Switched server systems whose parameters are normal numbers in base 4

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.