The problem of averaging of the relativistic motion equations of electron in the intense laser radiation, caused by the decreasing of the rate of wave phase change due to the Doppler’s effect, is considered. As a result the phase can go from the “fast” to “slow” variables of the motion, so averaging over the phase becomes impossible. An analysis is presented of the conditions which are necessary for averaging of the relativistic equations of motion over the “fast” phase of the intense laser radiation on the base of the general principles of the averaging method. Laser radiation is considered in the paraxial approximation, where the ratio of the laser beam waist to the Rayleigh length is accepted as a small parameter. It is supposed that the laser pulse duration is of the order if the laser beam waist. In this case first-order corrections to the vectors of the laser pulse field should be taken into account. The general criterion for the possibility of the averaging of the relativistic motion equations of electron in the intense laser radiation is obtained. It is shown that an averaged description of the relativistic motion of an electron is possible in the case of a fairly moderate (relativistic) intensity and relatively wide laser beams. The known in the literature analogical criterion has been obtained earlier on the base of the numerical results.

The asymptotic method is a very attractive area of applied mathematics. There are many modern research directions which use a small parameter such as statistical mechanics, chemical reaction theory and so on. The application of the Fokker-Planck equation (FPE) with a small parameter is the most popular because this equation is the parabolic partial differential equations and the solutions of FPE give the probability density function. In this paper we investigate the singularly perturbed Cauchy problem for symmetric linear system of parabolic partial differential equations with a small parameter. We assume that this system is the Tikhonov non-homogeneous system with constant coefficients. The paper aims to consider this Cauchy problem, apply the asymptotic method and construct expansions of the solutions in the form of two-type decomposition. This decomposition has regular and border-layer parts. The main result of this paper is a justification of an asymptotic expansion for the solutions of this Cauchy problem. Our method can be applied in a wide variety of cases for singularly perturbed Cauchy problems of Fokker-Planck equations.

The concept of cloud computing was created to better preserve user privacy and data storage security. However, the resources allocated for processing this data must be optimally allocated. The problem of optimal resource management in the loud computing environment is described in many scientific publications. To solve the problems of optimality of the distribution of resources of systems, you can use the construction and analysis of QS. We conduct an analysis of two-buffer queuing system with cross-type service and additional penalties, based on the literature reviewed in the article. This allows us to assess how suitable the model presented in the article is for application to cloud computing. For a given system different options for selecting applications from queues are possible, queue numbers, therefore, the intensities of transitions between the states of the system will change. For this, the system has a choice policy that allows the system to decide how to behave depending on its state. There are four components of such selection management models, which is a stationary policy for selecting a queue number to service a ticket on a vacated virtual machine each time immediately before service ends. A simulation model was built for numerical analysis. The results obtained indicate that requests are practically not delayed in the queue of the presented QS, and therefore the policy for a given model can be considered optimal. Although Poisson flow is the simplest for simulation, it is quite acceptable for performance evaluation. In the future, it is planned to conduct several more experiments for different values of the intensity of requests and various types of incoming flows.