  In this article we report on our results about the computation of the
elliptic flow of the quark-gluon-plasma produced in relativistic heavy ion
collisions, simulating the expansion of the fireball by solving the
relativistic Boltzmann equation for the parton distribution function tuned at a
fixed shear viscosity to entropy density ratio $\eta/s$. Our main goal is to
put emphasis on the role of a saturation scale in the initial gluon spectrum,
which makes the initial distribution far from a thermalized one. We find that
the presence of the saturation scale reduces the efficiency in building-up the
elliptic flow, even if the thermalization process is quite fast $\tau_{therm}
\approx 0.8 \,\rm fm/c$ and the pressure isotropization even faster
$\tau_{isotr} \approx 0.3 \,\rm fm/c$. The impact of the non-equilibrium
implied by the saturation scale manifests for non-central collisions and can
modify the estimate of the viscosity respect to the assumption of full
thermalization in $p_T$-space. We find that the estimate of $\eta/s$ is
modified from $\eta/s \approx 2/4\pi$ to $\eta/s \approx 1/4\pi$ at RHIC and
from $\eta/s \approx 3/4\pi$ to $\eta/s \approx 2/4\pi$ at LHC. We complete our
investigation by a study of the thermalization and isotropization times of the
fireball for different initial conditions and values of $\eta/s$ showing how
the latter affects both isotropization and thermalization. Lastly, we have seen
that the range of values explored by the phase-space distribution function $f$
is such that at $p_T<0.5\, \rm GeV$ the inner part of the fireball stays with
occupation number significantly larger than unity despite the fast longitudinal
expansion, which might suggest the possibility of the formation of a transient
Bose-Einstein Condensate.
