Cellular automaton simulation of pedestrian counter flow with different walk velocities

This paper presents a cellular automaton model without step back for pedestrian dynamics considering the human behaviors which can make judgments in some complex situations. This model can simulate pedestrian movement with different walk velocities through update at different time-step intervals. Two kinds of boundary conditions including periodic and open boundary for pedestrian counter flow are considered, and their dynamical characteristics are discussed. Simulation results show that for periodic boundary condition there are three phases of pedestrian patterns, i.e., freely moving phase, lane formation phase, and perfectly stopped phase at some certain total density ranges. In the stage of lane formation, the phenomenon that pedestrians exceed those with lower walk velocity through a narrow walkway can be found. For open boundary condition, at some certain entrance densities, there are two steady states of pedestrian patterns; but the first is metastable. Spontaneous fluctuations can break the first steady state, i.e., freely moving phase, and run into the second steady state, i.e., perfectly stopped phase.

Figure 1

Von Neumann neighbor setup in a two-dimensional rectangular grid.

Figure 2

(a)–(h) All the possible configurations of the right walker with walk velocity of $1.0\mathrm{m}∕\mathrm{s}$ (going to the right). The arrow in the right adjacent cell indicates the moving direction of the walker in that cell. The arrow from the core cell indicates the possible direction of the right walker in the core cell may select. The nomenclature of “$P$” is the transition probabilities to the nearest-neighbor sites, and the subscripts of “$a$,” “$b$,” and “$w$” means the above, bottom, and waiting labels, respectively.

Figure 3

(a)–(c) Adding possible configurations of the right walker with walk velocity of $1.5\mathrm{m}∕\mathrm{s}$ to those in Fig. 2.

Figure 4

(Color online) Schematic illustration of the pedestrian counter flow in the system, which is composed of top and bottom walls. The white and black full circles indicate the right walkers (going to the right) with walk velocities of 1.0 and $1.5\mathrm{m}∕\mathrm{s}$, respectively. The red and green full circles are the left walkers with walk velocities of 1.0 and $1.5\mathrm{m}∕\mathrm{s}$, respectively.

Figure 5

Plot of the mean velocity $⟨V⟩$ and the mean flow $⟨J⟩$ against the total density $p_p$ for the system size $W=60$ in the pedestrian counter flow with periodic boundary condition.

Figure 6

(Color online) Pedestrian patterns obtained at time $\mathrm{step}=20000$ for the system size $W=60$, (a) the freely moving phase obtained at $p_p=0.04$; (b) the lane formation phase obtained at $p_p=0.15$; (c) the perfectly stopped phase obtained at $p_p=0.40$.

Figure 7

Plot of the mean velocity $⟨V⟩$ and the mean occupancy $⟨\rho ⟩$ against the entrance density $p_o$ for the system size $W=60$ in the pedestrian counter flow with open boundary condition.

Figure 8

(Color online) Time evolution of (a) the velocity $V\left(t\right)$ and (b) the occupancy $p\left(t\right)$ for the system size $W=60$ in the pedestrian counter flow with open boundary condition at the entrance density $p_o=0.1$, 0.2, and 0.5.

Figure 9

(Color online) Time evolution of pedestrian patterns obtained at (a) time $\mathrm{step}=900$, (b) 1800, and (c) 2700 for the system size $W=60$ in the pedestrian counter flow with open boundary condition at the entrance density $p_o=0.2$.