Abstract: Non-equilibrium systems have been observed at almost all levels of natural and manmade systems, which are characterized by non-zero continuous traffic currents. The totally asymmetric simple exclusion process (TASEP) has been regarded as a paradigm for modeling and simulation of such non-equilibrium dynamic systems. These TASEP systems are simulated with Monte Carlo methods and analyzed using mean field approaches in this paper. Some interesting physical phenomena such as spontaneous symmetry breaking, finite-size effect, and flipping process, were observed via Monte Carlo simulations, and are discussed in detail. More careful Monte Carlo simulations are suggested to confirm the finite-size effect argued in the literature.