Totally asymmetric exclusion processes at constrained m-input n-output junction points

Totally asymmetric exclusion processes at constrained m-input n-output junction points under random update are studied by theoretical calculation and computer simulation in this paper. At the junction points, the hopping rate of particles from m-input parallel lattices to n-output parallel lattices is assumed to be equal to r/n (0 < r < 1). The mean-field approach and Monte Carlo simulations show that the phase diagram can be classified into three regions at any value of r. More interestingly, there is a threshold r_c = n( 1-√1-m/n)/m. In the cases of r > r_c and r < r_c, qualitatively different phases exist in the system. With the increase of the value of m/n, the regions of (LD, LD) and (MC, LD) or (HD, LD) decrease, and the (HD, HD) is the only phase that increases in the region (LD stands for low density, HD stands for high density, and MC for maximal current). Stationary current and density profiles are calculated, showing that they are in good agreement with Monte Carlo simulations.