Vol 28, No 4 (2020)

Queueing systems with different types of renovation mechanism and thresholds as the mathematical models of active queue management mechanism
Viana Carvalho Cravid H., Zaryadov I.S., Milovanova T.A.

This article is devoted to some aspects of using the renovation mechanism (different types of renovation are considered, definitions and brief overview are also given) with one or several thresholds as the mathematical models of active queue management mechanisms. The attention is paid to the queuing systems in which a threshold mechanism with renovation is implemented. This mechanism allows to adjust the number of packets in the system by dropping (resetting) them from the queue depending on the ratio of a certain control parameter with specified thresholds at the moment of the end of service on the device (server) (in contrast to standard RED-like algorithms, when a possible drop of a packet occurs at the time of arrivals of next packets in the system). The models with one, two and three thresholds with different types of renovation are under consideration. It is worth noting that the thresholds determine not only from which place in the buffer the packets are dropped, but also to which the reset of packets occurs. For some of the models certain analytical and numerical results are obtained (the references are given), some of them are only under investigation, so only the mathematical model and current results may be considered. Some results of comparing classic RED algorithm with renovation mechanism are presented.

Discrete and Continuous Models and Applied Computational Science. 2020;28(4):305-318
Stochastic analysis of a single server unreliable queue with balking and general retrial time
Boualem M.

In this investigation, we consider an M/G/1 queue with general retrial times allowing balking and server subject to breakdowns and repairs. In addition, the customer whose service is interrupted can stay at the server waiting for repair or leave and return while the server is being repaired. The server is not allowed to begin service on other customers until the current customer has completed service, even if current customer is temporarily absent. This model has a potential application in various fields, such as in the cognitive radio network and the manufacturing systems, etc. The methodology is strongly based on the general theory of stochastic orders. Particularly, we derive insensitive bounds for the stationary distribution of the embedded Markov chain of the considered system.

Discrete and Continuous Models and Applied Computational Science. 2020;28(4):319-326
On the realization of explicit Runge-Kutta schemes preserving quadratic invariants of dynamical systems
Ying Y., Malykh M.D.

We implement several explicit Runge-Kutta schemes that preserve quadratic invariants of autonomous dynamical systems in Sage. In this paper, we want to present our package ex.sage and the results of our numerical experiments. In the package, the functions rrk_solve, idt_solve and project_1 are constructed for the case when only one given quadratic invariant will be exactly preserved. The function phi_solve_1 allows us to preserve two specified quadratic invariants simultaneously. To solve the equations with respect to parameters determined by the conservation law we use the elimination technique based on Gröbner basis implemented in Sage. An elliptic oscillator is used as a test example of the presented package. This dynamical system has two quadratic invariants. Numerical results of the comparing of standard explicit Runge-Kutta method RK(4,4) with rrk_solve are presented. In addition, for the functions rrk_solve and idt_solve, that preserve only one given invariant, we investigated the change of the second quadratic invariant of the elliptic oscillator. In conclusion, the drawbacks of using these schemes are discussed.

Discrete and Continuous Models and Applied Computational Science. 2020;28(4):327-345
On methods of quantitative analysis of the company’s financial indicators under conditions of high risk of investments
Shchetinin E.Y.

The paper investigates the methods of quantitative analysis of hidden statistical relationships of the financial indicators of companies under conditions of high investment risk. A new semi-parametric method for estimating tail dependence indicators using BB1 and BB7 dependence structures is proposed. For a dataset containing the cost indicators of leading Russian companies, computer experiments were carried out, as a result of which it was shown that the proposed method has a higher stability and accuracy in comparison with other considered methods. Practical application of the proposed risk management method would allow financial companies to assess investment risks adequately in the face of extreme events.

Discrete and Continuous Models and Applied Computational Science. 2020;28(4):346-360
Single-mode propagation of adiabatic guided modes in smoothly irregular integral optical waveguides
Sevastianov A.L.

This paper investigates the waveguide propagation of polarized electromagnetic radiation in a thin-film integral optical waveguide. To describe this propagation, the adiabatic approximation of solutions of Maxwell’s equations is used. The construction of a reduced model for adiabatic waveguide modes that retains all the properties of the corresponding approximate solutions of the Maxwell system of equations was carried out by the author in a previous publication in DCM & ACS, 2020, No 3. In this work, for a special case when the geometry of the waveguide and the electromagnetic field are invariant in the transverse direction. In this case, there are separate nontrivial TEand TM-polarized solutions of this reduced model. The paper describes the parametrically dependent on longitudinal coordinates solutions of problems for eigenvalues and eigenfunctions - adiabatic waveguide TE and TM polarizations. In this work, we present a statement of the problem of finding solutions to the model of adiabatic waveguide modes that describe the stationary propagation of electromagnetic radiation. The paper presents solutions for the single-mode propagation of TE and TM polarized adiabatic waveguide waves.

Discrete and Continuous Models and Applied Computational Science. 2020;28(4):361-377
Solving the inverse problem for determining the optical characteristics of materials
Lovetski K.P., Zhukov A.A., Paukshto M.V., Sevastianov L.A., Tiutiunnik A.A.

The paper describes a methodology for determining the optical and physical properties of anisotropic thin film materials. This approach allows in the future designing multilayer thin-film coatings with specified properties. An inverse problem of determining the permittivity tensor and the thickness of a thin film deposited on a glass substrate is formulated. Preliminary information on the belonging of a thin-film coating to a certain class can significantly reduce the computing time and increase the accuracy of determining the permittivity tensor over the entire investigated range of wavelengths and film thickness at the point of reflection and transmission measurement Depending on the goals, it is possible to formulate and, therefore, solve various inverse problems: o determination of the permittivity tensor and specification of the thickness of a thick (up to 1 cm) substrate, often isotropic; o determination of the permittivity tensor of a thin isotropic or anisotropic film deposited on a substrate with known optical properties. The complexity of solving each of the problems is very different and each problem requires its own specific set of measured input data. The ultimate results of solving the inverse problem are verified by comparing the calculated transmission and reflection with those measured for arbitrary angles of incidence and reflection.

Discrete and Continuous Models and Applied Computational Science. 2020;28(4):378-397
Numerical modeling of laser ablation of materials
Amirkhanov I.V., Sarker N.R., Sarkhadov I.

In this paper, we report a numerical simulation of laser ablation of a material by ultrashort laser pulses. The thermal mechanism of laser ablation is described in terms of a one-dimensional nonstationary heat conduction equation in a coordinate system associated with a moving evaporation front. The laser action is taken into account through the functions of the source in the thermal conductivity equation that determine the coordinate and time dependence of the laser source. For a given dose of irradiation of the sample, the profiles of the sample temperature at different times, the dynamics of the displacement of the sample boundary due to evaporation, the velocity of this boundary, and the temperature of the sample at the moving boundary are obtained. The dependence of the maximum temperature on the sample surface and the thickness of the ablation layer on the radiation dose of the incident laser pulse is obtained. Numerical calculations were performed using the finite difference method. The obtained results agree with the results of other works obtained by their authors.

Discrete and Continuous Models and Applied Computational Science. 2020;28(4):398-405

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