use a well-developed apparatus of differential geometry. On this basis, we can examine the propagation of the electromagnetic field on the given parameters of the medium. It is also possible to find the parameters of the medium by a given law of propagation of electromagnetic fields.

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(1):81-90

81-90

## Articles

### Information about the authors

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(1):91-92

91-92

### Guidelines for Authors

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(1):93-94

93-94