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A heterogeneous lattice gas model for simulating pedestrian evacuation
Physica A 391 (2012) 582–592
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A heterogeneous lattice gas model for simulating pedestrian evacuation Xiwei Guo, Jianqiao Chen ∗ , Yaochen Zheng, Junhong Wei Department of Mechanics, Huazhong University of Science and Technology, Wuhan 430074, China Hubei Key Laboratory for Engineering Structural Analysis and Safety Assessment, Wuhan 430074, China
article
info
Article history: Received 17 December 2010 Received in revised form 15 July 2011 Available online 24 August 2011 Keywords: Heterogeneous lattice gas model Local population density Pedestrian evacuation Interactions between pedestrians Critical force of injury
abstract Based on the cellular automata method (CA model) and the mobile lattice gas model (MLG model), we have developed a heterogeneous lattice gas model for simulating pedestrian evacuation processes in an emergency. A local population density concept is introduced first. The update rule in the new model depends on the local population density and the exit crowded degree factor. The drift D, which is one of the key parameters influencing the evacuation process, is allowed to change according to the local population density of the pedestrians. Interactions including attraction, repulsion, and friction between every two pedestrians and those between a pedestrian and the building wall are described by a nonlinear function of the corresponding distance, and the repulsion forces increase sharply as the distances get small. A critical force of injury is introduced into the model, and its effects on the evacuation process are investigated. The model proposed has heterogeneous features as compared to the MLG model or the basic CA model. Numerical examples show that the model proposed can capture the basic features of pedestrian evacuation, such as clogging and arching phenomena. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Emergency evacuation in public places is a major public safety problem which has attracted much attention of researchers during the last decade. Emergency evacuation is a complex process. Due to the limits of undertaking such dangerous experiments and the absence of data from real evacuations, computer simulation is the main approach for studying emergency evacuation at present. Two kinds of model, i.e. discrete models or continuous models, are widely used for simulating pedestrian evacuation. A mobile lattice gas model (MLG model) is one of the discrete models developed from the cellular automata method. In the MLG model, the lattice positions around each pedestrian are mobile, depending upon the pedestrian’s real movement. The interaction force between every two pedestrians and that between a pedestrian and the building wall are determined by the distance between them and the pedestrians’ moving step sizes. The other main approaches in pedestrian dynamics studies are outlined below. Helbing [1,2] developed a social force model (SF model) to simulate panic situations in crowd evacuation. Zheng et al. [3] presented a model constructed by the combination of the SF model and a neural network, to simulate collective behaviors of pedestrians in various situations. Paris and Dorso [4] modified the SF model to simulate the evacuation of pedestrians from a room in a panic situation. The social force (SF) model is a continuous model for simulating pedestrian evacuation. In an SF model, for N pedestrians, the requirement of solving a couple of differential equations for each pedestrian implies that the calculation load is of order O(N 2 ), which is a heavy computation burden for large N. By contrast, a CA model can greatly improve the computational
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Corresponding author at: Department of Mechanics, Huazhong University of Science and Technology, Wuhan 430074, China. E-mail address: [email protected] (X. Guo).
0378-4371/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2011.07.055
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Fig. 1. Possible movements and probabilities for a pedestrian.
efficiency in simulating large-scale pedestrian evacuation since the calculation amount is of order O(N ). Several CA models have been proposed to study the exit dynamics of pedestrian in a room [5], simulate pedestrian counterflow in a channel [6], and simulate occupant evacuation from a fire room [7,8]. A two-dimensional CA model was developed to study the exit dynamics of occupant evacuation and the effect of obstacles on the evacuation process [9,10]. In addition, there are some CA models which have been used to study the friction between pedestrians [11–13]. The lattice gas model (LG model) is a special form of the CA model. Muramatsu et al. [14] have studied the counterflow of pedestrians within an underpass by using the LG model. It has been found that the dynamic jamming transition occurs at a critical density. Tajima [15] adopted a partial LG model to study the interaction between crowd flow in a T-shaped channel and crowd flow in a turnoff. In [16–18], the LG model was used to simulate the processes of pedestrian evacuation from a room full of smoke or in the dark. Weng et al. [19] modified the LG model by adopting behaviorism for a mobile robot. Jiang et al. [20] extended the LG model to allow large maximum velocity to simulate pedestrian behaviors. The interaction between pedestrians is an essential issue of crowd evacuation. Kirchner et al. [21] applied a bionics approach to describe the interaction between pedestrians using ideas of chemotaxis. Colin [22] introduced an explicit and individual-based physical force into a model proposed in Ref. [21] to improve the model quality. Based on the LG model, Song et al. [23] proposed a new evacuation model entitled the multi-grid model that uses a finer lattice. In the model, one pedestrian occupies multiple grids instead of only one, and overlapping of pedestrians is allowed. The model can simulate crowd evacuation more realistically. Guo et al. [24] developed a mobile lattice gas model (MLG model). Based on the MLG model, we have developed a heterogeneous lattice gas model to simulate pedestrian evacuation. The update rule in our model depends on the local population density and the exit crowded degree factor. A critical force of injury concept is introduced into the model, and the influence of possible casualty during evacuation is considered. The drift D (attraction strength to the exits) is allowed to change according to the local population density. Several numerical examples are worked out to display the characteristics of the model. 2. The model descriptions First, we will briefly survey the cellular automata model (CA model) and the mobile lattice gas model (MLG model) based on which the new evacuation model is developed. The CA model first proposed by John von Neumann and Ulam is defined in the cellular space constituted by discrete and finite state cells. The CA model is a dynamic system which is evolved according to a local rule on the discrete time dimension. The space of a room is divided into square grids when we use the CA model to simulate an evacuation process. These grids, like cells, have two states: to be free or to be occupied by one pedestrian. In each time step, the pedestrian can move into the free grids around it or stay in the current grid. Similar to the CA model, each grid in the classic lattice gas (LG) model has the same size, and each pedestrian just occupies a grid at each time step. The mobile lattice gas (MLG) model is one of the developed LG model. In the MLG model, the lattices around each pedestrian are dynamic, and are closely related to the pedestrian’s real movement. Fig. 1 (a) represents pedestrian n who is now at the center Cn and probably moves to one of the eight neighboring lattices (marked by black) at the next time step [24]. The center of each neighboring lattice is at a distance Sn from the center Cn ; that is, the pedestrian can move forward Sn in every time step. The transition probabilities from Cn to its neighboring grids are denoted by Pi,j and shown in Fig. 1 (b). 2.1. Basic updating equations for the new model Based on the CA model and MLG model, we have developed a new model entitled the ‘‘heterogeneous lattice gas model’’, considering the effect of the heterogeneous updating rule on crowd evacuation. Referring to the multiple-grid model and
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the MLG model [23,24], the transition probability Pi,j in our model is given by the following expression:
Pi,j = NIi,j δi,j
Di,j +
−
fi,mj
+
m
−
fi,Wj
− (1 − ψi,j )FWn ,
(1)
W
where δi,j = 0 if at least one pedestrian is in the grid at direction (i, j) and δi,j = 1 otherwise. The inertia factor Ii,j represents the will of a pedestrian for keeping his or her previous movement direction. The value of Ii∑ ,j is greater than 1 in the previous moving direction while Ii,j = 1 for the others. N is a normalization factor to ensure that (i,j) Pi,j = 1. After assigning the drift D that denotes the moving strength of pedestrian from the current position to the exit, we have [23] Di,j = D cos θi,j ,
(2) d0n,1 ,
d1n,1 ,
d1n,0
where Di,j is one of the three projections (along in Fig. 1(a)) of D, and θi,j is the angle between Di,j and D (the ∑ m ∑ W direction of the exit). m fi,j and W fi,j represent the interactions that pedestrian m and wall W exert on pedestrian n along direction (i, j). In crowd evacuation, injuries may occur in some cases which will exert a significant impact on the evacuation process. The last term in the Eq. (1) represents this effect. Here, ψi,j = 0 if at least one pedestrian is injured in direction (i, j) and ψi,j = 1 otherwise. The functions fi,mj , fi,Wj , and FWn are described in the next section. 2.2. Introduction of inhomogeneity by partly changing the drift D In our model, the drift D represents the strength with which a pedestrian rushes to the exit. We adjust the value of D according to the local population density in an evacuation process. The local population density (Jp ) is defined by the following equation: e Jp = , (3) π R2 where R is the radius of a circle whose center is the target pedestrian (in our numerical examples, R = 2 m), and e denotes the number of pedestrians in the circle. According to Ref. [25], Nelson and MacLennan analyzed experimental data and suggested that a pedestrian would move freely when the population density (Jp ) is less than 0.54 person/m2 and would find it hard to move when the population density is more than 3.8 person/m2 . Hence we propose a criterion to adjust the value of D according to the value of Jp . For Jp ≥ 3 person/m2 , we let the strength of the pedestrian rushing to the exit (D) increase or decrease, and keep those of the others unchanged. Increasing D means that the intention of the pedestrian moving towards the exit is very strong, although he or she may get injured or make the exit more crowded. On the other hand, if the local interaction between pedestrians becomes stronger and dominates the behaviors of pedestrians, the drift D may decrease, implying that the attraction of the exit is consequently weakened. 2.3. Interaction forces between pedestrians In Eq. (1), fi,mj and fi,Wj have the same meaning as in Ref. [24]. Let Fmn be an action that pedestrian m exerts on the pedestrian n along direction (0, 1). We have (Fig. 2) f0m,1 = −Fmn ,
f0m,−1 = Fmn ,
f−m1,0 = f1m,0 = −µFmn ,
√
f−m1,−1 = f1m,−1 = 0.5 2Fmn
√
f−m1,1 = f1m,1 = −0.5 2µFmn .
(4)
Here µ denotes the friction factor. The above relations still hold when pedestrian m is replaced by a wall, i.e., f0W,1 = −FWn , f0W,−1 = FWn , and so on. We propose the following expressions for the interaction forces Fmn and FWn : Fmn =
γ (Sn + Sm ) 2 k
FWn =
lmn
0,
γ Sn 2 k
0,
lWn
− k,
lmn ≤ γ (Sn + Sm )
(5)
lmn ≥ γ (Sn + Sm )
− k,
lWn ≤ γ Sn
(6)
lWn ≥ γ Sn ,
where lmn is the distance between pedestrian m and pedestrian n (see Fig. 2), and lWn is the distance between pedestrian n and the building wall. Sn and Sm denote the moving step sizes of pedestrian n and m, respectively. From Eqs. (5) and (6), we know that, when lmn and lWn decrease, Fmn and FWn increase sharply. As lmn or lWn approaches a very small value, the interaction force will become extremely large. In the proposed model Eq. (1), the value of parameter ψi,j is determined by the following criterion:
ψi,j =
1 0
F < Fo F ≥ Fo ,
(7)
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Fig. 2. Action that pedestrian m exerts on pedestrian n.
in which F denotes the scalar summation of forces a pedestrian in direction (i, j) sustained from his/her surrounding pedestrians or building wall, and Fo is the critical force of injury. For ψi,j = 0, the pedestrian in the direction (i, j) is injured. And then we deal with the injured pedestrian as a wall whose action exerted on a pedestrian is FWn ; that is, the injured pedestrian stops moving and remains in the position where he or she became injured. 2.4. The solution of overlapping confliction In our model, a pedestrian is represented by a circle whose radius is 0.2 m. First, we use the rules of the CA model to initialize pedestrians in a room to avoid overlapping confliction. The position of different pedestrians may however overlap at the next time step (different pedestrians may choose the same position to move to at the next time step). To overcome this confliction, the relative probability concept is adopted [26], i.e., pedestrians with larger relative probability choose the position to move to while others remain in their current position at the next time step. The relative probability of pedestrians 1 and 2 are calculated as p1 =
Pi1,j Pi1,j + Pi2,j
p2 =
Pi2,j Pi1,j + Pi2,j
,
(8)
where Pi1,j and Pi2,j respectively represent the probabilities of two pedestrians choosing the same position at the next time step. When there is more pedestrian confliction, the calculation of their relative probability is similar to Eq. (8). Once the pedestrian who has the largest relative probability is determined, he or she will move with probability Pi,j and the others conflicting with him or her will remain in their current positions. Thus all pedestrians change their positions synchronously. Up to now, we can know all pedestrians’ moving directions at the next time step by the method mentioned above. Then we can get the update positions of all pedestrians synchronously by Eq. (9): Pnt +1 = Pnt + sn αn
Pnt +1 =
xtn+1 ytn+1
[
]
Pnt =
[ t] xn ytn
αn =
[ α]
xn , yαn
(9)
where Pnt is the position of pedestrian n at time step t, Pnt +1 is the position of pedestrian n at time step t + 1, and αn denotes the most likely moving direction of pedestrian n at the next time step. In summary, our proposed model should match reality more closely since interactions between every two pedestrians and that between a pedestrian and the building wall are described by a nonlinear function of the corresponding distance, and a critical force of injury is introduced into the model. In addition, adjusting the value of D according to the local population density makes the model more flexible in describing the balance of the will of pedestrians to escape and the effect of local interaction between pedestrians. 3. Numerical simulation results 3.1. Influences of the critical force of injury on the evacuation process Referring to Ref. [2], the physical interactions in a jammed crowd add up and may cause dangerous pressures up to 4450 Nm−1 . Based on the model Eqs. (1)–(7), we first study the influence of the critical force of injury by setting Fo = ηMg ,
η = 1, 2, 3, . . . ,
where M denotes the mass of a pedestrian, and g represents acceleration due to gravity.
(10)
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Fig. 3. Influence of the critical force on the evacuation process.
We consider 200 pedestrians attempting to evacuate from a 16 m × 16 m room with one 2 m width door whose center is at the position (16, 8). First, 100 pedestrians with moving step size 0.3 m and the other half pedestrians with moving step size 0.4 m are randomly distributed in the room. The model parameters are as follows: the inertia I = 1.3, the drift D = 1.0, the friction coefficient µ = 0.3, k = 1, γ = 0.5, the length of the time step is 0.3 s, M = 60 kg, and the value of g is 9.8 m/s2 . In this simulation, the drift D is constantly unchanged regardless of the local density and the exit crowded degree factor. Fig. 3 shows the influence of the critical force of injury Fo on the evacuation process. We find that the evacuation time and the number of injured pedestrians decrease sharply as Fo increases from 1Mg to 4Mg. For 4Mg ≤ Fo ≤ 8Mg, both the average time and the maximal time reach a stable level, and the number of injuries is predicted to be 1–3 pedestrians. For Fo ≥ 9Mg, no injury occurs, which is similar to the situation in Ref. [24]. We conducted five simulations for each specific scenario and the averaged values are taken as the representative results of that scenario, so the number of injured persons may have a value other than an integer. Next, we set the critical force to be Fo = 4Mg, and investigate the evacuation process features. Four typical stages of pedestrian dynamics are shown in Fig. 4. Black circles and squares represent two classes of pedestrians with moving step 0.3 m and 0.4 m respectively, while ellipses represent the injured pedestrians. With the developing of the process, the crowd becomes increasingly dense near the exit, forming an arch. As more and more pedestrians leave the room, the arching disappears. The average evacuation time and the maximal evacuation time are about 17 s and 87 s for this scenario. Fig. 5 depicts the evacuation frequency during different time ranges. We can see that the evacuation is mostly concentrated in the range 10–30 s for the two cases Fo = 4Mg and Fo = ∞. As compared to the case Fo = ∞, more evacuation time is required for the former case. 3.2. The effect of local population density on the evacuation process In the following simulations, the basic scenario and related parameters are the same as those in Section 3.1, but casualties are not considered (Fo = ∞). All the pedestrians in the room are divided into two classes: half of the pedestrians’ moving
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Fig. 4. Four stages of pedestrian evacuation process considering casualties. (a) Initial, at time step t = 1. (b) At time step t = 15. (c) At time step t = 30. (d) At time step t = 55.
Fig. 5. Pedestrians evacuated during different time ranges.
step size is 0.3 m while the other half have step size is 0.4 m. The results of five simulations for each case are averaged to obtain the representative values of that case. First, we investigate the influence of the drift D on evacuation processes without considering the influence of local population density. We have simulated the following cases: (1) the drift D for all pedestrians is set to be 0.8, 1.0, and 1.2, respectively; (2) the drift D for half the pedestrians (randomly chosen) is set to be 0.8 while that for the other half is set to
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Fig. 6. Influences of the drift D on the average evacuation time.
be 1.0; (3) the drift D for half the pedestrians is set to be 1.0 while that for the other half is set to be 1.2. Fig. 6 shows that with the increase of D the average evacuation time decreases, implying that a clearer objective (with greater D) helps to make the evacuation efficient. Then, the influence of local population density on the evacuation process is investigated by introducing a heterogeneous updating rule. When the local population density defined in Eq. (3) is greater than 3 person/m2 , we let the strength of the target pedestrian rushing to the exit (D) increase or decrease while those of the others remain unchanged. Fig. 7 shows the results. The solid circle represents the homogeneous case D = 1.0, while others represent the cases of the heterogeneous updating rule. For example, the solid square corresponds to the case that, when Jp > 3 person/m2 , the target pedestrian will change his/her D from 1.0 to 0.6. The main findings are as follows. (i) An interesting finding from Fig. 7 is that the evacuation times of all the heterogeneous updating models are greater than that of the homogeneous case (D = 1.0). For the heterogeneous cases, the increase in D represents that the pedestrian has stronger will to move to the exit, probably creating serious clogging which will certainly block the evacuation process. In contrast, a reduction in D means that the interaction between pedestrians becomes stronger and the target pedestrian’s movement is limited, also resulting in a larger evacuation time. (ii) For the smaller population case (N < 270), increasing D can help pedestrians to evacuate more efficiently, while for the case of larger population (N ≥ 270), a strong will to move to the exit (increase in D) may create serious clogging near the exit, leading to a longer evacuation process. For the heterogeneous cases, we conclude that increasing of D is helpful to the evacuation efficiency with smaller population while decreasing of D is good for evacuation with a larger population. (iii) The effect of local population density on the maximal evacuation time is similar to the average time. We conclude that the local population density has a key influence on the evacuation process, and the heterogeneous updating which depends on the local density results in larger evacuation time as compared to the homogeneous case. Two other cases are simulated to test the robustness of our heterogeneous model. One is that we set D = 1.2 for the homogeneous case and it changes to 0.9 or 1.5 according to the heterogeneous updating rule described above. The other case is that we set D = 0.8 for the homogeneous case and adjust the value of D to 0.5 or 1.1 according to the heterogeneous updating rule. The same features as shown in Fig. 7 were observed for the test cases. We point out that heterogeneous updating can be regarded as a kind of modeling of a slight panic situation which will worsen the evacuation efficiency. 3.3. Evacuation from a two-door room through fixed target exits or the optimal exits We first simulate evacuation processes of pedestrians who have their own target exits. We set the critical force as Fo = ∞; the other parameters are the same as those in Section 3.1. There are two exits in adjacent sides of the room. At the beginning, we let 100 pedestrians, which are randomly selected from the total 200 pedestrians, evacuate through the exit at position (8, 0) while the other half evacuate through the exit at position (16, 8). Fig. 8 shows the typical stages of an evacuation process in which pedestrians evacuate through fixed exits. The circle denotes the pedestrians with a moving step size of 0.3 m while the square denotes the pedestrians with a moving step size of 0.4 m. From Fig. 8(a) and (b), we know that, during the cross-flow evacuation process, pedestrians gradually stratify, and a stripe appears. The appearance and disappearance of arching near the exit can be observed in Fig. 8(c) and (d). This shows that the model proposed can display the typical characteristics of pedestrian evacuation.
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Fig. 7. Influences of the heterogeneous updating on the evacuation process: (a) The average evacuation time. (b) The maximal evacuation time.
(a) Initial, at time step t = 1.
(b) At time step t = 20.
(c) At time step t = 35.
(d) At time step t = 45. Fig. 8. Typical stages of evacuation process for the fixed target exits case.
When pedestrians leave a room with multiple exits, they may choose the optimal exit to evacuate through. To simulate this situation, we introduce an exit crowded degree factor (J) as J =
8×q
πW2
.
(11)
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Fig. 9. Influence of heterogeneous updating on the evacuation based on the optimal exit criterion. (a) The average evacuation time. (b) The maximal evacuation time. Table 1 Calculated maximal evacuation time (s) (two exits in adjacent sides). (a) Fixed exits case. (b) Optimal exits case. Pedestrian number D D D D D D
(a)
(b)
= 1.0 = 1.0 → 0.8 = 1.0 → 1.2 = 1.0 = 1.0 → 0.8 = 1.0 → 1.2
100
200
300
400
28.2 36.6 33.0 25.8 29.4 26.4
53.6 62.4 59.4 48.8 58.2 52.2
91.2 96.6 103.2 85.2 92.4 93.6
122.4 133.2 141.6 114.6 124.8 132.0
In Eq. (11), W is the width of the exit, and q represents the number of pedestrians in the half circle of diameter W . Referring to Ref. [27], we define l(u) = 1 −
(V − 1)ru V ∑
(12)
ru
u=1
ρ ( u) = 1 −
(V − 1)Ju V ∑
(13)
Ju
u=1
exit (u) = l(u) + ρ (u) .
(14)
In Eqs. (12)–(14), V denotes the number of exits, ru is the distance between a pedestrian and exit u, and l(u) represents the contribution of the distance for choosing exit u. Ju expresses the crowded degree of exit u, and ρ(u) represents the contribution of the exit crowded degree factor in choosing exit u. A pedestrian will choose exit u with larger probability when exit (u) is greater. Fig. 9 shows the simulation results with two exits in adjacent sides. The scenario of the simulation is the same as in the fixed target exits case, but pedestrians can choose the optimal exit according to the rules presented above. We also see that the curve in Fig. 9 is similar to that in Fig. 7. Variations in D during the evacuation, no matter whether increasing or decreasing, will create worse effects on the evacuation. Table 1 compares the results of the two cases, the fixed target exits case and the optimal exits case. It can be found that the maximal evacuation time is larger for the former case no matter whether or not we consider the local population density. Choosing the optimal exit is helpful in shortening the maximal evacuation time. For the case of two doors on opposite sides, similar results are obtained, i.e., choosing the optimal exit to evacuate through can shorten the maximal evacuation time. 3.4. Influences of the exit configuration on the evacuation The position of the exits are set as follow: (a) at position (16, 8); (b) at position (16, 6) and position (16, 10); (c) at position (16, 8) and position (8, 0); (d) at position (16, 8) and position (0, 8). Pedestrians can choose the optimal exit to
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Fig. 10. Average evacuation time for different exit configurations.
evacuate through according to the rules mentioned in Section 3.3. The same scenarios as in Section 3.1 are simulated without considering injury. Fig. 10 gives the average evacuation time under different configurations of exits. The evacuation time for the one-exit case is the greatest among the four cases. For two exits, different configurations result in different evacuation times. When the two exits are located on opposite sides, the evacuation is the most efficient, while the simulation when two exits are located in the same side leads to the worst efficiency. Blockage in public boundary is observed in the simulations of evacuating from a room with two exits in one side or in adjacent sides, making the interaction between pedestrians stronger, and plays a certain role of obstacles. This will not happen when the two exits are located on opposite sides of a room. 4. Conclusions A heterogeneous lattice gas model is proposed to study pedestrians’ evacuation processes. The critical force of injury, local population density, and exit crowded degree factor are considered to establish the heterogeneous updating rule, making the simulation more reasonable. From the simulation results we obtain the following conclusions. (1) The critical force of injury Fo has a remarkable influence on the pedestrian evacuation process. For Fo < 4Mg, the average evacuation time largely depends on Fo . With decreasing of Fo , the number of the injured pedestrians and the average evacuation time increase rapidly. For 4Mg ≤ Fo ≤ 8Mg, the evacuation times reach a stable level, and Fo plays a weak role on the evacuation. For Fo ≥ 9Mg, no injury occurs, and the effect of Fo can be neglected. (2) The impact of local population density on the evacuation process is investigated by constructing and applying the heterogeneous updating rule. When the local density reaches a certain value, the target pedestrian can hardly move. Some pedestrians may stay still while the others may forcibly be moved following their neighbors’ step. There is no doubt that the former case (decreasing D) will increase the evacuation time. The latter case (increasing D) may cause clogging, which also increases the evacuation time. This is consistent with the simulation results in our model. As compared to homogeneous updating, heterogeneous updating can be regarded as a kind of panic modeling, which affects the efficiency of evacuation. (3) For the heterogeneous updating cases, we can find some interesting results. For the smaller population case, the stronger intensity of pedestrians moving toward the exit can make the evacuation efficient. For the larger population case, however, pedestrians should properly reduce the intensity of moving toward the exit to shorten the evacuation time. (4) For two exits, different configurations result in different evacuation times. The evacuation time of the situation with exits on opposite side is the least, while that with exits in one side is the highest. In the cases investigated, pedestrians choosing the optimal exit to evacuate through can improve the evacuation efficiency. In actual pedestrian evacuation, the pedestrians’ behaviors and the emergency are much more complex, and the evacuation model should be integrated into an emergency simulation model (such as a fire simulation model). Further work is being undertaken to simulate pedestrian evacuation under a fire emergency more realistically. Acknowledgments We thank two anonymous reviewers for their constructive comments and the Editor for his hard work. The work described in this paper is supported by the National Natural Science Foundation of China (Grant No. 50978113) which is greatly appreciated.
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A heterogeneous lattice gas model for simulating pedestrian evacuation Xiwei Guo, Jianqiao Chen ∗ , Yaochen Zheng, Junhong Wei Department of Mechanics, Huazhong University of Science and Technology, Wuhan 430074, China Hubei Key Laboratory for Engineering Structural Analysis and Safety Assessment, Wuhan 430074, China
article
info
Article history: Received 17 December 2010 Received in revised form 15 July 2011 Available online 24 August 2011 Keywords: Heterogeneous lattice gas model Local population density Pedestrian evacuation Interactions between pedestrians Critical force of injury
abstract Based on the cellular automata method (CA model) and the mobile lattice gas model (MLG model), we have developed a heterogeneous lattice gas model for simulating pedestrian evacuation processes in an emergency. A local population density concept is introduced first. The update rule in the new model depends on the local population density and the exit crowded degree factor. The drift D, which is one of the key parameters influencing the evacuation process, is allowed to change according to the local population density of the pedestrians. Interactions including attraction, repulsion, and friction between every two pedestrians and those between a pedestrian and the building wall are described by a nonlinear function of the corresponding distance, and the repulsion forces increase sharply as the distances get small. A critical force of injury is introduced into the model, and its effects on the evacuation process are investigated. The model proposed has heterogeneous features as compared to the MLG model or the basic CA model. Numerical examples show that the model proposed can capture the basic features of pedestrian evacuation, such as clogging and arching phenomena. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Emergency evacuation in public places is a major public safety problem which has attracted much attention of researchers during the last decade. Emergency evacuation is a complex process. Due to the limits of undertaking such dangerous experiments and the absence of data from real evacuations, computer simulation is the main approach for studying emergency evacuation at present. Two kinds of model, i.e. discrete models or continuous models, are widely used for simulating pedestrian evacuation. A mobile lattice gas model (MLG model) is one of the discrete models developed from the cellular automata method. In the MLG model, the lattice positions around each pedestrian are mobile, depending upon the pedestrian’s real movement. The interaction force between every two pedestrians and that between a pedestrian and the building wall are determined by the distance between them and the pedestrians’ moving step sizes. The other main approaches in pedestrian dynamics studies are outlined below. Helbing [1,2] developed a social force model (SF model) to simulate panic situations in crowd evacuation. Zheng et al. [3] presented a model constructed by the combination of the SF model and a neural network, to simulate collective behaviors of pedestrians in various situations. Paris and Dorso [4] modified the SF model to simulate the evacuation of pedestrians from a room in a panic situation. The social force (SF) model is a continuous model for simulating pedestrian evacuation. In an SF model, for N pedestrians, the requirement of solving a couple of differential equations for each pedestrian implies that the calculation load is of order O(N 2 ), which is a heavy computation burden for large N. By contrast, a CA model can greatly improve the computational
∗
Corresponding author at: Department of Mechanics, Huazhong University of Science and Technology, Wuhan 430074, China. E-mail address: [email protected] (X. Guo).
0378-4371/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2011.07.055
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Fig. 1. Possible movements and probabilities for a pedestrian.
efficiency in simulating large-scale pedestrian evacuation since the calculation amount is of order O(N ). Several CA models have been proposed to study the exit dynamics of pedestrian in a room [5], simulate pedestrian counterflow in a channel [6], and simulate occupant evacuation from a fire room [7,8]. A two-dimensional CA model was developed to study the exit dynamics of occupant evacuation and the effect of obstacles on the evacuation process [9,10]. In addition, there are some CA models which have been used to study the friction between pedestrians [11–13]. The lattice gas model (LG model) is a special form of the CA model. Muramatsu et al. [14] have studied the counterflow of pedestrians within an underpass by using the LG model. It has been found that the dynamic jamming transition occurs at a critical density. Tajima [15] adopted a partial LG model to study the interaction between crowd flow in a T-shaped channel and crowd flow in a turnoff. In [16–18], the LG model was used to simulate the processes of pedestrian evacuation from a room full of smoke or in the dark. Weng et al. [19] modified the LG model by adopting behaviorism for a mobile robot. Jiang et al. [20] extended the LG model to allow large maximum velocity to simulate pedestrian behaviors. The interaction between pedestrians is an essential issue of crowd evacuation. Kirchner et al. [21] applied a bionics approach to describe the interaction between pedestrians using ideas of chemotaxis. Colin [22] introduced an explicit and individual-based physical force into a model proposed in Ref. [21] to improve the model quality. Based on the LG model, Song et al. [23] proposed a new evacuation model entitled the multi-grid model that uses a finer lattice. In the model, one pedestrian occupies multiple grids instead of only one, and overlapping of pedestrians is allowed. The model can simulate crowd evacuation more realistically. Guo et al. [24] developed a mobile lattice gas model (MLG model). Based on the MLG model, we have developed a heterogeneous lattice gas model to simulate pedestrian evacuation. The update rule in our model depends on the local population density and the exit crowded degree factor. A critical force of injury concept is introduced into the model, and the influence of possible casualty during evacuation is considered. The drift D (attraction strength to the exits) is allowed to change according to the local population density. Several numerical examples are worked out to display the characteristics of the model. 2. The model descriptions First, we will briefly survey the cellular automata model (CA model) and the mobile lattice gas model (MLG model) based on which the new evacuation model is developed. The CA model first proposed by John von Neumann and Ulam is defined in the cellular space constituted by discrete and finite state cells. The CA model is a dynamic system which is evolved according to a local rule on the discrete time dimension. The space of a room is divided into square grids when we use the CA model to simulate an evacuation process. These grids, like cells, have two states: to be free or to be occupied by one pedestrian. In each time step, the pedestrian can move into the free grids around it or stay in the current grid. Similar to the CA model, each grid in the classic lattice gas (LG) model has the same size, and each pedestrian just occupies a grid at each time step. The mobile lattice gas (MLG) model is one of the developed LG model. In the MLG model, the lattices around each pedestrian are dynamic, and are closely related to the pedestrian’s real movement. Fig. 1 (a) represents pedestrian n who is now at the center Cn and probably moves to one of the eight neighboring lattices (marked by black) at the next time step [24]. The center of each neighboring lattice is at a distance Sn from the center Cn ; that is, the pedestrian can move forward Sn in every time step. The transition probabilities from Cn to its neighboring grids are denoted by Pi,j and shown in Fig. 1 (b). 2.1. Basic updating equations for the new model Based on the CA model and MLG model, we have developed a new model entitled the ‘‘heterogeneous lattice gas model’’, considering the effect of the heterogeneous updating rule on crowd evacuation. Referring to the multiple-grid model and
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the MLG model [23,24], the transition probability Pi,j in our model is given by the following expression:
Pi,j = NIi,j δi,j
Di,j +
−
fi,mj
+
m
−
fi,Wj
− (1 − ψi,j )FWn ,
(1)
W
where δi,j = 0 if at least one pedestrian is in the grid at direction (i, j) and δi,j = 1 otherwise. The inertia factor Ii,j represents the will of a pedestrian for keeping his or her previous movement direction. The value of Ii∑ ,j is greater than 1 in the previous moving direction while Ii,j = 1 for the others. N is a normalization factor to ensure that (i,j) Pi,j = 1. After assigning the drift D that denotes the moving strength of pedestrian from the current position to the exit, we have [23] Di,j = D cos θi,j ,
(2) d0n,1 ,
d1n,1 ,
d1n,0
where Di,j is one of the three projections (along in Fig. 1(a)) of D, and θi,j is the angle between Di,j and D (the ∑ m ∑ W direction of the exit). m fi,j and W fi,j represent the interactions that pedestrian m and wall W exert on pedestrian n along direction (i, j). In crowd evacuation, injuries may occur in some cases which will exert a significant impact on the evacuation process. The last term in the Eq. (1) represents this effect. Here, ψi,j = 0 if at least one pedestrian is injured in direction (i, j) and ψi,j = 1 otherwise. The functions fi,mj , fi,Wj , and FWn are described in the next section. 2.2. Introduction of inhomogeneity by partly changing the drift D In our model, the drift D represents the strength with which a pedestrian rushes to the exit. We adjust the value of D according to the local population density in an evacuation process. The local population density (Jp ) is defined by the following equation: e Jp = , (3) π R2 where R is the radius of a circle whose center is the target pedestrian (in our numerical examples, R = 2 m), and e denotes the number of pedestrians in the circle. According to Ref. [25], Nelson and MacLennan analyzed experimental data and suggested that a pedestrian would move freely when the population density (Jp ) is less than 0.54 person/m2 and would find it hard to move when the population density is more than 3.8 person/m2 . Hence we propose a criterion to adjust the value of D according to the value of Jp . For Jp ≥ 3 person/m2 , we let the strength of the pedestrian rushing to the exit (D) increase or decrease, and keep those of the others unchanged. Increasing D means that the intention of the pedestrian moving towards the exit is very strong, although he or she may get injured or make the exit more crowded. On the other hand, if the local interaction between pedestrians becomes stronger and dominates the behaviors of pedestrians, the drift D may decrease, implying that the attraction of the exit is consequently weakened. 2.3. Interaction forces between pedestrians In Eq. (1), fi,mj and fi,Wj have the same meaning as in Ref. [24]. Let Fmn be an action that pedestrian m exerts on the pedestrian n along direction (0, 1). We have (Fig. 2) f0m,1 = −Fmn ,
f0m,−1 = Fmn ,
f−m1,0 = f1m,0 = −µFmn ,
√
f−m1,−1 = f1m,−1 = 0.5 2Fmn
√
f−m1,1 = f1m,1 = −0.5 2µFmn .
(4)
Here µ denotes the friction factor. The above relations still hold when pedestrian m is replaced by a wall, i.e., f0W,1 = −FWn , f0W,−1 = FWn , and so on. We propose the following expressions for the interaction forces Fmn and FWn : Fmn =
γ (Sn + Sm ) 2 k
FWn =
lmn
0,
γ Sn 2 k
0,
lWn
− k,
lmn ≤ γ (Sn + Sm )
(5)
lmn ≥ γ (Sn + Sm )
− k,
lWn ≤ γ Sn
(6)
lWn ≥ γ Sn ,
where lmn is the distance between pedestrian m and pedestrian n (see Fig. 2), and lWn is the distance between pedestrian n and the building wall. Sn and Sm denote the moving step sizes of pedestrian n and m, respectively. From Eqs. (5) and (6), we know that, when lmn and lWn decrease, Fmn and FWn increase sharply. As lmn or lWn approaches a very small value, the interaction force will become extremely large. In the proposed model Eq. (1), the value of parameter ψi,j is determined by the following criterion:
ψi,j =
1 0
F < Fo F ≥ Fo ,
(7)
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Fig. 2. Action that pedestrian m exerts on pedestrian n.
in which F denotes the scalar summation of forces a pedestrian in direction (i, j) sustained from his/her surrounding pedestrians or building wall, and Fo is the critical force of injury. For ψi,j = 0, the pedestrian in the direction (i, j) is injured. And then we deal with the injured pedestrian as a wall whose action exerted on a pedestrian is FWn ; that is, the injured pedestrian stops moving and remains in the position where he or she became injured. 2.4. The solution of overlapping confliction In our model, a pedestrian is represented by a circle whose radius is 0.2 m. First, we use the rules of the CA model to initialize pedestrians in a room to avoid overlapping confliction. The position of different pedestrians may however overlap at the next time step (different pedestrians may choose the same position to move to at the next time step). To overcome this confliction, the relative probability concept is adopted [26], i.e., pedestrians with larger relative probability choose the position to move to while others remain in their current position at the next time step. The relative probability of pedestrians 1 and 2 are calculated as p1 =
Pi1,j Pi1,j + Pi2,j
p2 =
Pi2,j Pi1,j + Pi2,j
,
(8)
where Pi1,j and Pi2,j respectively represent the probabilities of two pedestrians choosing the same position at the next time step. When there is more pedestrian confliction, the calculation of their relative probability is similar to Eq. (8). Once the pedestrian who has the largest relative probability is determined, he or she will move with probability Pi,j and the others conflicting with him or her will remain in their current positions. Thus all pedestrians change their positions synchronously. Up to now, we can know all pedestrians’ moving directions at the next time step by the method mentioned above. Then we can get the update positions of all pedestrians synchronously by Eq. (9): Pnt +1 = Pnt + sn αn
Pnt +1 =
xtn+1 ytn+1
[
]
Pnt =
[ t] xn ytn
αn =
[ α]
xn , yαn
(9)
where Pnt is the position of pedestrian n at time step t, Pnt +1 is the position of pedestrian n at time step t + 1, and αn denotes the most likely moving direction of pedestrian n at the next time step. In summary, our proposed model should match reality more closely since interactions between every two pedestrians and that between a pedestrian and the building wall are described by a nonlinear function of the corresponding distance, and a critical force of injury is introduced into the model. In addition, adjusting the value of D according to the local population density makes the model more flexible in describing the balance of the will of pedestrians to escape and the effect of local interaction between pedestrians. 3. Numerical simulation results 3.1. Influences of the critical force of injury on the evacuation process Referring to Ref. [2], the physical interactions in a jammed crowd add up and may cause dangerous pressures up to 4450 Nm−1 . Based on the model Eqs. (1)–(7), we first study the influence of the critical force of injury by setting Fo = ηMg ,
η = 1, 2, 3, . . . ,
where M denotes the mass of a pedestrian, and g represents acceleration due to gravity.
(10)
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Fig. 3. Influence of the critical force on the evacuation process.
We consider 200 pedestrians attempting to evacuate from a 16 m × 16 m room with one 2 m width door whose center is at the position (16, 8). First, 100 pedestrians with moving step size 0.3 m and the other half pedestrians with moving step size 0.4 m are randomly distributed in the room. The model parameters are as follows: the inertia I = 1.3, the drift D = 1.0, the friction coefficient µ = 0.3, k = 1, γ = 0.5, the length of the time step is 0.3 s, M = 60 kg, and the value of g is 9.8 m/s2 . In this simulation, the drift D is constantly unchanged regardless of the local density and the exit crowded degree factor. Fig. 3 shows the influence of the critical force of injury Fo on the evacuation process. We find that the evacuation time and the number of injured pedestrians decrease sharply as Fo increases from 1Mg to 4Mg. For 4Mg ≤ Fo ≤ 8Mg, both the average time and the maximal time reach a stable level, and the number of injuries is predicted to be 1–3 pedestrians. For Fo ≥ 9Mg, no injury occurs, which is similar to the situation in Ref. [24]. We conducted five simulations for each specific scenario and the averaged values are taken as the representative results of that scenario, so the number of injured persons may have a value other than an integer. Next, we set the critical force to be Fo = 4Mg, and investigate the evacuation process features. Four typical stages of pedestrian dynamics are shown in Fig. 4. Black circles and squares represent two classes of pedestrians with moving step 0.3 m and 0.4 m respectively, while ellipses represent the injured pedestrians. With the developing of the process, the crowd becomes increasingly dense near the exit, forming an arch. As more and more pedestrians leave the room, the arching disappears. The average evacuation time and the maximal evacuation time are about 17 s and 87 s for this scenario. Fig. 5 depicts the evacuation frequency during different time ranges. We can see that the evacuation is mostly concentrated in the range 10–30 s for the two cases Fo = 4Mg and Fo = ∞. As compared to the case Fo = ∞, more evacuation time is required for the former case. 3.2. The effect of local population density on the evacuation process In the following simulations, the basic scenario and related parameters are the same as those in Section 3.1, but casualties are not considered (Fo = ∞). All the pedestrians in the room are divided into two classes: half of the pedestrians’ moving
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Fig. 4. Four stages of pedestrian evacuation process considering casualties. (a) Initial, at time step t = 1. (b) At time step t = 15. (c) At time step t = 30. (d) At time step t = 55.
Fig. 5. Pedestrians evacuated during different time ranges.
step size is 0.3 m while the other half have step size is 0.4 m. The results of five simulations for each case are averaged to obtain the representative values of that case. First, we investigate the influence of the drift D on evacuation processes without considering the influence of local population density. We have simulated the following cases: (1) the drift D for all pedestrians is set to be 0.8, 1.0, and 1.2, respectively; (2) the drift D for half the pedestrians (randomly chosen) is set to be 0.8 while that for the other half is set to
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Fig. 6. Influences of the drift D on the average evacuation time.
be 1.0; (3) the drift D for half the pedestrians is set to be 1.0 while that for the other half is set to be 1.2. Fig. 6 shows that with the increase of D the average evacuation time decreases, implying that a clearer objective (with greater D) helps to make the evacuation efficient. Then, the influence of local population density on the evacuation process is investigated by introducing a heterogeneous updating rule. When the local population density defined in Eq. (3) is greater than 3 person/m2 , we let the strength of the target pedestrian rushing to the exit (D) increase or decrease while those of the others remain unchanged. Fig. 7 shows the results. The solid circle represents the homogeneous case D = 1.0, while others represent the cases of the heterogeneous updating rule. For example, the solid square corresponds to the case that, when Jp > 3 person/m2 , the target pedestrian will change his/her D from 1.0 to 0.6. The main findings are as follows. (i) An interesting finding from Fig. 7 is that the evacuation times of all the heterogeneous updating models are greater than that of the homogeneous case (D = 1.0). For the heterogeneous cases, the increase in D represents that the pedestrian has stronger will to move to the exit, probably creating serious clogging which will certainly block the evacuation process. In contrast, a reduction in D means that the interaction between pedestrians becomes stronger and the target pedestrian’s movement is limited, also resulting in a larger evacuation time. (ii) For the smaller population case (N < 270), increasing D can help pedestrians to evacuate more efficiently, while for the case of larger population (N ≥ 270), a strong will to move to the exit (increase in D) may create serious clogging near the exit, leading to a longer evacuation process. For the heterogeneous cases, we conclude that increasing of D is helpful to the evacuation efficiency with smaller population while decreasing of D is good for evacuation with a larger population. (iii) The effect of local population density on the maximal evacuation time is similar to the average time. We conclude that the local population density has a key influence on the evacuation process, and the heterogeneous updating which depends on the local density results in larger evacuation time as compared to the homogeneous case. Two other cases are simulated to test the robustness of our heterogeneous model. One is that we set D = 1.2 for the homogeneous case and it changes to 0.9 or 1.5 according to the heterogeneous updating rule described above. The other case is that we set D = 0.8 for the homogeneous case and adjust the value of D to 0.5 or 1.1 according to the heterogeneous updating rule. The same features as shown in Fig. 7 were observed for the test cases. We point out that heterogeneous updating can be regarded as a kind of modeling of a slight panic situation which will worsen the evacuation efficiency. 3.3. Evacuation from a two-door room through fixed target exits or the optimal exits We first simulate evacuation processes of pedestrians who have their own target exits. We set the critical force as Fo = ∞; the other parameters are the same as those in Section 3.1. There are two exits in adjacent sides of the room. At the beginning, we let 100 pedestrians, which are randomly selected from the total 200 pedestrians, evacuate through the exit at position (8, 0) while the other half evacuate through the exit at position (16, 8). Fig. 8 shows the typical stages of an evacuation process in which pedestrians evacuate through fixed exits. The circle denotes the pedestrians with a moving step size of 0.3 m while the square denotes the pedestrians with a moving step size of 0.4 m. From Fig. 8(a) and (b), we know that, during the cross-flow evacuation process, pedestrians gradually stratify, and a stripe appears. The appearance and disappearance of arching near the exit can be observed in Fig. 8(c) and (d). This shows that the model proposed can display the typical characteristics of pedestrian evacuation.
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Fig. 7. Influences of the heterogeneous updating on the evacuation process: (a) The average evacuation time. (b) The maximal evacuation time.
(a) Initial, at time step t = 1.
(b) At time step t = 20.
(c) At time step t = 35.
(d) At time step t = 45. Fig. 8. Typical stages of evacuation process for the fixed target exits case.
When pedestrians leave a room with multiple exits, they may choose the optimal exit to evacuate through. To simulate this situation, we introduce an exit crowded degree factor (J) as J =
8×q
πW2
.
(11)
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Fig. 9. Influence of heterogeneous updating on the evacuation based on the optimal exit criterion. (a) The average evacuation time. (b) The maximal evacuation time. Table 1 Calculated maximal evacuation time (s) (two exits in adjacent sides). (a) Fixed exits case. (b) Optimal exits case. Pedestrian number D D D D D D
(a)
(b)
= 1.0 = 1.0 → 0.8 = 1.0 → 1.2 = 1.0 = 1.0 → 0.8 = 1.0 → 1.2
100
200
300
400
28.2 36.6 33.0 25.8 29.4 26.4
53.6 62.4 59.4 48.8 58.2 52.2
91.2 96.6 103.2 85.2 92.4 93.6
122.4 133.2 141.6 114.6 124.8 132.0
In Eq. (11), W is the width of the exit, and q represents the number of pedestrians in the half circle of diameter W . Referring to Ref. [27], we define l(u) = 1 −
(V − 1)ru V ∑
(12)
ru
u=1
ρ ( u) = 1 −
(V − 1)Ju V ∑
(13)
Ju
u=1
exit (u) = l(u) + ρ (u) .
(14)
In Eqs. (12)–(14), V denotes the number of exits, ru is the distance between a pedestrian and exit u, and l(u) represents the contribution of the distance for choosing exit u. Ju expresses the crowded degree of exit u, and ρ(u) represents the contribution of the exit crowded degree factor in choosing exit u. A pedestrian will choose exit u with larger probability when exit (u) is greater. Fig. 9 shows the simulation results with two exits in adjacent sides. The scenario of the simulation is the same as in the fixed target exits case, but pedestrians can choose the optimal exit according to the rules presented above. We also see that the curve in Fig. 9 is similar to that in Fig. 7. Variations in D during the evacuation, no matter whether increasing or decreasing, will create worse effects on the evacuation. Table 1 compares the results of the two cases, the fixed target exits case and the optimal exits case. It can be found that the maximal evacuation time is larger for the former case no matter whether or not we consider the local population density. Choosing the optimal exit is helpful in shortening the maximal evacuation time. For the case of two doors on opposite sides, similar results are obtained, i.e., choosing the optimal exit to evacuate through can shorten the maximal evacuation time. 3.4. Influences of the exit configuration on the evacuation The position of the exits are set as follow: (a) at position (16, 8); (b) at position (16, 6) and position (16, 10); (c) at position (16, 8) and position (8, 0); (d) at position (16, 8) and position (0, 8). Pedestrians can choose the optimal exit to
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Fig. 10. Average evacuation time for different exit configurations.
evacuate through according to the rules mentioned in Section 3.3. The same scenarios as in Section 3.1 are simulated without considering injury. Fig. 10 gives the average evacuation time under different configurations of exits. The evacuation time for the one-exit case is the greatest among the four cases. For two exits, different configurations result in different evacuation times. When the two exits are located on opposite sides, the evacuation is the most efficient, while the simulation when two exits are located in the same side leads to the worst efficiency. Blockage in public boundary is observed in the simulations of evacuating from a room with two exits in one side or in adjacent sides, making the interaction between pedestrians stronger, and plays a certain role of obstacles. This will not happen when the two exits are located on opposite sides of a room. 4. Conclusions A heterogeneous lattice gas model is proposed to study pedestrians’ evacuation processes. The critical force of injury, local population density, and exit crowded degree factor are considered to establish the heterogeneous updating rule, making the simulation more reasonable. From the simulation results we obtain the following conclusions. (1) The critical force of injury Fo has a remarkable influence on the pedestrian evacuation process. For Fo < 4Mg, the average evacuation time largely depends on Fo . With decreasing of Fo , the number of the injured pedestrians and the average evacuation time increase rapidly. For 4Mg ≤ Fo ≤ 8Mg, the evacuation times reach a stable level, and Fo plays a weak role on the evacuation. For Fo ≥ 9Mg, no injury occurs, and the effect of Fo can be neglected. (2) The impact of local population density on the evacuation process is investigated by constructing and applying the heterogeneous updating rule. When the local density reaches a certain value, the target pedestrian can hardly move. Some pedestrians may stay still while the others may forcibly be moved following their neighbors’ step. There is no doubt that the former case (decreasing D) will increase the evacuation time. The latter case (increasing D) may cause clogging, which also increases the evacuation time. This is consistent with the simulation results in our model. As compared to homogeneous updating, heterogeneous updating can be regarded as a kind of panic modeling, which affects the efficiency of evacuation. (3) For the heterogeneous updating cases, we can find some interesting results. For the smaller population case, the stronger intensity of pedestrians moving toward the exit can make the evacuation efficient. For the larger population case, however, pedestrians should properly reduce the intensity of moving toward the exit to shorten the evacuation time. (4) For two exits, different configurations result in different evacuation times. The evacuation time of the situation with exits on opposite side is the least, while that with exits in one side is the highest. In the cases investigated, pedestrians choosing the optimal exit to evacuate through can improve the evacuation efficiency. In actual pedestrian evacuation, the pedestrians’ behaviors and the emergency are much more complex, and the evacuation model should be integrated into an emergency simulation model (such as a fire simulation model). Further work is being undertaken to simulate pedestrian evacuation under a fire emergency more realistically. Acknowledgments We thank two anonymous reviewers for their constructive comments and the Editor for his hard work. The work described in this paper is supported by the National Natural Science Foundation of China (Grant No. 50978113) which is greatly appreciated.
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