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Necessity of guides in pedestrian emergency evacuation
Physica A 442 (2016) 397–408
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Physica A journal homepage: www.elsevier.com/locate/physa
Necessity of guides in pedestrian emergency evacuation Xiaoxia Yang a , Hairong Dong a,∗ , Xiuming Yao b , Xubin Sun b , Qianling Wang a , Min Zhou a a
State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, China
b
School of Electronic and Information Engineering, Beijing Jiaotong University, Beijing 100044, China
highlights • • • •
The influence of mass behavior on evacuation dynamics is investigated. An extended crowd model is proposed for the guided crowd. The effect of position of guide on evacuation is studied. The effect of the quantity of guides on evacuation is studied.
article
info
Article history: Received 5 May 2015 Received in revised form 21 July 2015 Available online 21 August 2015 Keywords: Social force model Guided crowd Uninformed pedestrians Desired direction Position of guide Quantity of guides
abstract The role of guide who is in charge of leading pedestrians to evacuate in the case of emergency plays a critical role for the uninformed people. This paper first investigates the influence of mass behavior on evacuation dynamics and mainly focuses on the guided evacuation dynamics. In the extended crowd model proposed in this paper, individualistic behavior, herding behavior and environment influence are all considered for pedestrians who are not informed by the guide. According to the simulation results, herding behavior makes more pedestrians evacuate from the room in the same period of time. Besides, guided crowd demonstrates the same behavior of group dynamics which is characterized by gathering, conflicts and balance. Moreover, simulation results indicate guides with appropriate initial positions and quantity are more conducive to evacuation under a moderate initial density of pedestrians. © 2015 Elsevier B.V. All rights reserved.
1. Introduction The efficiency of pedestrian evacuation is a key aspect of the safety performance evaluation in public places such as subway stations, airports, and stadiums. Appropriate evacuation measures developed in advance can play important roles in improving the evacuation efficiency when fire, explosion, earthquake and other emergencies occur. In 2008, more than two thousand students and teachers all realized the safe evacuation in 96 s when earthquake occurred in Zaosang middle school, Sichuan Province, China, since emergency evacuation drills were organized in each semester. A large number of facts have proven that the prevention beforehand works much better than a temporary response [1]. Moreover, the poor emergency warning system and unreasonable and temporary evacuation strategy are the root causes of catastrophe. Earthquake tsunami disaster having adverse impacts to more than 10 countries occurred in Indian Ocean in 2004, caused about 30 million deaths, and 14 billion dollars loss because of the poor communication of information. The establishment of
∗
Corresponding author. Tel.: +86 10 51685749; fax: +86 10 51685749. E-mail address: [email protected] (H. Dong).
http://dx.doi.org/10.1016/j.physa.2015.08.020 0378-4371/© 2015 Elsevier B.V. All rights reserved.
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the predictive evacuation system is therefore very important, and many scholars began to investigate pedestrian dynamics to better understand the evacuation procedure from the 1970s [2]. Generally, pedestrian models to predict pedestrians’ behaviors and dynamics in the normal and emergency situations are studied from the macroscopic and microscopic perspectives [3]. Macroscopic pedestrian model considers the pedestrian flow as a whole by using the flow density and velocity to describe its motion [4]. The most remarkable advantage of this model is to save the computation amount and enhance the real-time ability [5]. Fluid dynamics model as a typical macroscopic model treats the flow as the continuous liquid and is described based on the Navier–Stokes equation [6]. However, the macroscopic model neglects the detailed behaviors among pedestrians, and this is one reason why microscopic pedestrian model continues to emerge and has been widely applied in reality. Microscopic pedestrian model considers the interactions among pedestrians such as the cellular automata model [7–10], the social force model [11–13], and the agent based model [7]. Wang et al. [8] proposed a cellular automata model considering the game strategy to study the pedestrian evacuation in a hall. Frank and Dorso [9] studied the room evacuation in the presence of an obstacle based on the social force model and found ‘‘clever is not always better’’ effect. This paper chooses to use the microscopic social force model as the reference model which not only qualitatively represents self-organization behaviors such as lane formation [10] and strip formation [11] but also introduces the physical and psychology forces that make the model more realistic. During the major events in large public places, we always find some staffs wearing the clothes with bright colors to maintain order. These staffs who are familiar with the building structure may play guides when emergency occurs. Hou et al. [12] investigated the effect of the number and positions of guides on evacuation dynamics in rooms with limited visibility range. However, the walking rules set for pedestrians without following the guide do not consider the mass behavior which are not corresponding to reality. Yang et al. [13] developed the guided crowd model to study the crowd dynamics via modified social force model, but all pedestrians are assumed to have obtained the information from the guide, and they do not give the description for pedestrians who cannot get the information. This paper extends the model proposed by Yang et al. [13] to present the detailed behaviors not only for pedestrians who follow the guide but also for those who cannot be influenced by the guide. This contribution presents the dynamic behaviors of the guided crowd. It continues with the description of the extended crowd model in Section 2, where the mathematical models for the guide and all pedestrians whether are informed or not are developed. Section 3 gives the scenario setup, effects of mass behaviors in the cases of no guides, evacuation dynamic due to the guide, and the effect of position and quantity of guides on evacuation. After analyzing the simulation results, the key discoveries are reviewed and the future work is looked into in Section 4. 2. An extended crowd model This section mainly focuses on modeling of the guided crowd in a smoky room, where the normal pedestrians are not familiar with the surroundings except the guides who know exactly where the exit is. Generally, the vision field is limited in a smoky room, the pedestrians who locate in the influence field of the guide may follow the movement of the guide. However, for those who neither belong to the influence area of the guide nor the exit is visible, a random moving direction will be chosen. 2.1. Social force model The social force model is proposed by Helbing et al. [14], in which pedestrians are driven by the desired force, ⃗ f i0 ; the
interaction force between pedestrians i and j, ⃗ fij ; and the interaction force between pedestrian i and walls w , ⃗ fiw . This model is represented by mi
dv ⃗i (t ) dt
= ⃗f i0 +
⃗fij +
j(̸=i)
w
⃗fiw ,
(1)
where mi denotes the mass of pedestrian i, and v ⃗i (t ) denotes the actual walking velocity at time instant t. ⃗f i0 shows pedestrian’s willingness to reach the desired velocity: 0 ⃗0 ⃗f i0 = mi vi (t ) e i − v⃗i (t ) , τi
(2)
where vi0 denotes the desired speed, and ⃗ e 0i denotes the desired walking direction. τi is the adaptation time for adjusting the current velocity to the desired one. ⃗fij presents the pedestrian’s psychological tendency to move away from others and the physical force that occurs only when the distance between two pedestrians’ centers dij is less than the sum of these two pedestrians’ radii rij = ri + rj :
⃗fij = Ai exp rij − dij /Bi n⃗ij + kg rij − dij n⃗ij + κ g rij − dij 1vjit ⃗tij .
(3)
⃗ij = ( , ) = (⃗ri −⃗rj )/dij is the Here, Ai is the interaction strength and Bi is the magnitude of the repulsive interactions. n normalized vector pointing from pedestrian j to i, and ⃗ ri is the position of pedestrian i. k is the body compression coefficient, n1ij
n2ij
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Table 1 Parameters of social force model. Symbol
Meaning
Value
m r A B
Pedestrian mass Pedestrian radius Strength of social repulsive force Characteristic distance of social repulsive force Coefficient of sliding friction Body compression coefficient Pedestrian reaction time
80 kg 0.25 m 2000 N 0.08 m 240 000 kg m−1 s−1 120 000 kg s−2 0.5 s
κ k
τ
and κ is the coefficient of sliding friction. ⃗tij = (−n2ij , n1ij ) is the tangential direction, and 1vjit = v ⃗j − v⃗i · ⃗tij is the velocity difference along the tangential direction. The function g (x) is zero if pedestrians do not touch each other, i.e. (dij > rij ), otherwise it is equal to the argument x. ⃗fiw is modeled analogously:
⃗fiw = Ai exp[(ri − diw )/Bi ]⃗niw + kg (ri − diw )⃗niw + κ g (ri − diw )1vwt i⃗tiw .
(4)
The parameters of the original social force model are specified in Table 1 [14]. 2.2. The modeling for the guide and pedestrians following the guide Guides as an important part during the evacuation undertake the responsibilities of leading the pedestrians to evacuate as soon as possible. Yang et al. [13] introduced a modeling method for the guide who exactly knows the position of the exit and the pedestrians who follow this guide. This paper adopts the similar modeling method for the guide and followers. The motion of the guide who provides information to the surrounding pedestrians is described as mr
dv ⃗r (t ) dt
= ⃗f r0 +
⃗frj +
w
j
⃗fr w ,
⃗f r0 = mr vr (t )⃗e r − v⃗r (t ) . τr 0
0
(5)
Here, the guide’s desired direction ⃗ e 0r points to the exit. If pedestrians who are not familiar with the environment are in a guide’s influence area, they would tend to follow this guide. The corresponding motion equation is expressed as mi
dv ⃗i (t ) dt
= βi · ⃗f i0 +
j(̸=i)
⃗fij +
w
⃗fiw + ⃗fir + ⃗fir ,
0 0 ⃗f i0 = mi vi (t )⃗e i − v⃗i (t ) , τi ⃗ ⃗ r ⃗fir = mi · −b1 i (t ) − rr (t ) − b2 v⃗i (t ) − v⃗r (t ) , τi τi2
b1 , b2 > 0,
(6)
where βi ∈ [0, 1]. Pedestrians’ desired direction, ⃗ e 0i , points to the guide before they could see the exit. ⃗ fir is the interaction force between pedestrian i and the guide r. ⃗ fir is the navigational force provided by the guide. b1 and b2 are both positive constants reflecting the weights of navigational feedback. According to Ref. [15], the influence area of the guide with a certain radius Rinf is usually larger than the vision area with a certain radius Rvisual because of the phenomenon of flow with the stream. In this paper, it is assumed that all pedestrians in the guide’s influence area follow the guide in the evacuation. 2.3. The modeling for pedestrians uninformed by the guide In Ref. [13], it is assumed that all pedestrians can see the guide and are led by this guide. However, there is a limit of the vision field because of pedestrians themselves or the environmental uncertainty such as fire or smoke. It is, therefore, necessary to study the modeling method of those pedestrians who cannot see the guide or the exit. In this paper, the motion equation of pedestrians who have no information of the guide or exit is also based on the social force model given by Eq. (1), our work focuses on setting their desired directions. According to Ref. [12], pedestrians who cannot see the guide or exit are set to choose a direction randomly and follow this direction until they see a wall and then turn to a random direction again. This method, however, overlooks the influences of some factors such as the herding behavior which is very important during the evacuation. Helbing et al. [14] investigated the mass behavior in a smoky room, and
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Fig. 1. The diagram of the preferred individual desired direction.
pointed that each pedestrian may either choose an individually preferred desired direction ⃗ ei , or follow the mean walking direction ⟨⃗ e 0j (t )⟩i of pedestrians in his or her vision field, or have a mixture of both given in
⃗e 0i (t ) = Norm[(1 − pi ) · ⃗ei + pi · ⟨⃗e 0j (t )⟩i ].
(7)
Here, Norm(⃗ z ) = ∥⃗⃗zz ∥ . pi reflects the weight of the herding behavior, and 1 − pi reflects the weight of the individual behavior. Generally, the surrounding environment such as the noise can also affect the desired direction to some extent. This paper further introduces ξ⃗1 (t ) to reflect the influence of environment on pedestrians’ desired direction which is given by
⃗e 0i (t ) = Norm[χ1 · ⃗ei + χ2 · ⟨⃗e 0j (t )⟩i + χ3 · ξ⃗1 (t )],
(8)
where ξ⃗1 (t ) is a unit vector in a certain direction according to the actual situations, χ1 , χ2 and χ3 are positive constants between 0 and 1, and χ1 + χ2 + χ3 = 1. χ1 reflects the weight of the individualistic behavior, χ2 gives the weight of herding behavior, and χ3 is the weight of the influence of the surrounding environment. Generally, the selection of the preferred individual desired direction has some bases according to the visible walls and obstacles. With limited practical observations, we find when the wall belongs to a pedestrian’s vision field, this pedestrian prefers walking along the wall or moving away from the wall, but not directly heading towards the wall in order to find the exit as soon as possible. In Fig. 1, pedestrian i can only see a section of wall from a to b as the limit of vision field, which causes the sets of γ and η to come into being. It is worth noting that the set γ contains the pedestrian’s preferred individual desired directions along the wall or away from the wall, while the set η defines the corresponding directions heading towards the wall. This paper, therefore, considers the effect of the walls on the decision of the preferred individual desired direction once the wall is in pedestrians’ vision field, and proposes the preferred individual desired walking direction ⃗ ei choosing from the set γ with a large probability p1 or from the set η with a small probability p2 shown in Fig. 1. Note that the different areas A1 , A2 , A3 and A7 are developed according to the different walls in a pedestrian’s vision field. For example, pedestrians in area A1 cannot see any wall, and pedestrians in area A3 can only see the left wall, while pedestrians in area A7 can see both the left wall and the lower wall. The method of the determination of ⃗ ei in different areas, however, is the same. In Fig. 1, D cos ψ = R wall , and Dwall is the distance from the current pedestrian to the wall. Some rules are set as follows: visual
(a) If a pedestrian finds the exit, ⃗ ei will directly point to the exit. (b) If the distance between the guide and the current pedestrian is less than the radius of the guide’s influence area, that is, the pedestrian is in the guide’s influence area, he or she will follow this guide. (c) If a pedestrian cannot see the guide, the wall or the exit for example in the area A1 in Fig. 1, the preferred individual desired direction ⃗ ei will be randomly chosen and kept the same value until the guide, the wall or the exit is in his or her vision field. Moreover, if the wall is in a pedestrian’s vision field such as in the area A2 or A3 , his or her preferred individual desired direction will be chosen from ⃗ ei = (cos γ ′ , sin γ ′ ) (γ ′ ∈ γ ) with probability p1 or ⃗ ei = (cos η′ , sin η′ ) (η′ ∈ η) with probability p2 which will be kept for a fixed period, and then will be determined again according to the pedestrian’s current position. Note that γ ′ is a direction choosing from the set γ , and η′ is a direction choosing from the set η. Besides, ⟨⃗e 0j (t )⟩i and ξ⃗1 (t ) change according to the real situations.
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Fig. 2. The layout of the room.
3. Simulations and results In this section, the setup of the simulation scenario is first given, where the preferred individual desired direction set is presented. Then, the effects of mass behavior in the case of no guides are investigated. Moreover, the influences of the guide are studied, and some detailed phenomena are found. Finally, the influences of positions and the number of the guides on evacuation efficiency are given. 3.1. Scenario setup The above extended crowd model is applied to the evacuation with a guide in a room whose layout is shown in Fig. 2. The length and width of the room are both 50 m, and the exit is set at (50 m, 25 m) with the width equals to 1 m. The room is divided into areas A0 , A1 , . . . , A9 depending on whether or not the wall or the exit is in the pedestrian’s vision field. Take pedestrians in the area A0 as an example, they can see the exit and accordingly their desired walking directions point to the exit. For pedestrians who have no information of the guide or the exit in a certain area, for example, in the area A5 , their desired walking directions are determined by the individual behaviors affected by the right wall, pedestrians around this current pedestrian, and the surrounding environment according to Eq. (8). Moreover, their preferred individual desired direction set γ is given in Table 2. It is worth noting that the critical angles of the set γ in Table 2 are obtained according to wall . ψ = arccos RDvisual Besides, we adopt the same parameters with Ref. [13] that have been calibrated and validated: χ1 = 0.6, χ2 = 0.3, χ3 = 0.1, b1 = 0.0125, b2 = 0.025, βi = 0.6. 3.2. Effects of mass behavior in the case of no guides In order to investigate the effects of individualistic and herding behaviors on evacuation efficiency, we assume that 200 pedestrians randomly distribute inside of the room, and these pedestrians need to evacuate from this room as soon as possible because of the fire or other emergencies. The radius of the vision area Rvisual is set to 10 m in this paper. In Fig. 3, χ1 = 0.8, χ2 = 0.1 and χ3 = 0.1 indicate the individualistic behavior plays a dominant role. In Fig. 4, χ1 = 0.1, χ2 = 0.8 and χ3 = 0.1 reflect the herding behavior occupies the dominant position. The red dots represent pedestrians, and the length and point of blue arrows stand for the relative magnitude and direction of the desired walking velocity. In Fig. 3, the most obvious phenomenon is the consistency of pedestrians’ desired walking direction pointing to the exit for those people who can get the information of the exit, the desired directions of other pedestrians, however, point randomly over time which may reduce the evacuation efficiency to a certain extent. In Fig. 4, the desired walking directions of pedestrians all have the consistency in a certain range, which means that pedestrians’ desired walking directions are far more influenced by people around themselves. By comparing these two figures, we can find that individualistic behavior makes pedestrians walk randomly inside of the room throughout the evacuation process. Conversely, herding behavior makes pedestrians develop into groups which separate distinctly. We are, however, still not very clear which behavior or the combination of these two behaviors according to a certain percentage plays more important roles in the escape environment. Therefore, different weights of the individualistic behavior are set in Fig. 5, and 10 times repeatable simulations are run for each weight to get the corresponding results. Note that the weight of the influence of the surroundings χ3 = 0.1 keeps the same for different simulation runs in Fig. 5, which means χ1 + χ2 = 0.9 all the time. From Fig. 5, we can find that the larger the parameter χ1 , the smaller the mean number of pedestrians escaping from this room especially obvious for the simulation results within 180 s. This is different from the findings in Ref. [14], one reason may be the difference in the
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X. Yang et al. / Physica A 442 (2016) 397–408 Table 2 The preferred individual desired direction set γ in different areas of the room. Area
The preferred individual desired direction set γ
A0 A1 A2
Point to the exit [0, 2π) [0, 32π − arccos RDwall ] [ 32π + arccos
A3 A4 A5 A6 A7 A8 A9
Dwall , 2π) Rvisual wall wall [0, π − arccos RDvisual ] [π + arccos RDvisual , 2π) wall wall [0, π2 − arccos RDvisual ] [ π2 + arccos RDvisual , 2π) visual
wall wall [arccos RDvisual , 2π − arccos RDvisual ] D wall wall wall wall [0, 32π − arccos Rvisual ] [ 32π + arccos RDvisual , 2π) [arccos RDvisual , 2π − arccos RDvisual ] wall wall wall wall [0, 32π − arccos RDvisual ] [ 32π + arccos RDvisual , 2π) [0, π − arccos RDvisual ] [π + arccos RDvisual , 2π) wall wall wall wall [0, π − arccos RDvisual ] [π + arccos RDvisual , 2π) [0, π2 − arccos RDvisual ] [ π2 + arccos RDvisual , 2π) wall wall wall wall [0, π2 − arccos RDvisual ] [ π2 + arccos RDvisual , 2π) [arccos RDvisual , 2π − arccos RDvisual ]
(a) t = 1 s.
(b) t = 100 s.
(c) t = 200 s.
(d) t = 300 s.
Fig. 3. Snapshots of the simulation with χ1 = 0.8, χ2 = 0.1, χ3 = 0.1 at different time instants. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
layouts. There are two exits in Ref. [14], and larger weight of herding behaviors can result in pedestrians neglecting one exit, which will lead to the congestion around another exit. Another reason may be different densities of pedestrians set in the room. In our simulations, the density is not very large that prevents the over-crowing phenomenon around the exit. 3.3. Evacuation dynamics due to the guide The radius of the influence area of the guide Rinf is set to be 1.1 · Rvisual , which is marked by the black circle in Fig. 6. We assume that the initial position of the guide is (10 m, 25 m) depicted by a red star, χ1 = 0.6, χ2 = 0.3 and χ3 = 0.1. Pedestrians who are led by the guide are expressed by the pink circles and others who cannot see the guide are marked by the blue dots.
X. Yang et al. / Physica A 442 (2016) 397–408
(a) t = 1 s.
(b) t = 100 s.
(c) t = 200 s.
(d) t = 300 s.
403
Fig. 4. Snapshots of the simulation with χ1 = 0.1, χ2 = 0.8, χ3 = 0.1 at different time instants. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 5. The mean number of evacuated pedestrians and its deviation under different weights of individualistic behaviors in the case of no guides.
Fig. 6(a) shows the initial states of pedestrians and the guide, where pedestrians distribute randomly inside of the room. Fig. 6(b) presents that pedestrians in the guide’s influence area walk closer to the guide since they do not know where exactly the exit is, and the only information they can obtain comes from the guide. Pedestrians who are not in the influence area of the guide would walk randomly according to their own personal willingness, surrounding pedestrians and environment. During the evacuation, the number of pedestrians affected by the guide gets larger and larger, as shown in Fig. 6(c), and pedestrians who have already been influenced before would still follow the guide to evacuate. Fig. 6(d) shows that the guided
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Fig. 6. Snapshots of crowd dynamics in the case of a guide. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 7. The candidate positions of guides in the room. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
pedestrians walk to the exit. Generally, once pedestrians led by the guide see the exit, they will directly walk towards the exit as soon as possible instead of following this guide. 3.4. Effects of positions of the guide on evacuation Different positions of a guide may have different impacts on evacuation efficiencies. Nine positions in the room are chosen as the candidate places of the guide shown in Fig. 7, and these nine guides are marked by G1 , G2 , . . . , G9 with red stars, respectively. Besides, 200 pedestrians are randomly distributed in the room with the same initial positions during all simulation runs. The mean number of evacuated pedestrians from the room within 180 s, and the standard deviation by doing 10 times repeatable simulation runs for the guide with different candidate positions are given in Fig. 8(a). Note that 180 s is long enough to study this problem, which can ensure the guide to go out of the room and there is still enough time to let pedestrians around the exit go out. As imagined, G1 , G4 and G7 play the biggest roles, followed by G2 , G5 and G8 , and finally G3 , G6 and G9 . This is because the guide can affect more pedestrians and only a few pedestrians still need to keep looking
X. Yang et al. / Physica A 442 (2016) 397–408
(a) The mean number of evacuated pedestrians and its standard deviation under the guides with different initial positions within 180 s.
(c) The outflows and cumulated outflows from the exit under guides G4 , G5 and G6 respectively.
405
(b) The outflows and cumulated outflows from the exit under guides G1 , G2 and G3 respectively.
(d) The outflows and cumulated outflows from the exit under guides G7 , G8 and G9 respectively.
Fig. 8. Effects of different positions of the guide on the evacuation within 180 s.
for the exit, if this guide is farther away from the exit. However, we cannot exactly figure out which position is the best one among the positions of G1 , G4 and G7 from our simulation results. Fig. 8(b)–(d) present the outflow and the cumulated outflow of pedestrians from the exit within 180 s in a simulation run, and Table 3 gives the corresponding time that the guide goes out of the room. By observing all of the subgraphs of the outflows over time, we can find that these curves usually have relatively large fluctuations because of the limited exit capacity and the density of pedestrians around the exit. Besides, we can obtain the following findings: the outflow is relatively high at the beginning time because pedestrians distributing around the exit in the initial can directly go out of the room; the outflow drops after a short time because pedestrians who can see the exit have already finished going out, and the guided group still needs some time to arrive at the exit; once the guided group gets to the area A0 in Fig. 2, more pedestrians can know where the exit is and directly go to the exit instead of following the guide, the outflow accordingly becomes larger; after the guided group goes out of the room, pedestrians still at the room need to try their best to find the exit and the outflow at this time begins to drop and later keeps a relatively low value. Take the pedestrian evacuation under the guide G2 as an example, the cumulated outflow in the case of having G2 at the beginning time is similar to those under other different guides observed from Fig. 8(b)–(d), this is because pedestrians around the exit escape from the room as soon as possible once emergency occurs as mentioned above; G2 escapes from the room at 38.4 s obtained from Table 3, and it is easily found that both the outflow and cumulated outflow of pedestrians
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Table 3 The leaving time of different guides from the room. Guide identification number
G1
G2
G3
G4
G5
G6
G7
G8
G9
Time (s)
56.6
38.4
30.3
50.2
35.1
22.3
56.2
44.4
25.6
Fig. 9. The comparison of simulation results for the evacuations without any guide and with one guide G7 .
increase before and after 38.4 s because of the coming of the guided group; once the informed pedestrians by the guide G2 successfully escape from the room, the outflow decreases and keeps for a relatively low value after 125 s, and the corresponding cumulated outflow also increases slowly. Similarly, the evacuation dynamics under other guides also have the above shifting trends. Besides, by comparing the curves in the cumulated outflow subgraph of Fig. 8(b), we can observe that the cumulated outflow under the guide G3 is higher than those under G2 and G1 , and the corresponding value under G2 is higher than that under G1 for the time period approximately from 50 to 100 s. The reason is that G3 is closest to the exit and this guided group can arrive at the exit earlier than the other two guided groups under the same simulation environment. By comparing Figs. 8(a) and 5, we can find that the guide can make more pedestrians escape from the room during the evacuation set in this section. This can also be reflected from Fig. 9, in which 5 simulation runs are carried out for each scenario. In Fig. 9, we can observe that the total numbers of pedestrians leaving the room are similar to each other for the time period from the beginning to 80 s whether there is a guide or not, and an important role of the guide shows out after 80 s especially around 150 s. 3.5. Effects of the number of guides on evacuation Fig. 10 gives the mean leaving number of pedestrians from the room during 300 s for different evacuation cases with the different number of guides, namely guides G1 , (G1 , G7 ), (G1 , G3 , G7 ), (G1 , G3 , G7 , G9 ). It is worth noting that 20 simulation runs are carried out for each corresponding case. Fig. 10 shows that guides (G1 , G7 ) lead a better evacuation effect, which means there is no need to add the number of guides in this evacuation situation and the personnel cost comes naturally to be reduced. This can also be achieved from Fig. 11. Fig. 11 shows the pedestrian evacuation under four guides which can lead more pedestrians to arrive at the exit during a certain time. However, too many pedestrians are around the exit at the same time and the number of people exceeds the load-bearing capacity of the exit, which results in the terrible congestion conditions. Generally, the evacuation efficiency does not only have relationship with the positions of the guide and the number of guides but also with the density of the pedestrians in the room. When designing a new building, early prediction of pedestrian traffic is very necessary. If pedestrian traffic may be very large, appropriately increasing guides and exits should be considered beforehand. 4. Conclusion Guide plays an important role during the evacuation in large public places. This paper focuses on investigating the guided crowd dynamics using a proposed model. The simulation results show that herding behavior is conducive to an efficient evacuation when the initial density of pedestrians in the room with only one exit is moderate. Group dynamics characterized
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Fig. 10. The mean number of pedestrians leaving the room versus the different numbers of guides.
Fig. 11. Snapshots of crowd dynamics in the case of four guides.
by gathering, conflicts and balance emerges for the guided crowd. The simulation results reflect that guide is necessary in the emergency management, and the position of guide should also be arranged properly. Moreover, the appropriate quantity of guides should be arranged beforehand in the large public places, and the larger number of guides may not cause the better evacuation effects. In this paper, we only consider the room with one exit and no obstacles inside. However, the evacuation environment in reality is very complex, we need to consider the guided crowd dynamics in complex configurations in the later work. Acknowledgments This work was supported jointly by Fundamental Research Funds for Central Universities (No. 2013JBZ007), National Natural Science Foundation of China (No. 61233001), Beijing Jiaotong University Research Program (No. RCS2012ZZ003). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
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Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Necessity of guides in pedestrian emergency evacuation Xiaoxia Yang a , Hairong Dong a,∗ , Xiuming Yao b , Xubin Sun b , Qianling Wang a , Min Zhou a a
State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, China
b
School of Electronic and Information Engineering, Beijing Jiaotong University, Beijing 100044, China
highlights • • • •
The influence of mass behavior on evacuation dynamics is investigated. An extended crowd model is proposed for the guided crowd. The effect of position of guide on evacuation is studied. The effect of the quantity of guides on evacuation is studied.
article
info
Article history: Received 5 May 2015 Received in revised form 21 July 2015 Available online 21 August 2015 Keywords: Social force model Guided crowd Uninformed pedestrians Desired direction Position of guide Quantity of guides
abstract The role of guide who is in charge of leading pedestrians to evacuate in the case of emergency plays a critical role for the uninformed people. This paper first investigates the influence of mass behavior on evacuation dynamics and mainly focuses on the guided evacuation dynamics. In the extended crowd model proposed in this paper, individualistic behavior, herding behavior and environment influence are all considered for pedestrians who are not informed by the guide. According to the simulation results, herding behavior makes more pedestrians evacuate from the room in the same period of time. Besides, guided crowd demonstrates the same behavior of group dynamics which is characterized by gathering, conflicts and balance. Moreover, simulation results indicate guides with appropriate initial positions and quantity are more conducive to evacuation under a moderate initial density of pedestrians. © 2015 Elsevier B.V. All rights reserved.
1. Introduction The efficiency of pedestrian evacuation is a key aspect of the safety performance evaluation in public places such as subway stations, airports, and stadiums. Appropriate evacuation measures developed in advance can play important roles in improving the evacuation efficiency when fire, explosion, earthquake and other emergencies occur. In 2008, more than two thousand students and teachers all realized the safe evacuation in 96 s when earthquake occurred in Zaosang middle school, Sichuan Province, China, since emergency evacuation drills were organized in each semester. A large number of facts have proven that the prevention beforehand works much better than a temporary response [1]. Moreover, the poor emergency warning system and unreasonable and temporary evacuation strategy are the root causes of catastrophe. Earthquake tsunami disaster having adverse impacts to more than 10 countries occurred in Indian Ocean in 2004, caused about 30 million deaths, and 14 billion dollars loss because of the poor communication of information. The establishment of
∗
Corresponding author. Tel.: +86 10 51685749; fax: +86 10 51685749. E-mail address: [email protected] (H. Dong).
http://dx.doi.org/10.1016/j.physa.2015.08.020 0378-4371/© 2015 Elsevier B.V. All rights reserved.
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the predictive evacuation system is therefore very important, and many scholars began to investigate pedestrian dynamics to better understand the evacuation procedure from the 1970s [2]. Generally, pedestrian models to predict pedestrians’ behaviors and dynamics in the normal and emergency situations are studied from the macroscopic and microscopic perspectives [3]. Macroscopic pedestrian model considers the pedestrian flow as a whole by using the flow density and velocity to describe its motion [4]. The most remarkable advantage of this model is to save the computation amount and enhance the real-time ability [5]. Fluid dynamics model as a typical macroscopic model treats the flow as the continuous liquid and is described based on the Navier–Stokes equation [6]. However, the macroscopic model neglects the detailed behaviors among pedestrians, and this is one reason why microscopic pedestrian model continues to emerge and has been widely applied in reality. Microscopic pedestrian model considers the interactions among pedestrians such as the cellular automata model [7–10], the social force model [11–13], and the agent based model [7]. Wang et al. [8] proposed a cellular automata model considering the game strategy to study the pedestrian evacuation in a hall. Frank and Dorso [9] studied the room evacuation in the presence of an obstacle based on the social force model and found ‘‘clever is not always better’’ effect. This paper chooses to use the microscopic social force model as the reference model which not only qualitatively represents self-organization behaviors such as lane formation [10] and strip formation [11] but also introduces the physical and psychology forces that make the model more realistic. During the major events in large public places, we always find some staffs wearing the clothes with bright colors to maintain order. These staffs who are familiar with the building structure may play guides when emergency occurs. Hou et al. [12] investigated the effect of the number and positions of guides on evacuation dynamics in rooms with limited visibility range. However, the walking rules set for pedestrians without following the guide do not consider the mass behavior which are not corresponding to reality. Yang et al. [13] developed the guided crowd model to study the crowd dynamics via modified social force model, but all pedestrians are assumed to have obtained the information from the guide, and they do not give the description for pedestrians who cannot get the information. This paper extends the model proposed by Yang et al. [13] to present the detailed behaviors not only for pedestrians who follow the guide but also for those who cannot be influenced by the guide. This contribution presents the dynamic behaviors of the guided crowd. It continues with the description of the extended crowd model in Section 2, where the mathematical models for the guide and all pedestrians whether are informed or not are developed. Section 3 gives the scenario setup, effects of mass behaviors in the cases of no guides, evacuation dynamic due to the guide, and the effect of position and quantity of guides on evacuation. After analyzing the simulation results, the key discoveries are reviewed and the future work is looked into in Section 4. 2. An extended crowd model This section mainly focuses on modeling of the guided crowd in a smoky room, where the normal pedestrians are not familiar with the surroundings except the guides who know exactly where the exit is. Generally, the vision field is limited in a smoky room, the pedestrians who locate in the influence field of the guide may follow the movement of the guide. However, for those who neither belong to the influence area of the guide nor the exit is visible, a random moving direction will be chosen. 2.1. Social force model The social force model is proposed by Helbing et al. [14], in which pedestrians are driven by the desired force, ⃗ f i0 ; the
interaction force between pedestrians i and j, ⃗ fij ; and the interaction force between pedestrian i and walls w , ⃗ fiw . This model is represented by mi
dv ⃗i (t ) dt
= ⃗f i0 +
⃗fij +
j(̸=i)
w
⃗fiw ,
(1)
where mi denotes the mass of pedestrian i, and v ⃗i (t ) denotes the actual walking velocity at time instant t. ⃗f i0 shows pedestrian’s willingness to reach the desired velocity: 0 ⃗0 ⃗f i0 = mi vi (t ) e i − v⃗i (t ) , τi
(2)
where vi0 denotes the desired speed, and ⃗ e 0i denotes the desired walking direction. τi is the adaptation time for adjusting the current velocity to the desired one. ⃗fij presents the pedestrian’s psychological tendency to move away from others and the physical force that occurs only when the distance between two pedestrians’ centers dij is less than the sum of these two pedestrians’ radii rij = ri + rj :
⃗fij = Ai exp rij − dij /Bi n⃗ij + kg rij − dij n⃗ij + κ g rij − dij 1vjit ⃗tij .
(3)
⃗ij = ( , ) = (⃗ri −⃗rj )/dij is the Here, Ai is the interaction strength and Bi is the magnitude of the repulsive interactions. n normalized vector pointing from pedestrian j to i, and ⃗ ri is the position of pedestrian i. k is the body compression coefficient, n1ij
n2ij
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Table 1 Parameters of social force model. Symbol
Meaning
Value
m r A B
Pedestrian mass Pedestrian radius Strength of social repulsive force Characteristic distance of social repulsive force Coefficient of sliding friction Body compression coefficient Pedestrian reaction time
80 kg 0.25 m 2000 N 0.08 m 240 000 kg m−1 s−1 120 000 kg s−2 0.5 s
κ k
τ
and κ is the coefficient of sliding friction. ⃗tij = (−n2ij , n1ij ) is the tangential direction, and 1vjit = v ⃗j − v⃗i · ⃗tij is the velocity difference along the tangential direction. The function g (x) is zero if pedestrians do not touch each other, i.e. (dij > rij ), otherwise it is equal to the argument x. ⃗fiw is modeled analogously:
⃗fiw = Ai exp[(ri − diw )/Bi ]⃗niw + kg (ri − diw )⃗niw + κ g (ri − diw )1vwt i⃗tiw .
(4)
The parameters of the original social force model are specified in Table 1 [14]. 2.2. The modeling for the guide and pedestrians following the guide Guides as an important part during the evacuation undertake the responsibilities of leading the pedestrians to evacuate as soon as possible. Yang et al. [13] introduced a modeling method for the guide who exactly knows the position of the exit and the pedestrians who follow this guide. This paper adopts the similar modeling method for the guide and followers. The motion of the guide who provides information to the surrounding pedestrians is described as mr
dv ⃗r (t ) dt
= ⃗f r0 +
⃗frj +
w
j
⃗fr w ,
⃗f r0 = mr vr (t )⃗e r − v⃗r (t ) . τr 0
0
(5)
Here, the guide’s desired direction ⃗ e 0r points to the exit. If pedestrians who are not familiar with the environment are in a guide’s influence area, they would tend to follow this guide. The corresponding motion equation is expressed as mi
dv ⃗i (t ) dt
= βi · ⃗f i0 +
j(̸=i)
⃗fij +
w
⃗fiw + ⃗fir + ⃗fir ,
0 0 ⃗f i0 = mi vi (t )⃗e i − v⃗i (t ) , τi ⃗ ⃗ r ⃗fir = mi · −b1 i (t ) − rr (t ) − b2 v⃗i (t ) − v⃗r (t ) , τi τi2
b1 , b2 > 0,
(6)
where βi ∈ [0, 1]. Pedestrians’ desired direction, ⃗ e 0i , points to the guide before they could see the exit. ⃗ fir is the interaction force between pedestrian i and the guide r. ⃗ fir is the navigational force provided by the guide. b1 and b2 are both positive constants reflecting the weights of navigational feedback. According to Ref. [15], the influence area of the guide with a certain radius Rinf is usually larger than the vision area with a certain radius Rvisual because of the phenomenon of flow with the stream. In this paper, it is assumed that all pedestrians in the guide’s influence area follow the guide in the evacuation. 2.3. The modeling for pedestrians uninformed by the guide In Ref. [13], it is assumed that all pedestrians can see the guide and are led by this guide. However, there is a limit of the vision field because of pedestrians themselves or the environmental uncertainty such as fire or smoke. It is, therefore, necessary to study the modeling method of those pedestrians who cannot see the guide or the exit. In this paper, the motion equation of pedestrians who have no information of the guide or exit is also based on the social force model given by Eq. (1), our work focuses on setting their desired directions. According to Ref. [12], pedestrians who cannot see the guide or exit are set to choose a direction randomly and follow this direction until they see a wall and then turn to a random direction again. This method, however, overlooks the influences of some factors such as the herding behavior which is very important during the evacuation. Helbing et al. [14] investigated the mass behavior in a smoky room, and
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Fig. 1. The diagram of the preferred individual desired direction.
pointed that each pedestrian may either choose an individually preferred desired direction ⃗ ei , or follow the mean walking direction ⟨⃗ e 0j (t )⟩i of pedestrians in his or her vision field, or have a mixture of both given in
⃗e 0i (t ) = Norm[(1 − pi ) · ⃗ei + pi · ⟨⃗e 0j (t )⟩i ].
(7)
Here, Norm(⃗ z ) = ∥⃗⃗zz ∥ . pi reflects the weight of the herding behavior, and 1 − pi reflects the weight of the individual behavior. Generally, the surrounding environment such as the noise can also affect the desired direction to some extent. This paper further introduces ξ⃗1 (t ) to reflect the influence of environment on pedestrians’ desired direction which is given by
⃗e 0i (t ) = Norm[χ1 · ⃗ei + χ2 · ⟨⃗e 0j (t )⟩i + χ3 · ξ⃗1 (t )],
(8)
where ξ⃗1 (t ) is a unit vector in a certain direction according to the actual situations, χ1 , χ2 and χ3 are positive constants between 0 and 1, and χ1 + χ2 + χ3 = 1. χ1 reflects the weight of the individualistic behavior, χ2 gives the weight of herding behavior, and χ3 is the weight of the influence of the surrounding environment. Generally, the selection of the preferred individual desired direction has some bases according to the visible walls and obstacles. With limited practical observations, we find when the wall belongs to a pedestrian’s vision field, this pedestrian prefers walking along the wall or moving away from the wall, but not directly heading towards the wall in order to find the exit as soon as possible. In Fig. 1, pedestrian i can only see a section of wall from a to b as the limit of vision field, which causes the sets of γ and η to come into being. It is worth noting that the set γ contains the pedestrian’s preferred individual desired directions along the wall or away from the wall, while the set η defines the corresponding directions heading towards the wall. This paper, therefore, considers the effect of the walls on the decision of the preferred individual desired direction once the wall is in pedestrians’ vision field, and proposes the preferred individual desired walking direction ⃗ ei choosing from the set γ with a large probability p1 or from the set η with a small probability p2 shown in Fig. 1. Note that the different areas A1 , A2 , A3 and A7 are developed according to the different walls in a pedestrian’s vision field. For example, pedestrians in area A1 cannot see any wall, and pedestrians in area A3 can only see the left wall, while pedestrians in area A7 can see both the left wall and the lower wall. The method of the determination of ⃗ ei in different areas, however, is the same. In Fig. 1, D cos ψ = R wall , and Dwall is the distance from the current pedestrian to the wall. Some rules are set as follows: visual
(a) If a pedestrian finds the exit, ⃗ ei will directly point to the exit. (b) If the distance between the guide and the current pedestrian is less than the radius of the guide’s influence area, that is, the pedestrian is in the guide’s influence area, he or she will follow this guide. (c) If a pedestrian cannot see the guide, the wall or the exit for example in the area A1 in Fig. 1, the preferred individual desired direction ⃗ ei will be randomly chosen and kept the same value until the guide, the wall or the exit is in his or her vision field. Moreover, if the wall is in a pedestrian’s vision field such as in the area A2 or A3 , his or her preferred individual desired direction will be chosen from ⃗ ei = (cos γ ′ , sin γ ′ ) (γ ′ ∈ γ ) with probability p1 or ⃗ ei = (cos η′ , sin η′ ) (η′ ∈ η) with probability p2 which will be kept for a fixed period, and then will be determined again according to the pedestrian’s current position. Note that γ ′ is a direction choosing from the set γ , and η′ is a direction choosing from the set η. Besides, ⟨⃗e 0j (t )⟩i and ξ⃗1 (t ) change according to the real situations.
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Fig. 2. The layout of the room.
3. Simulations and results In this section, the setup of the simulation scenario is first given, where the preferred individual desired direction set is presented. Then, the effects of mass behavior in the case of no guides are investigated. Moreover, the influences of the guide are studied, and some detailed phenomena are found. Finally, the influences of positions and the number of the guides on evacuation efficiency are given. 3.1. Scenario setup The above extended crowd model is applied to the evacuation with a guide in a room whose layout is shown in Fig. 2. The length and width of the room are both 50 m, and the exit is set at (50 m, 25 m) with the width equals to 1 m. The room is divided into areas A0 , A1 , . . . , A9 depending on whether or not the wall or the exit is in the pedestrian’s vision field. Take pedestrians in the area A0 as an example, they can see the exit and accordingly their desired walking directions point to the exit. For pedestrians who have no information of the guide or the exit in a certain area, for example, in the area A5 , their desired walking directions are determined by the individual behaviors affected by the right wall, pedestrians around this current pedestrian, and the surrounding environment according to Eq. (8). Moreover, their preferred individual desired direction set γ is given in Table 2. It is worth noting that the critical angles of the set γ in Table 2 are obtained according to wall . ψ = arccos RDvisual Besides, we adopt the same parameters with Ref. [13] that have been calibrated and validated: χ1 = 0.6, χ2 = 0.3, χ3 = 0.1, b1 = 0.0125, b2 = 0.025, βi = 0.6. 3.2. Effects of mass behavior in the case of no guides In order to investigate the effects of individualistic and herding behaviors on evacuation efficiency, we assume that 200 pedestrians randomly distribute inside of the room, and these pedestrians need to evacuate from this room as soon as possible because of the fire or other emergencies. The radius of the vision area Rvisual is set to 10 m in this paper. In Fig. 3, χ1 = 0.8, χ2 = 0.1 and χ3 = 0.1 indicate the individualistic behavior plays a dominant role. In Fig. 4, χ1 = 0.1, χ2 = 0.8 and χ3 = 0.1 reflect the herding behavior occupies the dominant position. The red dots represent pedestrians, and the length and point of blue arrows stand for the relative magnitude and direction of the desired walking velocity. In Fig. 3, the most obvious phenomenon is the consistency of pedestrians’ desired walking direction pointing to the exit for those people who can get the information of the exit, the desired directions of other pedestrians, however, point randomly over time which may reduce the evacuation efficiency to a certain extent. In Fig. 4, the desired walking directions of pedestrians all have the consistency in a certain range, which means that pedestrians’ desired walking directions are far more influenced by people around themselves. By comparing these two figures, we can find that individualistic behavior makes pedestrians walk randomly inside of the room throughout the evacuation process. Conversely, herding behavior makes pedestrians develop into groups which separate distinctly. We are, however, still not very clear which behavior or the combination of these two behaviors according to a certain percentage plays more important roles in the escape environment. Therefore, different weights of the individualistic behavior are set in Fig. 5, and 10 times repeatable simulations are run for each weight to get the corresponding results. Note that the weight of the influence of the surroundings χ3 = 0.1 keeps the same for different simulation runs in Fig. 5, which means χ1 + χ2 = 0.9 all the time. From Fig. 5, we can find that the larger the parameter χ1 , the smaller the mean number of pedestrians escaping from this room especially obvious for the simulation results within 180 s. This is different from the findings in Ref. [14], one reason may be the difference in the
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X. Yang et al. / Physica A 442 (2016) 397–408 Table 2 The preferred individual desired direction set γ in different areas of the room. Area
The preferred individual desired direction set γ
A0 A1 A2
Point to the exit [0, 2π) [0, 32π − arccos RDwall ] [ 32π + arccos
A3 A4 A5 A6 A7 A8 A9
Dwall , 2π) Rvisual wall wall [0, π − arccos RDvisual ] [π + arccos RDvisual , 2π) wall wall [0, π2 − arccos RDvisual ] [ π2 + arccos RDvisual , 2π) visual
wall wall [arccos RDvisual , 2π − arccos RDvisual ] D wall wall wall wall [0, 32π − arccos Rvisual ] [ 32π + arccos RDvisual , 2π) [arccos RDvisual , 2π − arccos RDvisual ] wall wall wall wall [0, 32π − arccos RDvisual ] [ 32π + arccos RDvisual , 2π) [0, π − arccos RDvisual ] [π + arccos RDvisual , 2π) wall wall wall wall [0, π − arccos RDvisual ] [π + arccos RDvisual , 2π) [0, π2 − arccos RDvisual ] [ π2 + arccos RDvisual , 2π) wall wall wall wall [0, π2 − arccos RDvisual ] [ π2 + arccos RDvisual , 2π) [arccos RDvisual , 2π − arccos RDvisual ]
(a) t = 1 s.
(b) t = 100 s.
(c) t = 200 s.
(d) t = 300 s.
Fig. 3. Snapshots of the simulation with χ1 = 0.8, χ2 = 0.1, χ3 = 0.1 at different time instants. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
layouts. There are two exits in Ref. [14], and larger weight of herding behaviors can result in pedestrians neglecting one exit, which will lead to the congestion around another exit. Another reason may be different densities of pedestrians set in the room. In our simulations, the density is not very large that prevents the over-crowing phenomenon around the exit. 3.3. Evacuation dynamics due to the guide The radius of the influence area of the guide Rinf is set to be 1.1 · Rvisual , which is marked by the black circle in Fig. 6. We assume that the initial position of the guide is (10 m, 25 m) depicted by a red star, χ1 = 0.6, χ2 = 0.3 and χ3 = 0.1. Pedestrians who are led by the guide are expressed by the pink circles and others who cannot see the guide are marked by the blue dots.
X. Yang et al. / Physica A 442 (2016) 397–408
(a) t = 1 s.
(b) t = 100 s.
(c) t = 200 s.
(d) t = 300 s.
403
Fig. 4. Snapshots of the simulation with χ1 = 0.1, χ2 = 0.8, χ3 = 0.1 at different time instants. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 5. The mean number of evacuated pedestrians and its deviation under different weights of individualistic behaviors in the case of no guides.
Fig. 6(a) shows the initial states of pedestrians and the guide, where pedestrians distribute randomly inside of the room. Fig. 6(b) presents that pedestrians in the guide’s influence area walk closer to the guide since they do not know where exactly the exit is, and the only information they can obtain comes from the guide. Pedestrians who are not in the influence area of the guide would walk randomly according to their own personal willingness, surrounding pedestrians and environment. During the evacuation, the number of pedestrians affected by the guide gets larger and larger, as shown in Fig. 6(c), and pedestrians who have already been influenced before would still follow the guide to evacuate. Fig. 6(d) shows that the guided
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Fig. 6. Snapshots of crowd dynamics in the case of a guide. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 7. The candidate positions of guides in the room. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
pedestrians walk to the exit. Generally, once pedestrians led by the guide see the exit, they will directly walk towards the exit as soon as possible instead of following this guide. 3.4. Effects of positions of the guide on evacuation Different positions of a guide may have different impacts on evacuation efficiencies. Nine positions in the room are chosen as the candidate places of the guide shown in Fig. 7, and these nine guides are marked by G1 , G2 , . . . , G9 with red stars, respectively. Besides, 200 pedestrians are randomly distributed in the room with the same initial positions during all simulation runs. The mean number of evacuated pedestrians from the room within 180 s, and the standard deviation by doing 10 times repeatable simulation runs for the guide with different candidate positions are given in Fig. 8(a). Note that 180 s is long enough to study this problem, which can ensure the guide to go out of the room and there is still enough time to let pedestrians around the exit go out. As imagined, G1 , G4 and G7 play the biggest roles, followed by G2 , G5 and G8 , and finally G3 , G6 and G9 . This is because the guide can affect more pedestrians and only a few pedestrians still need to keep looking
X. Yang et al. / Physica A 442 (2016) 397–408
(a) The mean number of evacuated pedestrians and its standard deviation under the guides with different initial positions within 180 s.
(c) The outflows and cumulated outflows from the exit under guides G4 , G5 and G6 respectively.
405
(b) The outflows and cumulated outflows from the exit under guides G1 , G2 and G3 respectively.
(d) The outflows and cumulated outflows from the exit under guides G7 , G8 and G9 respectively.
Fig. 8. Effects of different positions of the guide on the evacuation within 180 s.
for the exit, if this guide is farther away from the exit. However, we cannot exactly figure out which position is the best one among the positions of G1 , G4 and G7 from our simulation results. Fig. 8(b)–(d) present the outflow and the cumulated outflow of pedestrians from the exit within 180 s in a simulation run, and Table 3 gives the corresponding time that the guide goes out of the room. By observing all of the subgraphs of the outflows over time, we can find that these curves usually have relatively large fluctuations because of the limited exit capacity and the density of pedestrians around the exit. Besides, we can obtain the following findings: the outflow is relatively high at the beginning time because pedestrians distributing around the exit in the initial can directly go out of the room; the outflow drops after a short time because pedestrians who can see the exit have already finished going out, and the guided group still needs some time to arrive at the exit; once the guided group gets to the area A0 in Fig. 2, more pedestrians can know where the exit is and directly go to the exit instead of following the guide, the outflow accordingly becomes larger; after the guided group goes out of the room, pedestrians still at the room need to try their best to find the exit and the outflow at this time begins to drop and later keeps a relatively low value. Take the pedestrian evacuation under the guide G2 as an example, the cumulated outflow in the case of having G2 at the beginning time is similar to those under other different guides observed from Fig. 8(b)–(d), this is because pedestrians around the exit escape from the room as soon as possible once emergency occurs as mentioned above; G2 escapes from the room at 38.4 s obtained from Table 3, and it is easily found that both the outflow and cumulated outflow of pedestrians
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Table 3 The leaving time of different guides from the room. Guide identification number
G1
G2
G3
G4
G5
G6
G7
G8
G9
Time (s)
56.6
38.4
30.3
50.2
35.1
22.3
56.2
44.4
25.6
Fig. 9. The comparison of simulation results for the evacuations without any guide and with one guide G7 .
increase before and after 38.4 s because of the coming of the guided group; once the informed pedestrians by the guide G2 successfully escape from the room, the outflow decreases and keeps for a relatively low value after 125 s, and the corresponding cumulated outflow also increases slowly. Similarly, the evacuation dynamics under other guides also have the above shifting trends. Besides, by comparing the curves in the cumulated outflow subgraph of Fig. 8(b), we can observe that the cumulated outflow under the guide G3 is higher than those under G2 and G1 , and the corresponding value under G2 is higher than that under G1 for the time period approximately from 50 to 100 s. The reason is that G3 is closest to the exit and this guided group can arrive at the exit earlier than the other two guided groups under the same simulation environment. By comparing Figs. 8(a) and 5, we can find that the guide can make more pedestrians escape from the room during the evacuation set in this section. This can also be reflected from Fig. 9, in which 5 simulation runs are carried out for each scenario. In Fig. 9, we can observe that the total numbers of pedestrians leaving the room are similar to each other for the time period from the beginning to 80 s whether there is a guide or not, and an important role of the guide shows out after 80 s especially around 150 s. 3.5. Effects of the number of guides on evacuation Fig. 10 gives the mean leaving number of pedestrians from the room during 300 s for different evacuation cases with the different number of guides, namely guides G1 , (G1 , G7 ), (G1 , G3 , G7 ), (G1 , G3 , G7 , G9 ). It is worth noting that 20 simulation runs are carried out for each corresponding case. Fig. 10 shows that guides (G1 , G7 ) lead a better evacuation effect, which means there is no need to add the number of guides in this evacuation situation and the personnel cost comes naturally to be reduced. This can also be achieved from Fig. 11. Fig. 11 shows the pedestrian evacuation under four guides which can lead more pedestrians to arrive at the exit during a certain time. However, too many pedestrians are around the exit at the same time and the number of people exceeds the load-bearing capacity of the exit, which results in the terrible congestion conditions. Generally, the evacuation efficiency does not only have relationship with the positions of the guide and the number of guides but also with the density of the pedestrians in the room. When designing a new building, early prediction of pedestrian traffic is very necessary. If pedestrian traffic may be very large, appropriately increasing guides and exits should be considered beforehand. 4. Conclusion Guide plays an important role during the evacuation in large public places. This paper focuses on investigating the guided crowd dynamics using a proposed model. The simulation results show that herding behavior is conducive to an efficient evacuation when the initial density of pedestrians in the room with only one exit is moderate. Group dynamics characterized
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Fig. 10. The mean number of pedestrians leaving the room versus the different numbers of guides.
Fig. 11. Snapshots of crowd dynamics in the case of four guides.
by gathering, conflicts and balance emerges for the guided crowd. The simulation results reflect that guide is necessary in the emergency management, and the position of guide should also be arranged properly. Moreover, the appropriate quantity of guides should be arranged beforehand in the large public places, and the larger number of guides may not cause the better evacuation effects. In this paper, we only consider the room with one exit and no obstacles inside. However, the evacuation environment in reality is very complex, we need to consider the guided crowd dynamics in complex configurations in the later work. Acknowledgments This work was supported jointly by Fundamental Research Funds for Central Universities (No. 2013JBZ007), National Natural Science Foundation of China (No. 61233001), Beijing Jiaotong University Research Program (No. RCS2012ZZ003). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
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