Boundary Value Problems in Stationary Queueing Systems with Diffusion Intensity of the Input Flow

Abstract

The analysis of characteristics of information networks (IN) is often performed on the basis of the mathematical apparatus of queueing theory, which fairly adequately describes the operation of an entire network as well as its separate components. In the recent years, queueing systems (QS) whose parameters change in the course of time and, possibly, even in a random manner [1] have attracted more attention in connection with the development of new IN. Queueing systems with a doubly stochastic (DS) Poisson flow (PF) of requests whose intensity is a Markovian or semi-Markovian chain with finite set of states [2, 3] have been discussed in detail in literature. The case of QS in which the intensity of the DSPF is a Markovian chain with infinitely many states or a Markovian process with continuous state space has not been studied equally thoroughly. For example, approximate analysis methods of QS with DSPF whose intensity is a jump-like random process were suggested in [4, 5]. The method of generating functions [6] is commonly used in analyzing a QS with an infinite accumulator. The widening application domain and more complex structures of systems and networks for data processing demand that queueing systems in which the input flow is a diffusion process be analyzed. In this paper, we suggest a matrix method for analyzing stationary characteristics of the number of requests addressed to a QS with a finite buffer and diffusion intensity of the input flow.