Vol 25, No 1 (2017)

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.

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.

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.

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.