Stochastic Theory of a Data-Handling System with Multiple Sources

TLDR

This model is useful for a data‑handling switch in a computer network. We model a buffer receiving messages from multiple independent, identically distributed sources that alternate between exponentially distributed on/off periods, with uniform transmission during on periods and a capped output rate, and derive the equilibrium buffer distribution via differential equations. The analysis yields closed‑form expressions for the buffer distribution, its moments, all eigenvalues, and asymptotic behavior, eliminating the need for matrix inversion, and provides numerically useful results illustrated graphically.

Abstract

In this paper we consider a physical model in which a buffer receives messages from a finite number of statistically independent and identical information sources that asynchronously alternate between exponentially distributed periods in the ‘on’ and ‘off’ states. While on, a source transmits at a uniform rate. The buffer depletes through an output channel with a given maximum rate of transmission. This model is useful for a data-handling switch in a computer network. The equilibrium buffer distribution is described by a set of differential equations, which are analyzed herein. The mathematical results render trivial the computation of the distribution and its moments and thus also the waiting time moments. The main result explicitly gives all the system's eigenvalues. While the insertion of boundary conditions requires the solution of a matrix equation, even this step is eliminated since the matrix inverse is given in closed form. Finally, the simple expression given here for the asymptotic behavior of buffer content is insightful, for purposes of design, and numerically useful. Numerical results for a broad range of system parameters are presented graphically.