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 Paulis 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.
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