Rostyslav E. Yamnenko

Upper estimate of overrunning by Sub φ (Ω) random process the level specified by continuous function

In this paper we consider random process from the space Sub φ (Ω), which is defined on compact set, and the probability that supremum of this process exceeds some function. The class of Sub φ (Ω) random processes is more general than the class of Gaussian processes. By applying obtained estimation to a fluid queue fed by a process of Ornstein-Uhlenbeck from the space strictly Sub φ (Ω), where we show that for interval [a, b] there exist constants A, B, D and for large enough buffer capacity x.

On the distribution of storage processes from the class (

Estimates for the distribution of a storage process Q(t) = sup s≤t ( X(t)−X(s)− (f(t)− f(s)) ) are obtained in the paper, where (X(t), t ∈ T ) is a stochastic process belonging to the class V (φ, ψ) and where the service output rate f(t) is a continuous function. In particular, the results hold if (X(t), t ∈ T ) is a Gaussian process. Several examples of applications of the results obtained in the paper are given for sub-Gaussian stationary stochastic processes.

Bounds for the distribution of some functionals of processes with -sub-Gaussian increments

Bounds for the distribution of some functionals of a stochastic process {X(t), t ∈ T} belonging to the class V (φ,ψ) are obtained. An example of the functionals studied in the paper is given by F { sup s≤t;s,t∈B (X(t)−X(s)− (f(t)− f(s))) > x } , where f(t) is a continuous function that can be viewed as a service output rate of a queue formed by the process X(t). For the latter interpretation, the bounds can be viewed as upper estimates for the buffer overflow probabilities with buffer size x > 0. The results obtained in the paper apply to Gaussian stochastic processes. As an example, we show an application for the generalized fractional Brownian motion defined on a finite interval. Introduction This paper continues the line of investigation of queues formed by stochastic processes belonging to the class V (φ, ψ) (see, for example, [7, 8, 9]). The classes V (φ, ψ) are rather general families of stochastic processes containing Gaussian stochastic processes, among others. Therefore, by studying the processes belonging to the class V (φ, ψ), one obtains results for well-known stochastic processes as well as for some of their modern generalizations. Some extremal functionals of the increments of a stochastic process {X(t), t ∈ T} belonging to the class V (φ, ψ) are studied in the paper. In particular, we study the following functionals: sup s≤t;s,t∈B ( X(t)−X(s)− (f(t)− f(s)) ) , inf s≤t;s,t∈B ( X(t)−X(s)− (f(t)− f(s)) ) , sup s≤t;s,t∈B ∣∣X(t)−X(s)− (f(t)− f(s))∣∣, where f(t) is a certain continuous function. The problems concerning the distribution of such functionals appear, for example, in queue theory. Examples of such problems in queue theory are estimating the maximal length of a queue or the buffer overflow probabilities. The function f above is viewed in queue theory as the service intensity of a queue (see, for example, [1, 5]). 2000 Mathematics Subject Classification. Primary 60G07; Secondary 60K25.

Generalized sub-Gaussian fractional Brownian motion queueing model

It is well known that often the one-dimensional distribution of a queue content is not Gaussian but its tails behave like a Gaussian. We propose to consider a general class of processes, namely the class of

On distribution of the norm of deviation of a sub-Gaussian random process in Orlicz spaces

\varphi

An estimate of the probability that the queue length exceeds the maximum for a queue that is a generalized Ornstein–Uhlenbeck stochastic process

φ-sub-Gaussian random processes, which is more general than the Gaussian one and includes non-Gaussian processes. The class of sub-Gaussian random processes contains Gaussian processes also and therefore is of special interest. In this paper we provide an estimate for the queue content distribution of a fluid queue fed by

On Supremum Distribution of Averaged Deviations of Random Orlicz Processes from the Class

N