Traffic flow in a crowd of pedestrians walking at different speeds

This study investigates motion in a crowd of pedestrians walking at different speeds. Three pedestrian groups are considered (slow walkers, normal walkers, and fast walkers), and we design the experimental condition by mixing the normal walkers with either the slow or the fast walkers to create flows with different speed compositions. All the walkers in this experiment were instructed to walk along a circular course unidirectionally. Fundamental diagrams and multiple regression analysis show that the speed at which a particular pedestrian walks is determined by both the local density and the speed at which the surrounding pedestrians are walking. We also find that the spontaneous lane formation, that occurs in bidirectional flow, does not occur in flow in which the speed is heterogeneous, thereby resulting in a spatial density distribution with large variance. This corresponds to pedestrian clustering, which reduces both the mean speed and the flow rate.

Figure 1

(a) Snapshot of the experiment. (b) Experimental field.

Figure 2

Fundamental diagrams of normal pedestrians in different compositions: the blue circle represents normal walkers walking with other normal walkers (homogeneous runs), and the red triangle and green diamond represent normal walkers walking with fast and slow walkers, respectively (heterogeneous runs). (a) Flow rate versus density; (b) speed versus density; (c) average speed versus density: the shaded area represents one-sigma error. Inset: the red dotted line with triangles is the result of the $t$ test between fast-mix and homogeneous flows and the green dotted line with diamonds is that between slow-mix and homogeneous trials.

Figure 3

(a) Region of interest used to define the surrounding speed of a given pedestrian; $\theta$ is the central angle. (b) Correlation coefficient between local density and surrounding speed versus the local density for different values of central angle as indicated in the legend. (c) Relationship between surrounding speed and walking speed in different density regions: low- ($\rho \le 1.0$), middle- ($1.0<\rho \le 2.5$), and high- ($2.5<\rho$) density regions.

Figure 4

(a) Estimated mean speed compared with actual mean speed. The solid lines represent the measured mean speeds of the fast-mix and slow-mix runs, and the dotted lines represent the estimated speeds of these flows, each of which is calculated as the simple mean of the two homogeneous flows. The shaded areas represent one-sigma errors for each speed. (b) The effect size $d$ of the $t$ test to examine whether measured speed and estimated speed are significantly different.

Figure 5

Speed distribution of different runs: ${\rho }_g=0.9\text{and}1.5{\mathrm{m}}^{-2}$ for both (a) slow-mix condition and (b) fast-mix condition. The horizontal axis represents walking speed and the vertical axis represents frequency (namely, appearance probability). The error bar represents the chronological fluctuation in frequency during one experimental run. Blue dotted lines represent experiment under ${\rho }_g=0.9$, while red solid lines represent ${\rho }_g=1.5$.

Figure 6

Time-space diagrams for low-density (${\rho }_g=0.7{\mathrm{m}}^{-2}$) conditions. Red dotted lines represent the trajectories of normal walkers while the green and blue solid lines represent slow and fast walkers, respectively: (a) normal homogeneous runs; (b) heterogeneous slow-mix runs: red dotted lines for normal walkers and green for slow walkers; (c) heterogeneous fast-mix runs: red dotted for normal walkers and blue for fast walkers.

Figure 7

Time-space diagrams for high-density (${\rho }_g=1.5{\mathrm{m}}^{-2}$) conditions: (a) normal homogeneous runs; (b) heterogeneous slow-mix runs: red dotted lines for normal walkers and green for slow walkers; (c) heterogeneous fast-mix runs: red dotted for normal walkers and blue for fast walkers.

Figure 8

Numbers of passing. For both upper and lower figures, the blue squares represent the numbers of bypassing observed in slow-mix while the red circle represent those of fast-mix conditions.

Figure 9

Variance in density distribution. The density variance at each time is averaged through one whole run.

Figure 10

Congestion levels for different crowd types depending on walking-speed composition.

Figure 11

Crowd danger for different types of crowds depending on walking-speed composition. Each set of points is fitted using Eq. (9).

Figure 12

Variance in density distribution. The density variance at each time is averaged through one whole run.

Figure 13

Speed distribution of different runs: $0.5\le {\rho }_g\le 1.5$ for both (a) slow-mix condition and (b) fast-mix condition.