Beyond the faster-is-slower effect

The “faster-is-slower” effect arises when crowded people push each other to escape through an exit during an emergency situation. As individuals push harder, a statistical slowing down in the evacuation time can be achieved. The slowing down is caused by the presence of small groups of pedestrians (say, a small human cluster) that temporarily block the way out when trying to leave the room. The pressure on the pedestrians belonging to this blocking cluster increases for increasing anxiety levels and/or a larger number of individuals trying to leave the room through the same door. Our investigation shows, however, that very high pressures alter the dynamics in the blocking cluster and, thus, change the statistics of the time delays along the escaping process. A reduction in the long lasting delays can be acknowledged, while the overall evacuation performance improves. We present results on this phenomenon taking place beyond the faster-is-slower regime.

Figure 1

Snapshot of an evacuation process from a $20\mathrm{m}\times 20\mathrm{m}$ room, with a single door of 1.2 m width. The blocking structure is identified in red color (gray color in the printed version). The rest of the crowd is represented by white circles. It can be seen three individuals that have already left the room. The desired velocity for the individuals inside the room was $v_d=6\mathrm{m}$/s.

Figure 2

Evacuation time and blocking time as a function of the desired velocity $v_d$. Both data sets represent the mean values from 60 evacuation processes. The simulated room was $20\times 20$ m with a single door of 1.2 m width on one side. The number of individuals inside the room was 225 (no reentering mechanism was allowed). The simulation lasted until 160 individuals left the room.

Figure 3

Evacuation time per individual vs desired velocity for $\mathrm{N}=225, 529$, and $961$ (no reentering mechanism was allowed). The rooms sizes were $20\times 20, 30\times 30$, and $40\times 40$ m, respectively, with a single door of 1.2 m width on one side. Mean values were computed from 30 evacuation processes until 70% of pedestrians left the room.

Figure 4

Mean number of blocking delays for four different time intervals (see legend for the corresponding blocking times $t_b$) as a function of the desired velocity $v_d$. The simulated room was $20\times 20$ m with a single door of 1.2 m width on one side. The number of individuals inside the room was 225 (no reentering mechanism was allowed). Mean values were computed from 60 realizations. The simulation lasted until 160 individuals left the room.

Figure 5

Histogram of the position of the breakup of the blocking cluster. The room size was $20\times 20$ m with 225 pedestrians (no reentering mechanism was allowed). The door's width was 1.2 m (from $y=9.4$ to 10.6 m). The vertical red lines represent its limits. 30 evacuation processes were performed until 70% of pedestrians left the room. The desired velocity was $v_d=10$ m/s.

Figure 6

Force balance for a moving pedestrian between two still individuals. The moving pedestrian is represented by the white circle, while the gray circles correspond to the still individuals. The movement is in the $+x$ direction. $f_s$ represents the (mean) force due to other pedestrians pushing from behind. $f_d$ is the moving pedestrian's own desire. $f_g$ corresponds to the tangential friction (i.e., granular force) between the moving pedestrian and his (her) neighbors. $\mathcal{F}$ and $f$ are the forces actuating on the upper (still) pedestrian. $f$ corresponds to the social repulsive force due to the moving pedestrian, while $\mathcal{F}$ represents the counter force for keeping the pedestrian still.

Figure 7

Blocking time of the three pedestrians model (one moving pedestrian between two still ones) as a function of the desired velocity $v_d$. The initial velocity of the moving pedestrian was set to zero. The crowd pressure was set to $\mathcal{F}=f_s=\beta v_d$ ($\beta$ value indicated on the top right). Each blocking time was recorded when the moving pedestrian lost contact with the other individuals. Desired velocities of $v_d=1.75$, 3.5, and 11.25 m/s are indicated in red color (and squared symbols) for $\beta =2000$. The blocking time for $v_d=1.75$ and 11.25 m/s are the same. Only one realization was done for each $v_d$ value.

Figure 8

Ratio of positive forces (desire force and social repulsion) and negative force (granular) on the moving pedestrian as a function of time for three desired velocities (see text for details). The initial velocity of the moving pedestrian was zero. The simulation finished when he loses contact with the other individuals. One realization is done.

Figure 9

Time delay ($v_{\infty }^{-1}$) for a moving individual passing between two still pedestrians, as shown in Fig. 6. The time interval was measured along 10 m across the still pedestrians. The initial velocity was $v_d$. The continuous line corresponds to the measured delay for $\beta v_d=2000v_d$ and $f=Aexp\left[\right(2r-d)/B]$ (see text for details). The dashed line corresponds to the measured delay for $\beta v_d=2000v_d$ and $f=Aexp\left[\right(2r-d)/B]+k(2r-d)$ (see text for details). The minimum time delay for both lines takes place at $v_d=1\mathrm{m}/\mathrm{s}$. The maximum time delay for the continuous line takes place at $v_d=3.7\mathrm{m}/\mathrm{s}$, while for the dashed line takes place at $v_d=4.2\mathrm{m}/\mathrm{s}$.