Self-organized queuing and scale-free behavior in real escape panic

Abstract

Numerical investigations of escape panic of confined pedestrians have revealed interesting dynamical features such as pedestrian arch formation around an exit, disruptive interference, self-organized queuing, and scale-free behavior. However, these predictions have remained unverified because escape panic experiments with real systems are difficult to perform. For mice escaping out of a water pool, we found that for a critical sampling rate the escape behavior exhibits the predicted features even at short observation times. The mice escaped via an exit in bursts of different sizes that obey exponential and (truncated) power-law distributions depending on exit width. Oversampling or undersampling the mouse escape rate prevents the observation of the predicted features. Real systems are normally subject to unavoidable constraints arising from occupancy rate, pedestrian exhaustion, and nonrigidity of pedestrian bodies. The effect of these constraints on the dynamics of real escape panic is also studied.

Searches of disaster databases§ would readily show that the escape panic of confined pedestrians is costly in terms of fatalities and property loss. Despite the huge toll inflicted by these incidents to society, the dynamics of escape panic are not completely understood because studies have been largely confined to numerical simulations that revealed a number of interesting dynamical features such as pedestrian arch formation around an exit, herding, and interference between arches in multiple-exit rooms (1). Recently, additional features such as disruptive interference, self-organized queuing, and scale-free escape dynamics (2) were found. Experiments in genuine escape panic are difficult, especially with humans because of possible ethical and even legal concerns.

This work addresses the behavior of panicking groups and how it is influenced by the architecture of the space in which they are confined. It determines whether the dynamical features predicted earlier in numerical experiments are observed in a group of real biological (nonrigid) agents undergoing escape panic. Experimental results from mice escaping out of a water pool show that for a critical sampling interval their escape behavior agrees with the numerically predicted exponential and power-law frequency distributions of the exit burst size even for short time durations. Oversampling or undersampling of the mouse escape rate prevents the observation of the predicted features.

Escape panic could happen in different confinement sizes, from a rioting crowd in a packed stadium to stunned customers in a smoke-filled bar. It is characterized by strong contact interactions between selfish individuals that quickly gives rise to herding, stampede, and clogging (3–7). Escape panic is simulated by solving a set of coupled differential equations (1, 8) or by the cellular automata (CA) technique (2) where the movement of confined pedestrians is tracked over time. Both approaches yield consistent results at least for escape features like arching and dependence of exit throughput rate with panic level.

The CA technique is attractive for its conceptual simplicity and numerical efficiency (9). It could account for the discrete nature of pedestrians, which is crucial when their number is not statistically large. Failure to consider discrete features could lead to erroneous outcomes about the collective behavior of a complex system (10).

Real escape panic was tested in mice escaping from a water pool to a dry platform through an exit door of varying widths and numbers. Our specific choice follows from observations that rodents such as the mouse exhibit strong motivation to escape from the water (11, 12). The modified water maze task is unhampered by problems of abortive choices or errors of omission in expected behavior of the test subjects; mice will always attempt to reach dry land in every trial run without additional motivation (e.g., food or drinking water deprivation).

Animals have evolved an effective (optimal) strategy to escape from water to dry land via the shortest path possible for minimum effort in drowning avoidance (12). Studies with rodents are useful because mammals exhibit and share similar behavioral traits and capacities (13). We have also chosen the mouse because it is readily bred in sufficient numbers and maintained at low cost, and its movements are easily restricted within a 2D plane for straightforward comparison with CA simulations.

Methodology

CA Model of Escape Panic. We simulate with a 2D CA model the movement and exit of escaping pedestrians in a room that is divided into L × W cells. Each cell is either empty (designated 0) or occupied (designated 1) by no more than one rigid pedestrian at a time. At the start (at iteration step q = 0), the pedestrians are distributed randomly inside the room (see Fig. 1a), and their individual movements are tracked through time. The room has an exit door that is w cells wide so that at most w pedestrians could escape simultaneously.

Fig. 1.

CA model for escape panic and block diagram of mouse experiment set-up. (a) Single-exit room with randomly located pedestrians at q = 0 (L × W = 18 × 14 cells, occupancy rate = 11.9%). (b) Pedestrian P_k prefers to move to an available cell nearest to exit location (unfilled square). (c) Setup for mouse experiments with one exit of width w. To achieve the desired effect of randomness at q = 0, no pedestrian occupies a cell that belongs to the column containing the exit or the one before it (dotted line boundary in a). In b, P_k can move to an available cell only after satisfying a certain condition involving his panic parameter ϕ and neighbors on his left (l), right (r), and rear (b).

At any given step q, the kth confined pedestrian P_k knows the exit location and the shortest (Euclidean) path to it. All P_ks compete to leave the room ahead of each other and every P_k can move forward, back, left, or right (von Neumann neighborhood, see Fig. 1b). If conditions allow, P_k prefers to occupy the neighboring empty cell that brings him closest to the exit. If more than one cell satisfies this most preferred condition then P_k randomly selects among these cells. On the other hand, if such a cell is unavailable, then P_k randomly considers with a uniform probability distribution any of the remaining adjacent vacant cells. P_k faces the candidate vacant cell and occupies it if: L + R < B + ϕ, where L, R, and B represent the total number of (connected) neighbors to the left (l), right (r), and rear (b) directions of P_k, respectively. Otherwise, P_k stays.

We introduce a panic parameter ϕ (≥ 0) to indicate the anxiety level of P_k. A highly agitated P_k is described by a high ϕ value and exhibits a strong tendency to move. P_k is considered as having left the room upon occupying the exit cell. Per step q, the process is repeated for all P_ks in the room. In this work, all P_ks in the room have the same ϕ value.

P_k moves forward if: L + R < B + ϕ. This condition considers the effect of neighbors and the degree of panic on the ability of P_k to actually occupy a cell of choice. B + ϕ represents the forward push magnitude that P_k receives from the rear in addition to his own tendency to move forward to the cell that he is facing. L + R represents the restraining effect of squeezing by the neighbors from the left and right sides.

We have previously reported that escape is more difficult for highly agitated pedestrians in a room (2). Similar results indicating the “faster-is-slower” effect were reported by Helbing et al. (1), using a continuous model for panic escape that relied on a generalized force equation.

At any step q, the number of P_ks is held constant by introducing a new one at a randomly chosen vacant cell, every time another leaves the room. We have previously found that P_ks leave the room in bursts of different sizes S (2). We define the burst size S as the total number of pedestrians who have exited successively in time without break from one iteration step q to the next, q + 1, where Δq = (q + 1) – q. In the mouse experiments, we found that Δq is the average time (2 sec) needed for a mouse body to completely traverse the exit.

Keeping the pedestrian number constant permits us to generate burst-size frequency distributions F(S) over a wide range (3 orders of magnitude) of possible burst sizes. Possible power-law behavior of the burst-size frequency distributions could therefore be ascertained accurately at different step q and door width w. Power–law behavior could indicate the role of self-organized criticality in the exit dynamics (14).

Mouse System. ICR mice were purchased from the Research Institute for Tropical Medicine in Muntinlupa, Philippines and bred at the Laboratory of Molecular Cell Biology, National Institute of Molecular Biology and Biotechnology. The mice were reared in the laboratory at 25°C with 12-h light/12-h dark periods. Their ages varied from 6 to 12 months, and their (width) sizes ranged from 2.5 to 3.0 cm measured along the axial plane. They weighed from 25 to 35 g. During each experimental session, a group of 60 mice (male/female 1:1) were used with none of the female mice ovulating. Other sex ratios were not tested because of limitation in the number of available male and female mice. ICR mice achieve sexual maturity at an age of ≈6–8 weeks.

Fig. 1c shows a block diagram of the rectangular container (L = 50 cm, W = 34 cm, H = 29 cm) where the experiments were performed. The mice were released into a pool of tap water (L = 38 cm, W = 34 cm, H = 10.5 cm) and left to swim toward a dry platform. An exit of width w is located between the wet and dry areas.

The setup was designed to induce panic in mice and maintain directed flow (from wet to dry) toward the exit. A series of preliminary tests were performed to observe mouse behavior under the given situation. Directional swimming (where mice swim in a direction parallel to their release orientation) was observed with each mouse released in the pool. The swimming mouse then moved along the wall (wall-seeking behavior) until it found the exit. Generally, mice with a previous experience in the pool could escape more quickly in time (than those without) but this benefit of experience did not result in quicker exit times for the succeeding trials, most likely because of exhaustion.

The actual experiment was carried out in two parts. First, we varied the door size (w = 3.5, 7.0, 10.0, and 13.0 cm) and then the door separation distance (d = 3.5, 7.0, 14.0, and 21.0 cm). Experimental sessions were recorded with a digital video camera (Sony Digital Handycam DCR-TRV820 NTSC, digital sampling rate = 25 frames per sec). Each session was performed for 180 sec, which is 90 times the sampling interval of Δq = 2 sec. The experimental session is repeated four times for every door size/door separation (pictures and video footages of experiments are at www.nip.upd.edu.ph/ipl/panic). A larger number of trials per session could not be performed because of mouse exhaustion. Mice were treated following the guidelines on the humane treatment of laboratory animals in Administrative Order No. 40 (Department of Agriculture) under the Philippine Animal Welfare Act of 1998.

For consistency with CA experiments, the number of mice in the water pool was always maintained at 30 by adding a new mouse at a randomly chosen pool location every time one exited the door. The (wet) mice that already exited were first kept in a staging area for recuperation before being allowed to join the reserve group. The new input was taken randomly from the remaining (reserve) group of 30 mice. Video analysis was performed to obtain the time interval between successive mouse passage through the exit as well as the total number of mice that exit at a given time. Experimental data were collected from January to August 2002.

Experimental Results

Fig. 2 plots the burst-size frequency distribution F(S) obtained from our CA experiments, for a single-exit room (L = 30, W = 38) with an occupancy rate of 11.8%. The same occupancy rate is used in the mouse experiments. The exit is located at the center of the shorter wall. The escaping pedestrians were monitored for a time duration (sampling period) of q = 5 × 10⁴ iteration steps. Every F(S) value represents not just the number of individual burst of size S but also the N-number of S-sized component bursts in a larger burst of size (N + ξ)S, where 0 ≤ ξ < N. For example, a single burst of size 5 would contribute to the F(S) values in the following manner: F(1) = 5, F(2) = 2, F(3) = 1, F(4) = 1, and F(5) = 1. This method of burst counting is adapted to highlight the fine details of the exit dynamics.

Fig. 2.

CA experiments with a single-exit room (30 × 38 cells, occupancy rate: 11.8%). Burst size-frequency distribution F(S) at different exit widths w (in cell-width units), where ϕ = 5, q = 5 × 10⁴. Solid line is (w = 1): F(S) 27,970.8 exp(–0.456S). At w = 4 and 10, F(S) plots are well approximated by a truncated power law: F(S) ≈ exp(γ)βS^α, where α(w = 4) = –0.33, γ(4) = –0.025, β(4) = exp (3.25), and α(10) =–0.38, γ(10) =–0.01, β(10) = exp(3.45). At w ≥ 15, F(S) curves approximately exhibit a power-law behavior of the form: F(S) = βS^α, where β(w = 15) = 5,583, α(15) =–1.15 (S_c = 80); β(20) = 5,678, α(20) =–1.09 (S_c = 150); β(23) = 7,510, α(23) = –1.08 (S_c = 600); and β(25) = 7,049, α(25) = –1.06 (S_c = 1,000).

At door width w = 1, burst-size frequency distribution F(S) decreases exponentially with S. At w = 4 and 10, the distributions are well approximated by a truncated power law of the form: F(S) ≈ exp(γ) βS^α. At w > 10 and neglecting the unavoidable boundary effects, the distributions exhibit a (full) power-law behavior: F(S) = βS^α. For the same room size and occupancy rate, the burst size-range (1 ≤ S ≤ cut-off burst-size S_c), where the power-law behavior is observed, is restricted by the door width w. A burst size-range covering 3 orders of magnitude (S_c = 1,000) is possible only with width w ≥ 23. Cut-off S_c is not increased by increasing the period q and small burst sizes (S ≤ 10) are more likely to happen with w = 1 than with larger door widths.

Within a clog-time t_c (minimum duration = Δq), no pedestrian could escape from a room, and extended clog times are unlikely in a single-exit room with width w = 1. Fig. 3 shows clog-time frequency distributions F(t_c) with a different dynamical behavior at w = 1. With larger exits w ≥ 2, the t_cs are distributed over a wider range of possible values.

Fig. 3.

CA experiments with a single-exit room (30 × 38 cells, occupancy rate: 11.8%, ϕ = 5, and q = 5 × 10⁴). Frequency distribution F(t_c) of clogged times t_c (in iteration step units) at w = 1(□), 2 (⋄), 3 (○), and 4 (▵). Solid line: F(t_c) = 34,730 t ^–1.043 c.

In a single-exit room with w = 1, queuing emerges spontaneously permitting the pedestrians to “stream” out of the exit. Queuing, which leads to diffusive flow, is a manifestation of self-organization (2). It enables pedestrians to leave the room in larger numbers for the same period q. Unlike the burst-size frequency distributions F(S), which are sensitive to w (see Fig. 2), the clog-time distributions F(t_c)s are almost indistinguishable for w = 2, 3, and 4. This finding implies that t_c is not a good parameter to use for investigating the exit dynamics of a single-exit room.

Fig. 4 plots the scaled burst-size distribution F(S)/q for a single-exit room (door width w = 1) at different door widths. The F(S)/q plots collapse into a single line, indicating that diffusive flow persists at all time scales. We also investigated the clearing-time characteristic of a single-exit room when the number of escaping pedestrians is allowed to decrease in time. Fig. 5 presents the clearing-time frequency distribution as a function of room occupancy rate at w = 1, 2, 3, 4. For a given door width w, the room clearing time increases linearly with occupancy rate. In the absence of clogging at the exit door, rooms with larger doors are emptied more quickly as expected, which illustrates that clearing-time behavior does not reveal the emergence of queuing in a single-exit room with w = 1.

Fig. 4.

CA experiments with a single-exit room with door width w = 1(30 × 38 cells, occupancy rate: 11.8%, ϕ = 5). Scaled burst-size frequency distribution F(S)/q, at step q = 10² (□), 5 × 10² (⋄), 10³ (○), 10⁴ (▵), 2 × 10⁴ (crossed squares), 3 × 10⁴ (crossed circles), 4 × 10⁴ (crosses), and 4 × 10⁴ (▿). Solid line is 0.6025exp(–0.45S).

Fig. 5.

CA experiments with a smaller single-exit room (18 × 14 cells, ϕ = 5). Frequency distribution of average clearing times (in iteration step units) at various room occupancy rates with door width w = 1(□), 2 (♦), 3 (○), and 4 (▴). Clearing time is the duration that it takes for a room to be completely emptied of escaping pedestrians. Solid line (□): y = 3.33x + 4.97, where y represents the clearing time and x represents the occupancy rate. Solid line (▴): y = 1.34x + 9.55.

For the mouse experiments, burst size S is given by the number of mice that have exited within a chosen sampling interval Δq. Fig. 6 plots the elapsed-time histogram between successive mouse exits as a function of exit width w. We found that the equivalent of one door-width unit (w = 1) is 3.5 cm (height = 3.5 cm), which is 25% larger than the average mouse girth diameter. A single mouse would encounter difficulty exiting at smaller door widths.

Fig. 6.

Mouse experiments. Frequency distribution of elapsed times (in units of 500 msec) between successive escape of mice in a single-exit room with w = 1 (a), 2 (b), 3 (c), and 4 (d). At any instant, 30 mice are in the water pool (effective occupancy rate: 14.3%). One door width unit (w = 1) is equivalent to 3.5 cm.

The exit dimensions were selected consistent with the numerical model where a door width w could permit the simultaneous escape of not more than w pedestrians. The following equivalent-width dimensions were determined: w = 1 (3.5 cm), w = 2 (7 cm), w = 3 (10 cm), and w = 4 (13 cm). A mouse typically displayed the wall-seeking behavior when reacting to a panic situation in isolation but this propensity was reduced when other escaping mice were present. The range of possible clogged times is narrowest at w = 1 (Fig. 6a) and widest at w = 4 (Fig. 6d), which is consistent with the findings of our CA experiments.

To determine the effect of door width w on the pedestrian escape rate, we measured the dependence of throughput Q as a function of door width w (Fig. 7a) for the CA and mouse experiments, where Q is the average number of pedestrians who have exited within the sampling interval Δq. For the mouse experiments, the sampling period is q = 180 sec, and Δq is equivalent to 2 sec. Because of mouse exhaustion, observation beyond 3 min was not possible. Even at a relatively short period of q = 90, the mouse results agree well with the CA predictions for Q at exit widths w ≤ 3 (Fig. 7a). At w = 4, a discrepancy occurred between the CA and mouse data; it is possible that at larger widths the maximum number of mice that could escape within Δq exceeded w because mouse bodies are compressible.

Fig. 7.

Mouse data and CA predictions. (a) Throughput Q vs. exit width w for CA (φ = 5, L = 18, W = 14) for q = 90 (○) and 5 × 10⁴ (□) and mouse experiments (•, Δq = 2 sec). (b) CA results for qQ vs. q with w = 1 (□), 2 (▪), 3(•), 4(○), 5(⋄), and 6 (♦). (c) Q vs. door separation d for CA (φ = 5) and mouse (•, Δq = 2 sec) experiments for q = 90 (○) and 5 × 10⁴ (□). Each mouse data point represents the average of four trials. Curves in b: qQ = 74.6q – 2.9 (solid black); qQ = –4.1 + 78.3q – 30.9 (dotted red); qQ = –1.5 + 80.5q – 46.6 (dotted blue). In c, room has two exits (w = 1) separated by d. At d = 0, one door is used with w = 2. Lines represent the computed Q values for a room with one (w = 1) door at q = 90 (solid line, Q = 0.7) and q = 5 × 10⁴ (dotted line, Q = 0.9).

Except at width w = 1, the CA predictions at q = 5 × 10⁴, which is that: Q(2 ≤ w ≤ 4) < Q(1), disagree with those obtained at q = 90. This is explained by the presence of queuing at long sampling periods, which favors the formation of small burst sizes in a single-exit room with w = 1 (exponential size-frequency distribution) than in w > 1 (power-law size-distributions). At q = 90, queuing is insignificant and Q(w = 1) < Q(2) < Q(3) < Q(4).

To illustrate the emergence of queuing in escape panic, we plotted the dependence of product qQ with iteration step (sampling period) q for different exit widths (see Fig. 7b). Product qQ represents the total number of pedestrians who have escaped within the sampling period q. For a single-exit room, the qQ values for w = 2 and 3 are less than those with w = 1 when step q becomes more than q = q_c ≈ 700. Note that q_c value can provide us with a working definition of a short observation time in escape panic, which is q < q_c.

We also investigated the effect of exit number on the pedestrian escape rate. Fig. 7c plots the dependence of throughput Q on the door separation d in a two-exit room. Regardless of d, CA results reveal that increasing the exit number to 2 does not double the throughput Q. Disruptive interference between the two pedestrian arches that are formed around the two exits restricts the pedestrian escape rate (2).

The CA predictions and mouse data agree with each other for a room with one large door (w = 2, d = 0). However, the two data sets disagree for a room with two exits (w = 1, d > 0); the mouse experiments yielded lower throughputs caused by herding (1) and to a lesser extent disruptive interference. Herding prevented the full utilization of the two exits by the escaping mice. Herding is not considered in our current CA simulations where the pedestrians are not allelomimetic (15).

We also determined the burst-size frequency distributions for the mouse (Fig. 8a) and CA experiments at a short sampling period of q = 90 (Fig. 8b). The burst size frequency distributions from the mouse data are consistent with the predictions of the CA experiments. The expressions for the solid (best-fit) curves in Fig. 8 are given in Table 1. For both the mouse and CA (q = 90) experiments, the amplitude and exponent values of the burst-size frequency distributions increases and decreases with exit width w, respectively.

Fig. 8.

Comparison of burst-size frequency distributions. (a) Mouse data with Δq = 2 sec. (b) CA predictions with q = 90 and φ = 5 for w = 1 (circles), 2 (squares), 3 (triangles), and 4 (diamonds). Description of solid curves are in Table 1.

Table 1.

Solid curve expressions of burst size-frequency distributions in Fig. 8

Door size w	Mice (Δq = 2 sec)	CA (q = 90, Φ = 5)	CA (q = 50,000, Φ = 5)
1	79 exp(-0.78S)	84 exp(-0.61S)	202,388 exp(-0.51S)
2	125 S-1.91	109 S-1.32	13,355 S-1.53
3	149 S-1.72	121 S-1.25	17,178 S-1.33
4	222 S-1.146	133 S-1.16	51,522 S-1.28

Discussion

We have studied the dynamics of escape panic in mice that are escaping from a water pool. Genuine escape panic is normally subject to unavoidable constraints concerning room size, pedestrian exhaustion and compressibility, and the effect of allelomimesis among escaping pedestrians (15). In smaller rooms, overcrowding happens more quickly and the f low of panic information is also faster. Hence, clogging occurs suddenly, making the observation of escape panic more difficult. Mouse exhaustion also limits the duration of escape panic while the compressibility of their bodies permits the simultaneous escape of more than one animal via a door of width w = 1. At the height of panic, allelomimetic tendencies dominate over individual decisions, giving rise to herding. The above constraints were neglected in our CA simulations to reduce computational complexity.

CA simulations revealed that pedestrians leave a single-exit room in bursts of different sizes S (see Fig. 2). At exit width w = 1, the burst-size frequency-distribution exhibits an exponential decay (diffusion-like) behavior. This behavior was observed in the mouse experiments even at short sampling periods q < q_c, and in the absence of self-organized queuing that emerges only at long periods (see Fig. 7). Queuing allows the pedestrian escape throughput to increase in a single-exit room with width w = 1. Discrepancies in the absolute amplitude and exponent values of the burst size-frequency distributions (Fig. 8) between the mouse data and the CA predictions (step q = 90), is caused by the compressibility of mouse bodies and the susceptibility of the mice to exhaustion and herding.

The burst-size frequency distributions first exhibit a truncated power-law behavior (w = 4, 10) and then finally a full (3 orders of magnitude range) power-law characteristic at door width w = 23 and 25 (see Fig. 2). Power-law behavior of the distributions varies with room size/door width (LW/w) ratio and is independent of sampling period.

In the mouse experiments, the choice of the sampling interval Δq affects the characteristics of the burst-size frequency distributions that could be generated from the same 3-min video footage. Because the sampling period is constant, a change in Δq affects the step value q in the CA simulation according to: q = (180 s)/Δq. Table 2 indicates that correlation between the CA predictions and the mouse data is sensitive to the choice of Δq. Sampling at a higher resolution of Δq = 1.5 sec results in a strong disagreement caused by possible erroneous overcounting of the mice that have escaped. Slighter disagreements were observed at a lower Δq of 2.5 sec, which could lead to possible undercounting. Dynamical features are lost by oversampling (or convolution) when viewed at higher (or lower) resolution. Tables 1 and 2 indicate that the suitable Δq value is 2 sec, which is the average time that a mouse body needs to traverse completely an exit. This choice also overcomes the difference between the shape (square-like) of pedestrians in the CA and mouse (approximately cylindrical) experiments.

Table 2. Expressions of burst size-frequency distributions for two other Δq values.

	Δq = 1.5 sec; q = 120		Δq = 2.5 sec; q = 72
w	CA	Mice	CA	Mice
1	98 S-1.68	111 exp(-0.78S)	48 exp(-0.46S)	75 exp(-0.66S)
2	153 S-1.35	101 exp(-0.60S)	92 S-1.50	120 S-1.50
3	167 S-1.26	150 exp(-0.58S)	99 S-1.35	133 S-1.44
4	183 S-1.23	179 exp(-0.40S)	105 S-1.20	213 S-1.26

Conclusions

The dynamics of genuine escape panic have been studied. Experimental results with escaping mice are consistent with the CA predictions for escape panic. We observed diffusive flow at exit width w = 1 and scale-free (power law) behavior at larger widths, for burst-size frequency distributions even at relatively short sampling periods and small confinements. Streaming is not a precondition for diffusive flow. It is also not apparent in the behavior of the clog-time and clearing-time frequency distributions. We have established the critical sampling rate required to unmask the true escape dynamics of real biological agents. This information is important in the formulation of effective strategies for disaster mitigation.