  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.
