  We investigate and map out the non-equilibrium phase diagram of a
generalization of the well known Aubry-Andr\'e-Harper (AAH) model. This
generalized AAH (GAAH) model is known to have a single-particle mobility edge
which also has an additional self-dual property akin to that of the critical
point of AAH model. By calculating the imbalance, we get hints of a rich phase
diagram. We find a fascinating connection between single particle wavefunctions
near the mobility edge of GAAH model and the wavefunctions of the critical AAH
model. By placing this model far-from-equilibrium with the aid of two baths, we
investigate the open system transport via system size scaling of
non-equilibrium steady state (NESS) current. Current is calculated by fully
exact non-equilibrium Green's function (NEGF) formalism. The critical point of
the AAH model now generalizes to a 'critical' line separating regions of
ballistic and localized transport. Like critical point of AAH model, current
scales sub-diffusively with system size on the 'critical' line ($I\sim
N^{-2\pm0.1}$). However, remarkably, the scaling exponent on this line is
distinctly different from that obtained for the critical AAH model (where
$I\sim N^{-1.4\pm0.05}$). All these results can be understood from the
above-mentioned connection between states near mobility edge of GAAH model and
those of critical AAH model. A very interesting high temperature
non-equilibrium phase diagram of the GAAH model emerges from our calculations.
