Switched server systems whose parameters are normal numbers in base 4

TL;DR

This paper demonstrates that certain switched server systems with parameters based on normal or rational numbers in base 4 have finite global attractors, contrasting with previously known fractal attractors, by analyzing their topological dynamics.

Contribution

It proves that if parameters are derived from normal or rational numbers in base 4, the switched server system's Poincaré map has a finite attractor, extending understanding of system behavior.

Findings

Systems with parameters from normal or rational base-4 numbers have finite attractors.

The approach involves studying topological dynamics of piecewise affine contractions.

The result contrasts with earlier findings of fractal attractors in similar systems.

Abstract

Switched server systems are mathematical models of manufacturing, traffic and queueing systems. Recently, it was proved in (Eur. J. Appl. Math. 31(4) (2020), pp. 682-708) that there exist switched server systems with $3$ buffers (tanks), a server, filling rates $\rho_1=\rho_2=\rho_3=\frac13$ and parameters $d_1, d_2, d_3>0$ whose global attractor is a fractal set. In this article, we prove that if $x_1$ in $(0,\frac13)$, $x_2$ in $(\frac13,\frac23)$ and $x_3$ in $(\frac23,1)$ are rational numbers or normal numbers in base $4$ (or more generally, rich numbers to base $4$) and $(d_1,d_2,d_3)$ is the vector with positive entries satisfying $$d_1=\frac{1}{3x_1}-1,\quad d_2=\frac{2-3x_2}{3x_2-1}, \quad d_3=\frac{3-3x_3}{3x_3-2},$$ then the corresponding switched server has no fractal attractor. More precisely, the Poincar\'e map of the system has a finite global attractor. The approach we use is to study the topological dynamics of a family of piecewise $\lambda$-affine contractions that includes the Poincar\'e map of the switched server system as a particular case.