# Vol 29, No 3 (2021)

**Year:**2021**Articles:**7**URL:**https://journals.rudn.ru/miph/issue/view/1473**DOI:**https://doi.org/10.22363/2658-4670-2021-29-3

## Full Issue

## Articles

### Modeling and design of an re-configurable isolated remote for plasma experiments with hard-real-time synchronization

#### Abstract

The purpose of this paper is to present the design and implementation of a reconfigurable remote control for performing plasma experiments with Hard-Real-Time (HRT) synchronization under jitter less than 1 microsecond. An additional requirement for a multichannel synchronization system is the use of high-speed optical converters to provide galvanic isolation between powerful modules of the setup and remote control in order to exclude any possibility of disruption of the physical experiment control system. Modeling and development of the software part of the maser remote control panel was performed in the LabVIEW application development environment with Real Time and FPGA modules. The hardware part of the control panel is implemented on a real-time controller working in conjunction with the Xilinx FPGA module. To ensure the optical isolation of synchronization signals, boards of electron-optical converters based on LED lasers with fiber-optic terminals were developed and manufactured. The control program is implemented in a two-module architecture with a HOST application and an FPGA application that exchange data over a 1000BASE-T Ethernet network.

**Discrete and Continuous Models and Applied Computational Science**. 2021;29(3):205-220

### Evaluation of the firewall influence on the session initiation by the SIP multimedia protocol

#### Abstract

Firewalls is one of the major components to provide network security. By using firewalls, you can solve such problems as preventing unauthorized access, and deleting, modifying and/or distributing information under protection. The process of information flows filtration by a firewall introduces additional time delays, thus possibly leading to disruption of stable operation of the protected automated system or to inaccessibility of the services provided by the system. Multimedia services are particularly sensitive to service time delays. The main purpose of the work presented in this paper is to evaluate the influence of the firewall on the time delays in data transmission process in the automated system with multimedia data transmission protocols. The evaluation is provided by the queuing theory methods while a session is initiated between two users by the Session Initiation Protocol (SIP) with firewall message filtration. A firewall is a local or functional distributing tool that provides control over the incoming and/or outgoing information in the automated system (AS), and ensures the protection of the AS by filtering the information, i.e., providing analysis of the information by the criteria set and making a decision on its distribution.

**Discrete and Continuous Models and Applied Computational Science**. 2021;29(3):221-229

### Evaluation of firewall performance when ranging a filtration rule set

#### Abstract

This article is a continuation of a number of works devoted to evaluation of probabilistic-temporal characteristics of firewalls when ranging a filtration rule set. This work considers a problem of the decrease in the information flow filtering efficiency. The problem emerged due to the use of a sequential scheme for checking the compliance of packets with the rules, as well as due to heterogeneity and variability of network traffic. The order of rules is non-optimal, and this, in the high-dimensional list, significantly influences the firewall performance and also may cause a considerable time delay and variation in values of packet service time, which is essentially important for the stable functioning of multimedia protocols. One of the ways to prevent decrease in the performance is to range a rule set according to the characteristics of the incoming information flows. In this work, the problems to be solved are: determination and analysis of an average filtering time for the traffic of main transmitting networks; and assessing the effectiveness of ranging the rules. A method for ranging a filtration rule set is proposed, and a queuing system with a complex request service discipline is built. A certain order is used to describe how requests are processed in the system. This order includes the execution of operations with incoming packets and the logical structure of filtration rule set. These are the elements of information flow processing in the firewall. Such level of detailing is not complete, but it is sufficient for creating a model. The QS characteristics are obtained with the help of simulation modelling methods in the Simulink environment of the matrix computing system MATLAB. Based on the analysis of the results obtained, we made conclusions about the possibility of increasing the firewall performance by ranging the filtration rules for those traffic scripts that are close to real ones.

**Discrete and Continuous Models and Applied Computational Science**. 2021;29(3):230-241

### Towards the analysis of the performance measures of heterogeneous networks by means of two-phase queuing systems

#### Abstract

Due to a multistage nature of transmission processes in heterogeneous 4G, 5G mobile networks, multiphase queuing systems become one of the most suitable ways for the resource allocation algorithms analysis and network investigation. In this paper, a few scientific papers that approached heterogeneous networks modelling by means of multiphase queuing systems are reviewed, mentioning the difficulties that arise with this type of analytical analysis. Moreover, several previously investigated models are introduced briefly as an example of two-phase systems of finite capacity and a special structure in discrete time that can be used for analysing resource allocation schemes based on the main performance measures obtained for wireless heterogeneous networks. One of the model presents a two-phase tandem queue with a group arrival flow of requests and a second phase of the complex structure that consists of parallel finite queues. The second model is a two-phase tandem queue with Markov modulated geometric arrival and service processes at the first phase and exhaustive service process at the second phase, which solves a cross-layer adaption problem in a heterogeneous network.

**Discrete and Continuous Models and Applied Computational Science**. 2021;29(3):242-250

### Asymptotically accurate error estimates of exponential convergence for the trapezoidal rule

#### Abstract

In many applied problems, efficient calculation of quadratures with high accuracy is required. The examples are: calculation of special functions of mathematical physics, calculation of Fourier coefficients of a given function, Fourier and Laplace transformations, numerical solution of integral equations, solution of boundary value problems for partial differential equations in integral form, etc. For grid calculation of quadratures, the trapezoidal, the mean and the Simpson methods are usually used. Commonly, the error of these methods depends quadratically on the grid step, and a large number of steps are required to obtain good accuracy. However, there are some cases when the error of the trapezoidal method depends on the step value not quadratically, but exponentially. Such cases are integral of a periodic function over the full period and the integral over the entire real axis of a function that decreases rapidly enough at infinity. If the integrand has poles of the first order on the complex plane, then the Trefethen-Weidemann majorant accuracy estimates are valid for such quadratures. In the present paper, new error estimates of exponentially converging quadratures from periodic functions over the full period are constructed. The integrand function can have an arbitrary number of poles of an integer order on the complex plane. If the grid is sufficiently detailed, i.e., it resolves the profile of the integrand function, then the proposed estimates are not majorant, but asymptotically sharp. Extrapolating, i.e., excluding this error from the numerical quadrature, it is possible to calculate the integrals of these classes with the accuracy of rounding errors already on extremely coarse grids containing only ∼ 10 steps.

**Discrete and Continuous Models and Applied Computational Science**. 2021;29(3):251-259

### Shifted Sobol points and multigrid Monte Carlo simulation

#### Abstract

Multidimensional integrals arise in many problems of physics. For example, moments of the distribution function in the problems of transport of various particles (photons, neutrons, etc.) are 6-dimensional integrals. When calculating the coefficients of electrical conductivity and thermal conductivity, scattering integrals arise, the dimension of which is equal to 12. There are also problems with a significantly large number of variables. The Monte Carlo method is the most effective method for calculating integrals of such a high multiplicity. However, the efficiency of this method strongly depends on the choice of a sequence that simulates a set of random numbers. A large number of pseudo-random number generators are described in the literature. Their quality is checked using a battery of formal tests. However, the simplest visual analysis shows that passing such tests does not guarantee good uniformity of these sequences. The magic Sobol points are the most effective for calculating multidimensional integrals. In this paper, an improvement of these sequences is proposed: the shifted magic Sobol points that provide better uniformity of points distribution in a multidimensional cube. This significantly increases the cubature accuracy. A significant difficulty of the Monte Carlo method is a posteriori confirmation of the actual accuracy. In this paper, we propose a multigrid algorithm that allows one to find the grid value of the integral simultaneously with a statistically reliable accuracy estimate. Previously, such estimates were unknown. Calculations of representative test integrals with a high actual dimension up to 16 are carried out. The multidimensional Weierstrass function, which has no derivative at any point, is chosen as the integrand function. These calculations convincingly show the advantages of the proposed methods.

**Discrete and Continuous Models and Applied Computational Science**. 2021;29(3):260-270

### Richardson-Kalitkin method in abstract description

#### Abstract

An abstract description of the Richardson–Kalitkin method is given for obtaining a posteriori estimates for the proximity of the exact and found approximate solution of initial problems for ordinary differential equations (ODE). The problem ${\rm P}$ is considered, the solution of which results in a real number $u$. To solve this problem, a numerical method is used, that is, the set $H\subset \mathrm{\mathbb{R}}$ and the mapping ${u}_{h}:H\to \mathrm{\mathbb{R}}$ are given, the values of which can be calculated constructively. It is assumed that 0 is a limit point of the set $H$ and ${u}_{h}$ can be expanded in a convergent series in powers of $h:{u}_{h}=u+{c}_{1}{h}^{k}+...$. In this very general situation, the Richardson–Kalitkin method is formulated for obtaining estimates for $u$ and $c$ from two values of ${u}_{h}$. The question of using a larger number of ${u}_{h}$ values to obtain such estimates is considered. Examples are given to illustrate the theory. It is shown that the Richardson–Kalitkin approach can be successfully applied to problems that are solved not only by the finite difference method.

**Discrete and Continuous Models and Applied Computational Science**. 2021;29(3):271-284