The experiments were performed respectively on November 16, 2014 (50 graduate students, of them 39 male, 11 female) and on June 7, 2015 (100 undergraduates, of them 85male, 15 female) in Hefei University of Technology. Experiments were conducted in an artificial room with one exit. The sizes of the virtual room were 7 m×7 m and 10 m×10 m respectively, so that the pedestrian density is approximately the same. The width of the exit is 0.8 m. At the boundary of the virtual room some chairs and two rostrum tables were placed as walls, see Fig. 1.

Participants knew clearly where the exit was, and were asked to escape from the room as soon as possible, but not in a panic environment. It can be regarded as the case that the participants take part in evacuation drills. Before the evacuation, the individuals had uniform random distribution. If too many participants gather in one small area, the experimental commander would ask them to disperse to other place. Pushing by the arm and slight squeezing by the body were allowed in the experiments. Before each round of the experiment, the commander reminded that the participants should escape as if real disaster happened. But if there were some safety hazards, the commander would stop the experiment immediately. Fortunately, no danger happened in two experiments.

The two sets of experiments were conducted with 12 and 20 replications respectively. The motion of each participant was recorded by video camera (SONY HDR-CX510E), and the evacuation times were recorded manually.

Fig. 1.

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	Fig. 1. (color online) Snapshots at the beginning of experiments: (a) 50 pedestrians and (b) 100 pedestrians.

Tables 1 and 2 show the evacuation times in the two sets of experiments. The average evacuation time in the first set of experiments is 16.96 s, and the standard deviation is 0.89 s. The second set of experiments has larger average evacuation time 32.02 s, which is nearly twice that of the first set of experiments, and the standard deviation (1.79 s) is larger.

Table 1.

Table 1.

Table 1. Evacuation times (unit: second) in the first set of experiments. . First set of experiments No. Time No. Time No. Time No. Time 1 16.96 4 16.02 7 16.00 10 17.64 2 17.97 5 16.70 8 16.01 11 15.80 3 18.44 6 17.72 9 17.50 12 16.78 Average time 16.96 Standard deviation 0.89 Simulation Average time 16.92 Standard deviation 1.02

Table 1. Evacuation times (unit: second) in the first set of experiments. .

Figure 2 shows the distribution of evacuation time gaps. It can be seen that the distribution for each set of experiments keeps basically consistent, and has a peak value around 0.3 s. As a result, the standard deviations of the evacuation time gap are 0.1776 s and 0.1815 s respectively. The small proportion of low evacuation time gap (0.1 s) means that few pedestrians can escape from the room simultaneously, due to the exit width. In addition, the small proportion of high evacuation time gap (> 0.6 s) shows that few arches emerge in the evacuation process.

Table 2.

Table 2.

Table 2. Evacuation times (unit: second) in the second set of experiments. . Second set of experiments No. Time No. Time No. Time No. Time 1 32.25 6 32.46 11 35.03 16 33.38 2 28.4 7 30.72 12 32.10 17 34.14 3 29.92 8 33.28 13 33.39 18 33.98 4 30.62 9 30.37 14 32.85 19 31.41 5 33.40 10 29.38 15 30.29 20 33.1 Average time 32.02 Standard deviation 1.79 Simulation Average time 31.96 Standard deviation 1.90

Table 2. Evacuation times (unit: second) in the second set of experiments. .

Fig. 2.

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	Fig. 2. (color online) Evacuation time gap distribution.

Evacuation time gap between two consecutive pedestrians varies in the evacuation process. Figure 3 shows the average evacuation time gap between the i-th escaped pedestrian and the (i + 1)-th escaped pedestrian. The average time gap is an indicator to reflect concentration trend, which can undermine the randomness and fluctuation due to the differences in individuality and condition. The average is a common method in evacuation study, such as in the Garcimartín et al.’s work.^[15] The time gap is essentially constant in both sets of experiments. This implies that the evacuation time gap essentially does not depend on the number of pedestrians that have not yet escaped. Note that the panic condition cannot be built in the experiment, so no faster-is-slower or deadlock emerges near the exit. In the real panic, the life instinct may cause different dynamic characteristics. The experiment is more similar to the real case, when pedestrians leave the stadium in a hurry after one public activity. Therefore, the experimental conclusion has limited application range.

Fig. 3.

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	Fig. 3. (color online) Variations of average evacuation time gap between consecutive pedestrians with sequence number in (a) first set N = 50 and (b) second set of experiments N = 100. The results are averaged over 12 replications and the 20 replications respectively. The dash lines are linear fitting results and are guides for the eyes.

We simulate the evacuation dynamics of the 50-pedestrian experiment by using the original social force model. In the social force model, the motion of a pedestrian is described by three different components: driving force, repulsive force between pedestrians, and repulsive force between pedestrian and wall. The equation of motion is as follows: 1 Here m_i is the mass of pedestrian i. The driving force represents the pedestrian motivation to walk in a given direction with desired speed, which reads 2 Here and are respectively the magnitude and direction of the pedestrian desired speed. v_i(t) is the current speed, and τ denotes the relaxation time.

The repulsive force f_ij describes the interaction between pedestrian i and other pedestrians, which includes socio-psychological force to stay away from others and physical contact force, and is expressed as 3 4 In Eq. (3), the first term on the right-hand side denotes the socio-psychological force, and the second term represents the physical contact force. Pedestrians i and j are regarded as cylinders with radius r_i and r_j, respectively. The r_ij denotes the sum of the radius of the two pedestrians, namely r_ij = r_i + r_j. The d_ij is the distance between the centers of the two pedestrians. n_ij represents the unit normal vector from pedestrian j to i, and q_ij means the unit tangential vector. They are expressed as follows: 5 6 where x_i, x_j and y_i, y_i are respectively the horizontal and vertical coordinates of the centers of the two pedestrians. The means the tangential velocity, 7 Δvjiq=(vj−vi)⋅qij, where kn_ij and are body pressure and friction force respectively. A, B, k, and κ are four parameters.

The repulsive force from the wall is modeled analogously as 8 Here, means vertical distance to the wall, denotes the unit normal vector from the wall to pedestrian i, and is the unit vector tangential to the wall. The tangential velocity , since speed of the wall is zero.

In the simulation, we set A = 0, i.e., we neglect the socio-psychological repulsive force, because pedestrians do not care to stay away from others in the evacuation process. Other parameters are calibrated and they are m_i = 60 kg, m/s, r_i = 0.225 m, τ = 0.5 s, k = 7.2 × 10⁴ kg·s⁻², and κ = 60 kg·m⁻¹·s⁻¹. The uniform random distribution is used. With these parameters, the average evacuation time of 50 pedestrians from a 7 m×7 m room is 17.44 s, which is in good agreement with the experimental result. However, as can be seen from Fig. 4, instead of being essentially constant, the evacuation time gap gradually increases. Also, the average evacuation time of 100 pedestrians is 25.87 s, which is significantly different from the experimental result. It is due to the accumulation effect of driving force. The driving force transmits to the exit area by the body contact, such as pushing or squeezing. More pedestrians in the room generate larger accumulation, facilitating the evacuation process. When the pedestrians are fewer, the accumulation effect weakens, slowing down evacuation efficiency.

Fig. 4.

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	Fig. 4. Variation of average evacuation time gap between consecutive pedestrians with sequence number in simulation results of escaping individually in the original social force model. The simulation results are averaged over 50 replications.

To quantify the effect, we define a pressure-like parameter P as the magnitude of repulsive physical contact forces between pedestrian i and other pedestrians, and it is expressed as 9 where N is the number of pedestrians still not yet escaped.

Fig. 5.

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	Fig. 5. (color online) Simulation results of distribution of the pressure-like parameter P (unit: m/s2). Initially there are 50 pedestrians in the room simulated by the original social force model.

Figures 5(a) and 5(b) show the distributions of P, which are averaged over 50 replications. Generally speaking, the closer to the door, the larger P is. Moreover, with more pedestrians escaped from the room (Fig. 5(b)), the magnitude of P in the vicinity of the door remarkably decreases. When P is large, pedestrians near the door will be pushed by large force. Therefore, they escape quickly. With the decrease of P, the pushing force decreases, lowering down the movement speed. To some extent, this result is like the fluid dynamic problem. Suppose that there is a tap at the bottom of a barrel. The less the water in the barrel, the smaller the flow speed from the tap is, because the pressure near the tap lowers down. Here the driving force of pedestrian is analogous to the gravity of water, and the pressure-like parameter P is analogous to the pressure of water in the barrel.

Now we propose an improved social force model to overcome the deficiency presented above. When squeezed in the crowd, a pedestrian can be pushed away if the resultant contact force from other pedestrians and walls is large. Under this circumstance, the pedestrian would not be able to perform the driving force temporarily. Based on this fact, the driving force is modified as follows: 10 Here 11 means the resultant physical force. The f_c is a threshold parameter. Note that Parisi et al.^[16] have proposed a self-stopping mechanism in a normal situation, in which pedestrians prefer to keep a distance from others so that they proactively stop performing the driving force when getting close to others. In contrast, in our model, escaping pedestrians are not able to perform the driving force passively when getting squeezed.

Next, we present simulation results of the improved social force model. Calibration shows that f_c = 600 N, m/s, and other parameters are the same as those used before. As can be seen from Tables 1 and 2, the average evacuation time is 17.48 s and 32.01 s, which are in pretty good agreement with the experimental results. Figure 2 shows that simulation results of the distribution of evacuation time gaps are in consistent with the experimental ones.

More importantly, Figure 6 shows that the evacuation time gap is essentially a constant as observed in the experiments. The distribution of the pressure-like parameter P is shown in Fig. 7, which is remarkably different from that in the original model. One can see that magnitude of P in the vicinity of the door is essentially independent of the number of pedestrians that have not yet escaped. If the pushing force is temporarily too large, the pedestrian has to keep the body balance to avoid possible falling-down. In this condition, the pedestrian has no mind going forward, and breaks the increase of the pushing force from congestion boundary to congestion center. Therefore, the pushing force keeps constant, and time gap keeps constant, too. This result is analogous to the grain flow problem. The flow rate of sandglass remains constant. As shown in Fig. 8, the more the pedestrians in the room, the more the pedestrians not able to perform the driving force is. Consequently, the evacuation time gap essentially does not change.

Fig. 6.

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	Fig. 6. (color online) Simulation results of average evacuation time gap between consecutive pedestrians by the improved social force model, in which pedestrians escape individually in the cases of (a) N = 50 and (b) N = 100.

Fig. 7.

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	Fig. 7. (color online) Simulation results of distribution of the pressure-like parameter P (unit: m/s2). Initially there are 50 pedestrians in the room simulated by the improved model.

Fig. 8.

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	Fig. 8. (color online) Two snapshots of the evacuation process in one replication of the simulation. Initially there are 50 pedestrians in the room, and they escape individually. The red dots (blue circles) mean that pedestrians are able (unable) to perform the driving force at the moment.