Risk Analysis, Vol. 33, No. 8, 2013
DOI: 10.1111/j.1539-6924.2012.01923.x
Risk of Large-Scale Evacuation Based on the Effectiveness of Rescue Strategies Under Different Crowd Densities Jinghong Wang,1,2 Siuming Lo,2,3 Qingsong Wang,1 Jinhua Sun,1,2,∗ and Honglin Mu1,2
Crowd density is a key factor that influences the moving characteristics of a large group of people during a large-scale evacuation. In this article, the macro features of crowd flow and subsequent rescue strategies were considered, and a series of characteristic crowd densities that affect large-scale people movement, as well as the maximum bearing density when the crowd is extremely congested, were analyzed. On the basis of characteristic crowd densities, the queuing theory was applied to simulate crowd movement. Accordingly, the moving characteristics of the crowd and the effects of typical crowd density—which is viewed as the representation of the crowd’s arrival intensity in front of the evacuation passageways—on rescue strategies was studied. Furthermore, a “risk axle of crowd density” is proposed to determine the efficiency of rescue strategies in a large-scale evacuation, i.e., whether the rescue strategies are able to effectively maintain or improve evacuation efficiency. Finally, through some rational hypotheses for the value of evacuation risk, a three-dimensional distribution of the evacuation risk is established to illustrate the risk axle of crowd density. This work aims to make some macro, but original, analysis on the risk of large-scale crowd evacuation from the perspective of the efficiency of rescue strategies. KEY WORDS: Characteristic crowd density; evacuation risk; large-scale evacuation; queuing simulation; rescue strategies
1. INTRODUCTION
trol have been issued by the governments of various countries, many disasters caused by stampeding crowds still occur, such as: the 1996 Guatemala City disaster in which 84 people died on a football field; the 2001 Johannesberg disaster in which 43 people died in Ellis Park Stadium; the 2001 Ghana disaster in which 126 people died in a football match; the 2003 Chicago disco disaster in which 21 evacuees were crushed to death while exiting a club; the 2004 Beijing Miyun disaster in which 37 people died in a stampede on the Mihong Bridge; the 2010 India Kunda disaster in which 71 people were killed in the Ram Junki Temple; and the 2010 Khmer Water Festival disaster in which 347 people died in a stampede in Cambodia. This indicates that when a huge number of people gather together, crowd movement, especially in an emergency situation, may be
Forty years ago, the Glasgow Ibrox Stadium tragedy happened in which 66 people were crushed to death. The tragedy led the government of the United Kingdom to review sports ground safety and corresponding safety measures were recommended. Although various similar guidelines for crowd con1 State
Key Laboratory of Fire Science, University of Science and Technology of China, Hefei 230026, China. 2 USTC-CityU Joint Advanced Research Centre (Suzhou), Suzhou 215123, China. 3 Department of Civil and Architectural Engineering, City University of Hong Kong, Hong Kong. ∗ Address correspondence to Jinhua Sun, State Key Laboratory of Fire Science, University of Science and Technology of China, Hefei 230026, China and USTC-CityU Joint Advanced Research Centre (Suzhou), Suzhou 215123, China;
[email protected].
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1554 potentially hazardous. The uncoordinated motion of the crowd may lead to jamming, pushing, crushing, and trampling. In such a circumstance, an individual’s control over the movement becomes impossible and shock waves may propagate through the crowd mass and cause uncontrollable surges. This may be even more evident when evacuation is initiated in places where large numbers of people gather together, such as emporiums, temple fairs,4 stadiums, etc. Accordingly, establishment of rational guidance for crowd management in a large-scale evacuation is of prime importance. In addition to engineers, building designers and planners are also studying to alleviate the scale of such crowd-crushing disasters that occur especially in densely populated areas and are generally related to crowd characteristics and conditions of the moving paths. Crowd characteristics refer to the structure of the crowd, such as age, cultural diversity, crowd density, speed of movement, the crowd’s motivation, and so on. The conditions of the moving paths refer to whether there are obstacles or bottlenecks in the paths. Focusing on these factors, many representative studies have been reported. R. A. Smith reviewed some works on the relationship between crowd velocities, flow rates, and densities for unidirectional motion. From his review, it is collectively believed that crowd density should be a key factor that dominates the moving characteristics of the crowd.(1) Fang et al. demonstrated that crowd movement is significantly affected by the people at the front and those at the back, and that the movement speed will be largely affected by the front and back interperson distance.(2) Wang et al. analyzed the February 2004 trample disaster at the Mihong Bridge in Beijing, China, and pointed out that the high density of the crowd on the bridge was the main reason that led to asphyxia and even death in some people.(3) Therefore, effective control of crowd density is extremely important at major entertainment or sporting events that attract tens of thousands of people. Canetti(4) stated that individual behaviors in public places are greatly affected by surrounding crowds. Such collective behavior may dominate an individual’s reaction and can be considered as “herd behavior” in a crowd. When crowd density is low, the crowd movement can be viewed as free flow rather 4
The temple fair is a social activity in Chinese society in which a large group of people gather around a famous temple to pray to “gods” for fortune. Nowadays, it has evolved into a marketplace for people to buy and sell local products and a place for cultural performances.
Wang et al. than continuum flow. At free flow stage, a trampling disaster is unlikely to happen. When crowd density is high and large numbers of people are moving the same way at the same time, thereby involving herd behavior, such crowd movement can be viewed as continuum flow. It is, therefore, reasonable to treat a large-scale evacuation involving large numbers of evacuees as continuum flow. In 2002, Hughes(5,6) proposed a continuum theory for the flow of pedestrians. He described the movement pattern and herd behavior, related to path choice, of the crowd through a series of nonlinear partial differential equations involving crowd density and movement speed, which were analogous to the continuity equations of fluid. Nowadays, Hughes’s continuum theory for pedestrian flow has been widely adopted in the research on macroscopic models of pedestrians’ movement in emergency evacuation.(7−10) Many cities in China, such as Beijing, Shanghai, Chongqing, and Hong Kong, are densely populated. With the increase in the number and size of mass events, such as the Olympic Games, World Expo, World Football Cup, etc., and serious disastrous incidents, such as earthquakes and tsunamis, proper management of large-scale evacuation in densely populated cities will be of major concern to the Chinese government. Currently, a great deal of research on large-scale evacuations focuses on the macroscopic characteristics of crowd flow,(11−14) such as the clearance time or planning of rescue strategies.(15−17) However, not much research takes into consideration crowd density when integrating crowd movement and rescue strategies. In other words, the change of crowd densities relating to the real-time adjustment of rescue strategies is a critical issue that needs to be solved. In this article, we present a study concerning the safety problem of large-scale evacuation by: (1) analyzing a series of characteristic densities that affect the large-scale crowd flow (see Sections 2.1 and 2.2); and (2) applying queuing theory on the basis of characteristic densities of a crowd to simulate the moving characteristics of the crowd in respect to different rescue strategies (see Sections 3.1 and 3.2). Through the integration of crowd movement and rescue strategies, the efficiency of the rescue strategies, i.e., whether the rescue strategies are able to effectively maintain or improve the evacuation efficiency in large-scale evacuation, can be determined (see Section 3.3). This article aims to generate some macro, yet original, analysis on the risk of large-scale crowd evacuation from the perspective of the efficiency of the rescue strategies.
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2. CHARACTERISTIC DENSITIES OF LARGE-SCALE CROWD EVACUATION 2.1. Characteristic Densities Based on the Continuum Theory for Pedestrian Flow In the continuum theory, Hughes proposed three hypotheses for pedestrian flow in which only a single type of pedestrian is involved.(5) Hypothesis (1): The walking speed of pedestrians is determined solely by the density of the surrounding pedestrian flow and the behavioral characteristics of the pedestrians. Thus, for a single type of pedestrian, the velocity components are given by u = f (ρ)φˆ x v = f (ρ)φˆ y ,
(1)
where f (ρ) is the speed and φˆ x and φˆ y are the direction cosines of the motion. Hypothesis (2): The direction of the motion of any pedestrian is perpendicular to the potential, which is defined by the speed (u, v). Pedestrians have a common objective (termed as potential) to reach their common destination and they always exert every effort to change the existing potential, thus the motion of any pedestrian is in the direction perpendicular to the potential. Therefore, the expression of the direction cosines of the motion can be written as φˆ x = √
−(∂φ/∂ x)
φˆ y = √
−(∂φ/∂ y)
(∂φ/∂ x)2 +(∂φ/∂ y)2
(2) . 2
(∂φ/∂ x)2 +(∂φ/∂ y)
Hypothesis (3): Pedestrians seek to avoid extremely high densities and minimize their estimated travel time so that the distance between potentials must be proportional to the speed. In other words, the product of pedestrian travel time and speed (a function of the density) is at a minimum. Thus,
g(ρ) ·
1 u2 + v 2 = , (∂φ/∂ x)2 + (∂φ/∂ y)2
(3)
where g(ρ) represents the tempering behavior at high densities in accordance with this hypothesis and for most densities it is equal to one.
Based on the above three hypotheses and in accordance with the conservation equation of continu+ ∂ρu + ∂ρv = 0, the final motion ous pedestrians ∂ρ ∂t ∂x ∂y equations can be derived as: ⎧ ∂ρ ∂ ∂ ∂φ ∂φ ⎪ + ρg(ρ) f 2 (ρ) + ρg(ρ) f 2 (ρ) =0 − ⎪ ⎨ ∂t ∂ x ∂x ∂y ∂y , 1 ⎪ g(ρ) · f (ρ) = ⎪
⎩ 2 2 ∂φ ∂ x + ∂φ ∂ y (4)
Fig. 1. The relationship between crowd density and crowd flow rate before crowd motion stagnates.
which can be adopted for describing the flow situation of different types of crowds while under congestion. As stated before, crowd speed is intensely related to crowd density. Fig. 1 shows the relationship between crowd density and crowd flow rate. When crowd density is zero, there is no flow. When crowd density is quite low, the crowd can move freely with a maximum speed, which in general can be viewed as the free walking speed. When crowd density increases to a certain value ρtran , crowd speed begins to decrease but crowd flow continues to increase. When crowd density continues to increase to a certain critical value ρcrit , crowd flow reaches the maximum state and begins to decrease with the further increase of crowd density until the density reaches the value of ρmax , when crowd flow decreases to zero. Here, ρmax is viewed as the maximum crowd density under a normal motion situation. When crowd density keeps increasing, the crowd has already been in a state of squeezing stagnation. Thus, the continuum theory for the flow of pedestrians will not be suitable for analyzing the motion characteristics of the crowds when crowd density is higher than ρmax . Accordingly, the continuum theory for the flow of pedestrians will only be used to analyze characteristic crowd densities before crowd motion stagnates and these characteristic densities then will be used in the subsequent queuing simulation. For the crowd that flows normally, previous work found some empirical formulas in which crowd speed can be expressed as the function of crowd density, f1 (ρ) = A− Bρ 1 2
f2 (ρ) = A ρtran ρ /
ρtran ρcrit f3 (ρ) = A ρmax − ρcrit
ρ ≤ ρtran .(18)
(5)
ρtran ≤ ρ ≤ ρcrit .(5) (6) 1/2
(ρmax − ρ)1/2 ρ
ρcrit ≤ ρ ≤ ρmax . 5
(7)
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In these formulas, A represents the highly free walking speed and B is a constant whose value is related with the age, stature, weight, and behavioral character of the crowd. Taking into account the general condition of the crowd, we set: A = 1.4 m/s
B = 0.25 m3 (p · s) ρtran = 0.8 p m2 ρcrit = A 2B = 2.8 p m2 ρmax = A B = 5.6 p m2 . Recently, a number of empirical studies that can provide the relationship between crowd speed and crowd density were performed. Ando et al.(19) and Togawa(20) found that an individual in the crowd could have a comfortable walking speed when crowd density was less than 0.8 p/m2 . Older(21) proposed that the comfortable walking speed (free walking speed) was 1.4 m/s when crowd density was low, but this speed would rapidly decrease with the increase in crowd density. Through experimental study, Polus et al.(22) found that when crowd density was less than 0.1 p/m2 , crowd speed was around 1.3 m/s. Predtechenskii and Milinski(23) indicated that the range of crowd speed was from the free walking speed at the density that was less than 0.2 p/m2 to 0.27 m/s at the density of 4.5 p/m2 . They referred to the situation in which crowd density was 4 p/m2 or more as “stagnation,” but they also stated that even when the density was 7.4 p/m2 , or even a little more, some slow movement was still possible. Lo et al.(24) adopted a lattice-gas model to study the relationship of crowd density and crowd moving speed ranging from 1.4 m/s at a density less than 0.75 p/m2 and approaching 0.1 m/s at a density greater than 4.2 p/m2 . Based on these results, it can be seen that the free walking speed of the crowd is about 1.4 m/s and maximum crowd density at which the crowd movement is nearly stagnated is 4–7 p/m2 . Therefore, the values of ρtran , ρcrit , and ρmax adopted in this article are essentially in accordance with those empirical results proposed in previous studies. In addition, when crowd density is more than ρcrit , crowd flow has already begun to decrease. At this moment, a tendency of squeezing in the crowd has indeed emerged. Previous work indicated that when crowd density was more than 4 p/m2 a kind of pressure wave would be formed in the crowd.(5) In this circumstance, we use ρh = 4 p/m2 as the characteristic density that represents
the emergence of the pressure wave inside a crowd in a large-scale evacuation. 2.2. Maximum Bearing Density of a Large-Scale Crowd In a large-scale evacuation when crowd density is quite high, crowd movement will nearly stagnate and, subsequently, the inner pressure of the crowd will go up with the increase of crowd density. Once the inner pressure of the crowd rises to a certain value, people in the crowd may be asphyxiated by interperson forces, i.e., some people may be seriously pressed and cannot breathe normally. Modern medical studies have indicated that, under normal conditions, lung capacity (the maximum gas volume breathed into the lung through deep breathing) of a healthy adult is about 4–5 L and the maximum bearing pressure of the chest is 20–30 kPa.(25) The value of the maximum pressure is related to an individual’s characteristics, such as age, gender, and physique. Crowd density can reflect how people are squeezed in a certain space and usually can be indicated by the number of people distributed on unit area, something mainly determined by an individual’s physical size within the crowd. In general, maximum crowd density is 7–8 p/m2 , reaching up to 13–15 p/m2 in extreme situations. In current crowd evacuation research, an individual’s maximum physical size is determined by body thickness, dp , and shoulder breadth, bp .(26) To simplify the calculation, usually the individual’s body is abstracted into an ellipse or a rectangle zone, thus the area occupied by the individual can be obtained, π (8) Ellipse area : Se = bp dp . 4 Rectangle area : Ss = bp dp .
(9)
Using the size of Chinese people in a crowd as an example, Table I shows body thickness and shoulder breadth of normal adults in China (Male: 18–60 years old; Female: 18–55 years old), thus the average value of dp and bp is 0.24 m and 0.45 m, respectively. Therefore, the minimum average area occupied by a Chinese individual is 0.11 m2 and the maximum bearing density is approximately 9 p/m2 . Similarly, the maximum bearing densities of the crowd in different countries or areas are also calculated, as shown in Table II. This table demonstrates that due to racial diversity the physical size of a crowd can be quite different; thus the maximum bearing
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Percentile Body thickness Shoulder breadth
Male Female Male Female
1
5
10
50
90
95
99
159 176 383 347
170 186 398 363
176 191 405 371
199 212 431 397
230 237 460 428
239 245 469 438
260 261 486 458
Table II. Physical Size of a Crowd in Different Countries or Areas and the Calculated Maximum Bearing Densities Countries or Areas England America Japan India Average
Body Thickness (mm)28 Male Female Male Female Male Female Male Female
Shoulder Breadth (mm)28
285 295 280 295 230 235 235 255 263.75
510 435 515 470 475 425 455 390 459.38
density under an extremely squeezed situation is 8–11 p/m2 , with an average value of approximately 9 p/m2 . As a result, in the following queuing simulation we use ρbear = 9 p/m2 as the maximum bearing density of a large-scale crowd.
3. THE EFFECTIVENESS OF RESCUE STRATEGIES IN A LARGE-SCALE EVACUATION BASED ON CHARACTERISTIC DENSITIES Herd behavior is expected to be serious in a large-scale evacuation. Once substantial stress has built up among the evacuees, and they are all simultaneously moving toward an escape route, panic may arise and the uncoordinated movement of the evacuees may lead to jamming, pushing, crushing, and trampling. This will seriously affect the evacuation as well as rescue efficiency. Typically, rescue guidance plays a significant role in a large-scale evacuation, providing useful information to evacuees, which can alleviate the crowd’s anxiety or panic and help to improve the evacuation efficiency through the reasonable arrangement of various kinds of rescue measures (such as guiding the crowd’s evacuation direction). To illustrate the proposed approach in this article, rescue strategy merely refers to the number of
Ellipse Area (m2 )
Rectangle Area (m2 )
Maximum Bearing Densities (p/m2 )
0.114 0.101 0.113 0.109 0.086 0.078 0.084 0.078 0.095
0.145 0.128 0.144 0.139 0.109 0.1 0.107 0.099 0.122
7.7 8.7 7.8 8.1 10.3 11.2 10.5 11.3 9.4
evacuation routes designed or adjusted under different crowd densities and the corresponding evaluation refers to the assessment of the efficiency of rescue guidance. To simplify the analysis, we suppose that when the crowd is able to move, people can pass in an orderly manner through evacuation passageways in lines without panic. As a result, this study applies the classic queuing theory to analyze the effectiveness of rescue strategies in large-scale evacuation under different characteristic densities. In queuing theory, the crowd’s arrival intensity is an important indicator, which in this study is reflected by crowd density because it is a main factor that affects the queuing. The higher crowd density is, the more people will use the passageways and the arrival intensity will be higher. To simplify the calculation, the crowd’s arrival intensity is considered to be proportional to crowd density.
3.1. Establishment of a Queuing Model for Infinite Crowd Flow Crossing a Bridge To illustrate the idea of this article, a scenario of infinite crowd flow crossing a bridge in lines is assumed. In this situation, the bridge in question has a certain width and at the bridge entrance, some passageways can be set up to facilitate the crowd
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moving on the bridge in an orderly fashion. The large-scale crowd can reach safe destinations through this bridge and the crowd with different densities will cause different arrival intensity in front of the bridge entrance. The main hypotheses of this queuing model are as follows: (1) There are n queuing teams (windows) organized by the rescue guidance at the bridge entrance. (2) The crowd flow is infinite and the individuals are mutually independent. (3) The crowd flow arriving at the entrance of the bridge follows the Poisson distribution with the arrival intensity λ. (4) The crowd has no preference for any of the passageways (i.e., the windows). (5) The first-come-first-serve principle is applied and all individuals can shift freely among the queues and always move to shorter queues. No one will leave due to the queues being too long. (6) The time taken by everyone to cross the bridge follows the exponential distribution and the average crossing time is 1.5 unit time (1 unit time is set as 60 seconds).
3.2. Results and Discussion 3.2.1. Evacuation Efficiency of the Crowds with Different Densities Under Different Rescue Strategies In general, the average waiting time and number of waiting agents are two main indicators to evaluate the efficiency of the system. Fig. 2 shows the influence of the number of passageways on the average waiting time and number of waiting agents at the bridge entrance under different crowd densities. It can be seen that with the increase of crowd density more passageways will be needed. Moreover, under a certain number of passageways, the higher the crowd density, the greater the average waiting time and number of waiting agents. These results correspond with common sense.
Based on the above hypotheses, the formulas of the system efficiency indicators for the infinite crowd flow crossing the bridge can be established as follows: (1) Crowd’s arrival intensity λ: to simplify the calculation, it is considered that λ ∝ ρ. (2) Average crossing time of every individual: t¯ , which fits the exponential distribution. (3) Service capability of the passageways (windows): μ = 1t¯ . (4) Service intensity of the system, i.e., the proportion of the system’s service time in a unit λ . time: θ = nμ (5) Free probability of the system: P0 = n i
θ n+1 ]−1 . [( θi! ) + n!(n−θ ) i=0
(6) Average number of waiting agents in the sysθ n+1 P0 tem: L = n·n!(1− θ 2. ) n
(7) Average waiting time: W = Lλ . (8) Average sojourn time: W0 = W + t¯ .
Fig. 2. The relationship between the number of passageways and the average waiting time/number of agents under different crowd densities.
Risk of Large-Scale Evacuation When crowd density is less than 5 p/m2 , the adjustment of rescue strategies, namely, the increase of passageways, can noticeably affect the average waiting time and number of waiting agents. Taking the crowd density of 4 p/m2 (when the pressure wave begins to emerge in the crowd), for example, when the number of passageways is seven, the average number of waiting agents is 50 people and the average waiting time is about 12.2 unit times. If the rescue guidance set up one more passageway, the average number of waiting agents will decrease to 10 people and the average waiting time will be reduced to 2.6 unit times. If the rescue guidance continues to set up more passageways, the decrease of the average number of waiting agents and waiting time will be inconspicuous. Furthermore, it should be pointed out that the initial point of every curve in Fig. 2 is not under the condition of the same number of passageways. This is because when the number of passageways is less than seven, still using crowd density of 4 p/m2 as an example, no flow can be simulated using the queuing model. In other words, under crowd density of 4 p/m2 , the queuing theory will be valid only if the number of passageways is seven or more. Therefore, the evacuation efficiency curve for the crowd density of 4 p/m2 begins at seven passageways. Similarly are other curves for different crowd densities. Actually, this phenomenon can also be deduced mathematically according to the formulas of the system efficiency indicators established above. The stationary probability condition of a queuing system should satisfy 0 ≤ θ < 1, otherwise the queue length will be infinite. As a result, the requirement of n for a valid flow should satisfy n > λ · t¯ , which is the inner reason why the queuing model fails to be valid when the number of passageways is under a certain threshold. It can be indicated that the higher the crowd density, the larger the minimum number of passageways for effective evacuation will be. If the rescue guidance fails to ensure such minimum need of passageways at the beginning of an evacuation, the possibility of the crowd falling into a situation of squeeze or stagnation is quite high. Because the minimum number of passageways can ensure that the crowd is evacuated effectively, it is still necessary to discover the optimal number of passageways to promote evacuation with optimal efficiency. We have assumed that people can orderly pass through the evacuation passageways in lines without panic, thereby sometimes eliminating the possibility of casualties. Therefore, the economic cost
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Fig. 3. The evacuation cost benefit of different crowd densities.
of rescue strategies becomes the main consideration. To illustrate the definition of “optimal evacuation efficiency” in this study, we adopt a concept of “cost benefit” that is generally used in the economic field. The cost benefit is a relative index, that involves comparing the total expected cost of each option or decision against the total expected benefits, to see whether the benefits outweigh the costs, and by how much.(29) Supposing the cost of setting up one passageway is constant, we define the “evacuation cost benefit” as follows: p , CB = n where p is the percentage of reduced average waiting time (or number of waiting agents) and n is the increased number of passageways. Fig. 3 shows the evacuation cost-benefit curves (based on the waiting time) of different crowd densities when the rescue strategies increase from one to nine passageways. When one more passageway is increased from the minimum need, the evacuation cost benefit is the largest. As a result, with Figs. 2 and 3, it is possible to discover the optimal number of passageways for different crowd densities. Fig. 4 shows the comparison of average waiting time of crowds with different densities under the condition of minimum and optimal number of passageways. It can be seen that when the number of passageways is designed to an optimal value, the average waiting time of crowds with different densities all noticeably decreases, i.e., the evacuation efficiency significantly increases. Such improvement is more noticeable for the crowd with lower density, whose evacuation efficiency can be increased up to
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Fig. 5. The risk axle of crowd density in a large-scale evacuation.
Fig. 4. The average waiting time of crowds with different densities under the condition of minimum and optimal number of passageways.
more than 80%. When the evacuation is valid, the positive effect of the increase of passageways on the improvement of evacuation efficiency is also closely related to crowd density. The lower the crowd density, the more noticeable the increase of evacuation efficiency is through such adjustments of rescue strategies. In addition, it is found that when crowd density is more than 5 p/m2 , no matter how many passageways are designed, no crowd flow can be formed, i.e., the queuing model becomes invalid under crowd density that is more than 5 p/m2 . As previously discussed, in the mathematical model, when ρmax equals 5.6 p/m2 crowd flow rate approaches 0. Therefore, it is reasonable to consider that, in practice, when crowd density is more than 5 p/m2 , crowd flow rate may have already decreased to an extremely low level and evacuees are nearly stagnated. At this time, once some critical incidents have occurred in the crowd, such as a person falling down, or the external environment changed unpredictably, such as a blocked escape passageway, the crowd would easily collapse. The critical invalid density obtained by our model, 5 p/m2 , is reasonable to some extent and can be considered as a criterion to evaluate whether the rescue strategies are able to play a positive role in a real evacuation. 3.2.2. Analysis of the Evacuation Risk on the Basis of the Effectiveness of Rescue Strategies Based on the above results and discussion, a “risk axle of crowd density” is proposed to determine the efficiency of the rescue strategies (such as the design of the number of passageways) in a large-scale evac-
uation. As shown in Fig. 5, with the stretch of the axle to the right, the risk of large-scale crowd evacuation increases. In this figure, the ρcrit represents critical density derived by the theoretical model, namely, crowd density at which crowd flow rate reaches the maximum value, demonstrating that evacuation efficiency will be the best at this time. The ρh is crowd density at which the pressure wave will emerge inside the crowd, as indicated in Hughes’s continuum theory for the flow of pedestrians. The ρe is the crowd density at which the rescue strategy becomes invalid based on the simulation via continuum theory and the queuing model. The ρmax is the maximum crowd density derived by the theoretical model, namely, the density at which the crowd speed is approaching zero. The ρbear is the criterion of whether there will be squeezing or an asphyxia accident inside the crowd under the congestion situation, i.e., the maximum bearing density. The three regions on the risk axle of crowd density represent three different kinds of crowd flow specific to the rescue strategies in a large-scale evacuation. • Effective flow means that the evacuation efficiency can be improved by some interventions from the rescue guidance, such as the real-time adjustment of evacuation passageways. In this region, the risk of a large-scale evacuation is under effective control. • Noneffective flow means that the crowd cannot move and the common interventions can no longer play positive role, thus making accidents such as a squeezing situation highly probable. In this region, the risk of a large-scale evacuation is extremely high. • Critical zone represents a buffer in which the rescue strategies will develop from valid to invalid, which is in fact the deviation of the theoretical model and queuing simulation. If the rescue strategies can be strengthened urgently before or just inside this critical zone, the subsequently invalid stage would perhaps be inevitable. Therefore, it can be viewed that in this critical region the risk of a large-scale evacuation is quite high and requires some urgent
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Fig. 6. The evacuation risk distribution for different crowd densities under different rescue strategies.
Fig. 7. The three regions of a large-scale evacuation risk.
rescue guidance to relieve the pressure caused by the high level of crowd aggregation. If we further consider that the evacuation risk (risk of evacuation failure) is between 0 and 1, consequently the value of evacuation risk for different crowd densities under different rescue strategies (represented by the number of passageways) can be given out with some qualitative comparison, meaning according to the corresponding change of the average waiting time. Fig. 6 shows the three-dimensional distribution of the evacuation risk, in which the highrisk region, with a risk value of more than 0.7, and the low-risk region, with a risk value of less than 0.3, are differentiated clearly. The platform in the high-risk region occurs when the crowd density is more than
5.6 p/m2 , therefore, there is no effective flow; thus, the evacuation risk is set as 1 for all crowd densities more than 5.6 p/m2 under any rescue strategy. Furthermore, to illustrate the three regions of the risk axle of crowd density, the projection of the evacuation risk is prepared on the basis of Fig. 6, as shown in Fig. 7. In the effective flow region, with crowd density less than 5 p/m2 , the improvement of rescue strategies, namely, the increase of passageways, is able to reduce the evacuation risk to quite a low level, i.e., below 0.3. In the noneffective flow, with crowd density greater than 5.6 p/m2 , no matter how the passageways increase, evacuation risk remains at a very high level, i.e., above 0.8. While in the critical zone, it is hard to reduce the evacuation risk to as low a level as that in the effective flow region.
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Nevertheless, it is still possible to reduce the evacuation risk to some extent, i.e., around 0.5, and in reverse, it is also very possible to cause the evacuation transform into the invalid stage if no urgent rescue guidance is activated.
system, such as crowd psychology and the feature of the disaster, which will be integrated with the rescue strategies. In addition, multiple simulation methods will be applied to analyze the risk of a large-scale evacuation.
4. CONCLUSION
ACKNOWLEDGMENTS
In this article, the macro features of crowd flow and rescue strategies are integrated to illustrate the risk of a large-scale evacuation. A series of characteristic densities that affect large-scale crowd flow, as well as the maximum bearing density when the crowd is extremely congested, is analyzed. Based on these characteristic densities, queuing theory is applied to simulate the moving characteristics of the crowd and an original model for the assessment of crowd congestion and recovery is presented. The effectiveness of rescue strategies was found to be strongly related to crowd density. The higher the crowd density, the larger the minimum number of passageways for effective evacuation will be. Such minimum requirement of effective evacuation under a certain value of crowd density significantly indicates that the rescue guidance should avoid crowd congestion—and even stagnation—at the beginning of the evacuation via setting up the prerequisite number of passageways. When crowd density is less than 5 p/m2 , the adjustment of rescue strategies, namely, increasing the number of passageways to a certain optimal value, can noticeably increase the evacuation efficiency, and such improvement is more conspicuous for lower crowd density. In addition, the queuing model fails to be valid when crowd density reaches a certain threshold, 5 p/m2 , which is recommended as a criterion to evaluate whether the rescue strategies are able to play a positive role in a real evacuation. Furthermore, based on the effects of typical densities on the rescue strategies, a “risk axle of crowd density” is proposed to determine the efficiency of the rescue strategies in a large-scale evacuation, i.e., whether the rescue strategies are able to effectively control the evacuation risk. To illustrate the idea of the risk axle of crowd density, a threedimensional distribution of the evacuation risk is established through some rational hypotheses for the value of evacuation risk. This article aims to make some macro, but original, analysis of the risk of a large-scale crowd evacuation from the perspective of the efficiency of the rescue strategies. Our future work will further consider a variety of uncertainties in a large-scale evacuation
This work was supported by the National Natural Science Foundation of China (91024025), the General Research Grant of the Research Grant Council, HKSAR No. CityU 118708, and the Key Technologies R&D Program of China during the 12th Five-Year Plan Period (Grant. No. 2012BAK13B01). The authors deeply appreciate their support. NOMENCLATURE ρ
f (ρ) u, v φˆ x φˆ y φ
g(ρ)
A B
ρtran ρcrit ρmax ρh ρe ρbear
dp bp
The density of the pedestrian flow, which is the number of individuals located within unit area at a given time t and location(x, y) The crowd walking speed, which is a function of the density The two velocity components on the x and y axis, respectively The direction cosine of the motion on the x axis The direction cosine of the motion on the y axis The potential of the motion; pedestrians have a common objective (termed as potential) to reach their common destination and they always exert every effort to change the existing potential A nondimensional factor of pedestrians to allow for discomfort at very high densities, which represents the tempering behavior at high densities; for most densities it is equal to one The highly free walking speed, m/s A constant and its value is related with the age, stature, weight, and behavioral character of the crowd, m3 /(p · s) The crowd density at which the crowd speed begins to decrease but the crowd flow still increases, p/m2 (persons/m2 ) The crowd density at which the crowd flow reaches the maximum state and begins to decrease with the further increase of the crowd density, p/m2 The crowd density at which the crowd flow decreases to zero; it can be viewed as the maximum crowd density under the situation of normal motion, p/m2 The crowd density at which the pressure wave begins to emerge inside the crowd, p/m2 The crowd density at which the rescue strategy becomes invalid, p/m2 The maximum bearing density, which is the criterion of whether there will be squeezing or asphyxia accident inside the crowd under the congestion situation, p/m2 The body thickness of the individual, m The shoulder breadth of the individual, m
Risk of Large-Scale Evacuation
Se Ss
n λ t¯ μ
θ
P0 L W W0 p n CB
The area occupied by the individual when the individual body is abstracted into an ellipse zone, m2 The area occupied by the individual when the individual body is abstracted into a rectangle zone, m2 The number of queuing teams (windows) organized by the rescue guidance at the entrance The crowd’s arrival intensity, p/s The average crossing time of every individual, which follows the exponential distribution The service capability of the passageways, which can be understood as the average number of people who have passed the passageways and left in a unit time, p/s The service intensity of the system, i.e., the proportion of the system’s service time in a unit time, which is a nondimensional ratio The free probability of the system The average number of waiting agents in the system The average waiting time The average sojourn time The percentage of reduced average waiting time after increasing the number of passageways The increased number of passageways The “evacuation cost benefit (CB),” which is a relative index comparing the total expected cost of increasing a certain number of passageways against the total expected benefits
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