Vol 218, No 2 (2016)

We consider a single server queue with an unreliable server and two Poisson input flows. For this system we find stability conditions and obtain the limit distribution of the number of customers in the system.

We prove pointwise upper estimates of a fundamental solution of a linear evolution equation with fractional Laplacian and with a singular drift term. Such a result has already been obtained in [Maekawa and Miura, Journal of Functional Analysis 264 (2013) 2245–2268] by generalizing the classical ideas of Carlen, Kusuoka, and Stroock. Here, we propose a different and direct method to show such type of estimates, which appears also to be more elementary.

In this paper we introduce some new classes of discrete stable random variables that are useful for understanding a new general notion of stability of random variables called casual stability. Some examples of casual and discrete stable random variables are given. We also propose a class of discrete stable random variables for describing the of rating of a scientific work.

An improved and corrected version of the functional limit theorem is proved establishing weak convergence of random walks generated by compound doubly stochastic Poisson processes (compound Cox processes) to Lévy processes in the Skorokhod space under more realistic moment conditions. As corollaries, theorems are proved on convergence of random walks with jumps having finite variances to Lévy processes with variance-mean mixed normal distributions, in particular, to stable Lévy processes, generalized hyperbolic and generalized variance-gamma Lévy processes.

The paper presents an empirical approach to the analysis of broadband Fourier-spectra of the low-frequency structural plasma turbulence based on a priori assumptions concerning the number of processes and the Gaussian form of the spectral components. This type of turbulence in toroidal plasma devices is described by the mathematical model of non-stationary continuous-time random walk, namely compound doubly stochastic Poisson process or compound Cox process. Measurements of low-frequency (in ranges of frequencies to 5 MHz) long-wave (3..6 cm) fluctuations spectra were obtained at the plasma edge in the L-2M stellarator by the technique of Doppler reflectometry. Under some experimental conditions, the efficiency of the methodology was demonstrated. The spectra were successfully decomposed into a few components. There are more than two harmonics in broadband Fourier-spectra besides a harmonic connected with poloidal rotation of the plasma. Those additional harmonics correspond to fluctuations rotating in the directions of electron and ion drift in toroidal plasma devices. In all modes it was possible to reveal the components corresponding to the poloidal rotation of the plasma (which could be defined by a radial electric field) and phase velocity of the two types of structural turbulence (which could be defined by electron gradient and ion gradient instabilities). The suggested description of probabilistic and spectral characteristics of low-frequency plasma turbulence allow us to formulate the correct problem of characterization of structural turbulence by a system of stochastic differential equations. The equations of the system should involve stochastic processes with densities in the form of mixtures of probability distributions. Such a comprehensive approach assumes a correct comparison of different structural turbulence models (caused by drift dissipative, ion-sound, gradient instabilities, etc.) with characteristics of the obtained stochastic processes.

We obtain central limit type theorems for the total number of edges in the generalized random graphs with random vertex weights under different moment conditions on the distributions of the weights.