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.

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).

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).

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.

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.

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.

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).

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.