  We present a simplified description for spin-dependent electronic transport
in honeycomb-lattice structures with spin-orbit interactions, using
generalizations of the stochastic non-equilibrium model known as the totally
asymmetric simple exclusion process. Mean field theory and numerical
simulations are used to study currents, density profiles and current
polarization in quasi- one dimensional systems with open boundaries, and
externally-imposed particle injection ($\alpha$) and ejection ($\beta$) rates.
We investigate the influence of allowing for double site occupancy, according
to Pauli's exclusion principle, on the behavior of the quantities of interest.
We find that double occupancy shows strong signatures for specific combinations
of rates, namely high $\alpha$ and low $\beta$, but otherwise its effects are
quantitatively suppressed. Comments are made on the possible relevance of the
present results to experiments on suitably doped graphenelike structures.
