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Team-moving effect in bi-direction pedestrian flow
Physica A 391 (2012) 3119–3128
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Team-moving effect in bi-direction pedestrian flow Ziyang Wang a,b,∗ , Bingxue Song c,d , Yong Qin b , Limin Jia b a
School of Traffic and Transportation, Beijing Jiaotong University, Beijing, 100044, China
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State Key Laboratory of Railway Control and Safety, Beijing Jiaotong University, Beijing, 100044, China
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Civil & Environment School, University of Science and Technology Beijing, Beijing, 100083, China
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Work safety Statistics & Analysis Center, Beijing Municipal Institute of Labour Protection, Beijing, 100054, China
article
info
Article history: Received 25 October 2010 Received in revised form 8 December 2011 Available online 20 January 2012 Keywords: Team-moving Pedestrian movement Cellular automation Corridor capacity Marginal utility of team-moving
abstract We propose a cellular automation model to simulate team-moving behavior in bidirectional pedestrian flow. The moving rules for double-pedestrian teaming include the constraint that pedestrians remain on adjacent cells. Phase transition, critical density — team number, velocity–density, and flow–density relationships are key component parts of the analysis. Simulations show that team-moving produces significant corridor capacity effects, and effects highly depend on the type of teaming behavior. In daily life, pedestrians prefer traverse teaming; under this bias, as teaming pedestrians increase in number, critical density reduces; that means traverse teaming will weaken the capacity of the corridor. The effect of traverse team-moving is nonlinear, and capacity will continually reduce as the team numbers increase; however, reduction rate will decay. We call this phenomenon, ‘‘the marginal utility of team-moving.’’ © 2012 Elsevier B.V. All rights reserved.
1. Introduction Cellular automation is widely used to simulate pedestrian movements. By setting simple local rules to each individual, an approximation of actual pedestrian behavior can be created, and collective pedestrian behavior emerges as an outgrowth interaction of the micro-simulation rules set [1–10]. This is very helpful in studying pedestrian movement law, as well as for analyzing and designing transportation facilities. In most current studies of pedestrian dynamics simulation, the main idea is taking the isolated individuals as the modeling objective. Related research work includes the Two Process Model [1], Lattice Gas Model [2–4], Floors Model [5–10], Pre-fixed Probabilities Model [11], Dynamic Parameters Model [12] and the Multi-grid Model [13,14]. In these studies usually the pedestrian flow is partitioned into three types according to the moving directions, which are bi-direction flow, laterallyinterfered flow and evacuation flow. Bi-direction flow restricts the moving direction to forward and backwards [15], and laterally-interfered flow restricts the pedestrian move laterally [16], while evacuation flow has little restriction on flow direction [17,18]. Among them the bi-direction flow model uses simple rules to reveal complex collective moving phenomena, such as ‘‘quick is low’’ [19,20], self-organized, jam formulation, and has been considerately used in the studying of pedestrian movement. Among the researches for the bi-directional model, Fang, et al. presented a set of four-direction moving rules for CA local moving [15], which allowed each pedestrian four latent positions for next step moving. Though the rules are simple, they can reveal most pedestrian dynamics, and the ‘‘back-step strategy’’ can reduce the jamming conditions significantly. Based on Fang’s work, Yue et al. modified the moving rules to eight-direction [21], and discussed the pedestrian dynamics under
∗
Corresponding author at: School of Traffic and Transportation, Beijing Jiaotong University, Beijing, 100044, China. E-mail address: [email protected] (Z.Y. Wang).
0378-4371/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2011.12.066
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the conditions of direction split and right-side walking habit. Similar work includes Refs. [22–25]. Unfortunately, most of the work did not consider the team-moving phenomenon, though it is very common in reality. For example, observations at Beijing’sWangFuJing Business Street, demonstrate more than 70% of pedestrians were in a team-moving state. In other public places, like parks and theaters, it is quite similar. Definition 1. Team-moving means that pedestrians join to form a team; in moving they keep close and keep together. Pedestrians in the same team tend to remain close to each other to avoid obstacles; it is quite different with isolated individual movement. The collective dynamics are different. Yang et al. discussed the kin behavior in evacuation flow [26]; the main work was using different kin behavior coefficients to generate the team-moving phenomenon. That study lacked the quantitative measurement of team-moving effects. Moussaïd et al. observed the team-moving phenomenon in public parks under natural conditions [27], and used a ‘‘social force model’’ to show that the transformation depends on two factors – communication – and crowd density. Unfortunately, their work did not measure the effect of different teaming manners. Summarily speaking, pedestrian dynamics studies of the team-moving phenomenon have been largely neglected. Interactions of individuals together with the collective dynamics under different teaming manners require additional deep study. A new bi-directional model with team-moving behavior is proposed here, which describes the individuals moving forwards and avoiding obstacles under team-moving condition. In this model, the probability of position transition reflects the pedestrian bias to different teaming manners. The simulation results show in the quantitative way how team-moving affects the collective dynamics, such as phase transition, velocity–density curve and flow–density curve. It is found that the effect is primarily decided by the bias to teaming manner. The collective dynamics under traverse teaming is quite different with that under lengthways teaming, while diagonal teaming will also play an important role to effect the collective dynamics. It is also found that there is some marginal utility in the effect of team-moving to the corridor capacity, the teaming effect will decay with the increase of team number. 2. Bi-direction model with team-moving 2.1. Space partition and moving rules In the model, pedestrian moving space is portioned into W × W grids in the plane. Every grid is a cell. Pedestrians move on the grids, and at each time step, each cell must be occupied by one pedestrian or must be empty. Under the doublepedestrian team-moving condition, each team occupies two cells. The size of a cell corresponds to 40 × 40 cm2 , and this the typical space occupied by a pedestrian in a dense crowd [28]. There are two type of moving directions. One is moving forward, the other is moving backwards; back stepping is inhibited. According to the manner of teaming, there are ten types of individuals in the system; five of them move forward and the other five backward. Figs. 1–5 show at k step the positions of the five forward moving types and their possible transition positions at k + 1 step. Fig. 1 shows the conditions for isolated individuals, at step k its state is T10 (k), it has 6 possible positions at step k + 1, those are T11 (k + 1), T12 (k + 1), . . . , T16 (k + 1) respectively. The possibility of transition to i those 6 positions are D11 /N1 , D21 /N1 , . . . , D61 /N1 respectively, and D11 , D21 , . . . , D61 are nonnegative numbers, N1 = i=1 D1 . Figs. 2–5 show conditions for teaming individuals. Taking Fig. 2 as an example, at step k the state is signed by T10 (k), by enumerating it has 14 possible positions at step k + 1, those are signed by T21 (k + 1), T22 (k + 1), . . . , T214 (k + 1) respectively.
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i The possibility of transition to those 14 positions are D2 /N2 , with D2 ≥ 0, j = 1, 2, . . . , 14 respectively, and N2 = i =1 D 2 . The position transition condition of the five backwards moving individuals is the center symmetry to Figs. 1–5, signed by 0 (k). The possibility values represent different moving and teaming biases. T60 (k), T70 (k), . . . , T10 The boundaries of upper and bottom are periodic, that means when the forward-moving pedestrian reaches the upper boundary, it will move back from the bottom boundary, and when the backwards-moving pedestrian arrives at the bottom it will return back from the upper boundary. The left and right boundaries are close. The total number of each type of individual is constant.
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2.2. Updating rules Five updating rules exist in the model. (1)At the initial step, all the isolated individuals and team-moving individuals are randomly distributed on the W × W grids. (2)The update is consequential. Every individual is assigned a number randomly, and at each time step the individual positions are updated according by the numbers. Isolated individuals each have a number, and each team has a number. (3) Transition positions are decided by Figs. 1–5, and are computed randomly by the possibility. When the potential positions are blocked by other pedestrians or by a wall, the corresponding possibility should be modified to 0. For example, 13 when at step k some traverse teaming individuals meet an obstacle as Fig. 6-a, D22 , D32 , D42 , D11 2 , D2 should be set to 0. When 8 13 1 11 the left wall is besides them as Fig. 6-b, D2 , D2 , D2 , D2 should be set to 0.
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Fig. 1. Possible transition positions for isolated forward moving individuals.
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Fig. 2. Possible transition positions for forward traverse teaming individuals. A1 A1
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Fig. 3. Possible transition positions for forward lengthways teaming individuals.
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Fig. 4. Possible transition positions for forward diagonal teaming individuals.
(4) The position transition should be consistent to team-moving habit in common sense, otherwise the corresponding possibility should be set to 0. (5) Team-moving habit definitions: (a) when one pedestrian is ahead of another, the front is the leader and the other is the follower. The follower should not violate the leader’s moving purpose; (b) when one moves to the left, the other one should not move to the right. So from these two team-moving habits in common practice, the following possibility values 9 13 16 5 9 15 9 10 15 14 11 12 14 1 12 14 should be set 0, D13 2 , D2 , D3 , D3 , D3 , D4 , D4 , D4 , D4 , D4 , D4 , D5 , D5 , D5 , D5 , D5 , D5 . 3. Simulation results N is the total number of pedestrians. M is the total number of teams. The total density p is defined as the value of N divided by the total number of cells, W × W . The velocity of pedestrians moving in one time-step is defined as the value
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Fig. 5. Possible transition positions for forward reverse-diagonal teaming individuals.
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Fig. 6. Transition possibility values modification: (a) front cell is occupied; (b) left side is boundary.
of the number of pedestrians moving ahead divided by the total number of walkers, N. The flow of pedestrians is the sum number of up walkers moving through the upper boundary plus down walkers moving through the bottom boundary in one time unit. ⟨v⟩ and ⟨f ⟩ are computed according to the mean values of velocity and flow. For each simulation, altogether 1500 steps were carried out. The first 1000 steps were used to start the simulation and were then discarded. The values of ⟨v⟩ and ⟨f ⟩ are computed according to the last 500 steps. 3.1. Parameter configuration The transition position possibility represents different moving and teaming bias. There are four kinds of teaming manners, traverse teaming (Fig. 2, T20 (k)), lengthways teaming (Fig. 3, T30 (k)), diagonal teaming (Fig. 4, T40 (k)), and reverse-diagonal teaming (Fig. 5, T50 (k)). Traverse teaming is favored because it is helpful for communication [27]. The police and army when patrolling often adopt lengthways teaming. In daily life, pedestrians will assume mixed teaming behaviors of the four types. For studying the effect of each manner, we decompose them as isolated behaviors, and then later recompose some of them. (1) Configuration 1 (C1): Traverse teaming is prior, diagonal teaming and reverse-diagonal teaming are allowed, and lengthways teaming is forbidden. D1 = D6 = [100 10000 100 1 1 1], D2 = D7 = [100 0 100 100000000 100 0 100 10000 10000 10000 1000000 1000000 0 0], D3 = D8 = [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0], D4 = D9 = [1000000 0 0 1000000 0 0 0 1 0 1000 0 100 0 0], D5 = D10 = [0 0 1000000 0 1000000 0 0 1 0 0 100 0 100 0 0]; In C1, traverse teaming is the biggest possibility, and in order to focus on the effect of traverse teaming, the possibilities of transition to lengthways teaming are set to 0. (2) Configuration 2 (C2): Traverse teaming is allowed, lengthways teaming, diagonal teaming and reverse-diagonal teaming are forbidden. D1 = D6 = [100 10000 100 1 1 1], D2 = D7 = [0 0 0 10000 0 0 0 1 1 1 100 100 0 0], D3 = D8 = [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0], D4 = D9 = [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0], D5 = D10 = [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]; C2 has an extreme bias to traverse teaming; it forbids all other teaming manners.
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Fig. 7. Effect on phase transition (a. N = 170, M = 0, b. N = 170, M = 20, c. N = 170, M = 40).
(3) Configuration 3 (C3): Lengthways teaming is prior, traverse is forbidden, diagonal teaming and reverse-diagonal teaming are allowed. D1 = D6 = [100 10000 100 1 1 1], D2 = D7 = [0 0 0 0 0 0 0 0 0 0 0 0 0 0], D3 = D8 = [0 100 100000000 100 0 1 1 10000 0 10000 1000000 0 0 10000 1000000 0], D4 = D9 = [0 100000000 100000000 0 0 1000000 1000000 1 0 100 0 0 10000 0 0], D5 = D10 = [0 100000000 0 100000000 0 1000000 1000000 1 0 0 10000 0 100 0 0]; In C3, lengthways teaming is given the biggest possibility, and in order to focus on the effect of lengthways teaming, the possibilities of transition to traverse teaming are set to 0. (4) Configuration 4 (C4): Lengthways teaming is allowed, traverse teaming, diagonal teaming and reverse-diagonal teaming are forbidden. D1 = D6 = [100 10000 100 1 1 1], D2 = D7 = [0 0 0 0 0 0 0 0 0 0 0 0 0 0], D3 = D8 = [0 0 10000 0 0 0 0 1 0 1 100 0 0 1 100 0], D4 = D9 = [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0], D5 = D10 = [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]; C4 has an extreme bias to lengthways teaming; it forbids all other teaming manners. C1–C4 are designed, refer to Refs. [11,15,19]. In these references whenever there is no obstacle in the front, pedestrians will move forward, because forward moving has the largest gain. When the front cell is occupied, a pedestrian will take avoidance actions. This indicates pedestrians will take the transition corresponding to the largest gain. Suggested configuration rules: 1. Possibility values should be consistent with the demanding bias, such as in C1, traverse teaming has higher priority. 2. Under the same teaming manner, the transitions corresponding to higher gain has higher priority. 3. Shared teaming manner and gain motivations increase the transitions in which the two pedestrians move forward as a team. 4. The possibility of one priority level is 100 times greater than its adjacent one-degree-lower priority level. 3.2. Effect on phase transition Experiment 1. Use C1 as the possibility configuration and set W = 20, N = 170. Change M from smaller values to bigger values, and observe the phase transition course. Consider Fig. 7. Red triples represent forward moving pedestrians, and green triples stand for backwards moving pedestrians. Two triples with a line means a team. When there is no team-moving (N = 170, M = 0), because of high density, the crowd begins to self-organize (Fig. 7-a). When there are a small amount of teams (N = 170, M = 20), as shown by Fig. 7-b, the self-organized straps change from narrow to wide. In order to overcome blockage, the self-organized straps change from narrow to wide. When there are more teams, the system will enter a jam state (N = 170, M = 40), as shown by Fig. 7-c. As Refs. [15,21–25,29] report that without team-moving there is a critical density value in the system. When the density is below its critical value, the pedestrians will move freely; when the density exceeds that value, the system will become jammed. This phase transition is quite similar to the phase transition shown by Fig. 7. This illuminates that the effect of merely adding teams is equal to adding the pedestrian number, they all create more blocks in the crowd. In experiment when M = 0 and M = 20, N = 170 is smaller than the critical density, while when M = 40, N = 170 is bigger than the critical density, this means adding teams will make the critical density decrease.
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Fig. 8. Relationship of critical density — team number when W = 20.
When using the configurations C2–C4 to repeat this experiment, the phenomenon of critical density point drift also occurs. Experiment 2 was designed to quantify the relationship of M with critical density under different configurations. 3.3. Effect on critical density Experiment 2. Set W = 20, and use C1–C4 respectively. Under each configuration, the critical point was recorded corresponding to different team numbers. As an example, take C1—first fix M = 0, and set N = 0, change the value of N to continually increase. The phase transition course was observed and critical density recorded. Then re-fix M = 10 and repeat. Finally, change M to larger numbers. Results are shown by Fig. 8. By Fig. 8, the below phenomena can be found. (1) In the experiments of C2 and C4, adding team numbers will decrease critical density; in other words, team-moving increases jamming. This means both the traverse teaming and lengthways teaming will create blocks in a crowd, and reduce corridor capacity. The curve of C2 is below the curve of C4, this means traverse-teaming makes more blocks than lengthways teaming. (2) The curve of C2 is below the curve of C1. This means for the same team numbers M the critical density of C1 is significantly higher than that of C2. This is because in C1, diagonal-teaming and reverse-diagonal-teaming are allowed. The effects of diagonal teaming and reverse-diagonal-teaming are like a lubricant. According to aerodynamics, diagonal teaming and reverse-diagonal-teaming will reduce blocks significantly. (3) Considering C3, adding team numbers will enlarge critical density. This is because under C3, an efficient avoid mode is formed for teaming pedestrians. In C3, the lengthways teaming is a priority, so the blocks are not as big as seen in C1 or C2. In lengthways teaming, one pedestrian is a leader, and the other is a follower. When they meet an obstacle, the leader will take avoidance action first; this action will change teaming manner to diagonal or reverse-diagonal, so the block will be reduced further. Very little space is needed to change teaming manner from diagonal and reverse-diagonal to lengthways; and it is easy to revert lengthways teaming. This avoid-mode has high efficiency. In C4, the diagonal and reverse-diagonal motions are forbidden, so the avoid efficiency is lower than that of C3. Summarily speaking the effect of team-moving on corridor capacity is mainly decided by bias to teaming-manner. When pedestrians are apt to lengthways teaming, diagonal teaming and reverse diagonal teaming, the negative effect on capacity is small, and sometimes it will even enlarge the capacity. While when pedestrians are apt to traverse teaming the capacity will reduce significantly and the jam is happens easily. 3.4. Effect on velocity–density curve and flow–density-curve Experiment 3. Use C1 configuration and set W = 20. Draw the velocity–density curves and flow–density-curves at the conditions that M = 0, 10, 20, . . . , 70 respectively. See Fig. 9.
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Fig. 9. Effect on velocity–density curve and flow–density curve when W = 20.
Fig. 10. Reduced capacity-added team number.
(1) Fig. 9 shows that the velocity–density curves (and flow–density) under different M values have the same varying tendency. There are critical density points in the curves. Below the critical density point, the velocity varies gently, and stays near v = 1.0 level, which means pedestrians can move freely; flow increases with the density, proportionally. When the density exceeds the critical point, both the velocity and flow will drop sharply. While the critical density points corresponding to different M are different. Fig. 9 shows that by increasing M, the critical density will decrease continually. The critical density is just the density corresponding to different M in the green solid line of Fig. 8 (the results under C1). Therefore, Fig. 9 is just another description of Fig. 8 when taking C1 as the configuration.
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Fig. 11. Team number of diagonal and reverse-diagonal teams.
Fig. 12. Relationship of critical density — team number when W = 30.
(2) Fig. 9 also shows when the density is small, team-moving has little effect on velocity and flow (Fig. 9-a and b, the curves when N ≤ 100). While from the range of N = 100 to the critical density, to the same density when M increases the velocity will decrease, and to each curve with the increase of N, the velocity will decrease, too. This shows that both the ways of merely increasing M and N will create blocks in crowd, and will decrease velocity. To flow, Fig. 9 shows a similar phenomenon. (3) Figs. 8 and 9 have a corresponding relationship. For the same M, the densities in the green solid line of Fig. 8 (the results under C1) equals critical density in Fig. 9. It is easy to see that for other configurations (C2–C4), the velocity–density curves and flow–density curves under different M are similar to Fig. 9, except that the critical densities for each configuration are different. 3.5. Marginal utility of effect on corridor capacity There is an interesting phenomenon seen in Fig. 9. The distance of critical points of the adjacent two lines will decrease when M increases. For example, the critical points distance of M = 10 and M = 20 is smaller than that of M = 0 and M = 10. The critical points distance of M = 20 and M = 30 is smaller than that of M = 10 and M = 20. This means that the effect of increasing M to corridor capacity is nonlinear. When adding the first 10 teams to the system, the capacity will
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Fig. 13. Course of evolution with N = 750 (a. M = 0, t = 250 step b. M = 0, t = 300 step c. M = 0, t = 400 step d. M = 20, t = 50 step e. M = 20, t = 100 step f. M = 20, t = 150 step g. M = 40, t = 50 step h. M = 40, t = 100 step i. M = 40, t = 150 step).
drop fast, while when adding the second 10 teams to the system, the capacity will not drop as fast. Fig. 10 is derived from Fig. 9; it shows that by adding each 10 teams to the system, capacity drops, while increasing M will reduce the effect on corridor capacity. This is quite similar to the ‘‘marginal utility’’ concept in economics. ‘‘Marginal utility’’ here – in this discussion – is hypothesized as follows: After C1 is used, increasing M will create more blocks in a crowd, so the capacity will drop when M increases. In C1, diagonal and reverse-diagonal teaming is allowed, so when the blocks increase, more pedestrians will take diagonal and reverse-diagonal teaming manners. Diagonal and reverse-diagonal teaming will reduce blocks significantly and slow the decrease of capacity. There is a negative feedback loop in the intersection; this is consistent with the concept of marginal utility in economics. To prove this hypothesis the below experiment was designed: Experiment 4. Use the configuration C1 and set W = 20. Fix N = 130, N = 150, N = 170 respectively. For each fixed N recording the number of diagonal and reverse-diagonal teams. More than 40 independent simulations were conducted for each set of parameters, and the mean values are provided in Fig. 11. It can be found that for each fixed N, the teams taking diagonal and reverse-diagonal manner will increase with the increase of M. While the curve of N = 170 is above that of N = 150 and the curve of N = 150 is above that of N = 130. Adding more teams and adding more pedestrians will stimulate more diagonal and reverse-diagonal teams. 3.6. Other phenomena analysis (a) Effect of space size Experiment 5. Set W = 30. Use configurations C1–C4 respectively to repeat Experiment 2. See Fig. 12.
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By comparing Fig. 12 with Fig. 8, when taking different space size, the curves of critical density — team numbers are quite similar. This shows that when W = 30, the results and laws discussed in 3.2–3.5 are still valid. (b) Effect on evolution course with time Experiment 6. Set W = 40, N = 750, use C1. Carry out 3 simulations with M = 0, M = 20, M = 40 respectively, and recording the revolution course with times for each simulation. In the three simulations, the initial pedestrian distributions are the same. For the simulation of M = 20, 20 randomly paired teams are used. In the simulation of M = 40 there are 40 teams. See Fig. 13. It can be found that the jam is first occurring locally (Fig. 13-(a), (d) and (g)), later it will diffuse to the whole system. Then a complete jam will form. When there is no team-moving (M = 0), more time passes from an initial condition to a local jam (Fig. 13-(a), (b)). With team-moving, much less time is required to form a jam (Fig. 13-(e), and (h)). This is another validation that team-moving facilitates jamming. It can also be found that when there is no team-moving or when there are only small amounts of team-moving, there is only one local jam point in the formation of complete jam (Fig. 13-(a), (d)). With increased team-moving, there may be more than one local jam point (Fig. 13-(g))). One jam point may enlarge incrementally, and others will join this jam point or disappear. When there are multi-jam points, it will spend a lot of time to form complete jam. 4. Conclusion A cellular automation model is established to simulate the pedestrian team-moving behavior, and the rules for doublepedestrian team-moving are defined. Effects of team-moving are discussed considering velocity, flow, corridor capacity and phase transitions. Simulation results show the significant effect of team-moving. The effect is further influenced by bias to teaming manners. When pedestrians join in traverse teaming, team-moving will create blocks in a crowd, and the critical density will decrease and the corridor capacity will weaken. When pedestrians utilize other teaming manners, such as lengthways, diagonal or reverse-diagonal teaming, fewer and smaller blockages result. When pedestrians take the mixed teaming behaviors including lengthways, diagonal, and reverse-diagonal, the corridor capacity will increase as the team number increases. The marginal utility of team-moving is also noted. It is reasonable to believe that this model has potential practical applications in future pedestrian dynamics studies. Acknowledgments We would like to thank the anonymous referees for their insightful suggestions. This work is funded by the Fundamental Research Funds of Ministry of Education of China (2011JBM064, 2011JBM058), National Natural Science Foundation of China (70901007), Doctoral Fund of Ministry of Education of China (20090009120014). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
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Physica A journal homepage: www.elsevier.com/locate/physa
Team-moving effect in bi-direction pedestrian flow Ziyang Wang a,b,∗ , Bingxue Song c,d , Yong Qin b , Limin Jia b a
School of Traffic and Transportation, Beijing Jiaotong University, Beijing, 100044, China
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State Key Laboratory of Railway Control and Safety, Beijing Jiaotong University, Beijing, 100044, China
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Civil & Environment School, University of Science and Technology Beijing, Beijing, 100083, China
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Work safety Statistics & Analysis Center, Beijing Municipal Institute of Labour Protection, Beijing, 100054, China
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Article history: Received 25 October 2010 Received in revised form 8 December 2011 Available online 20 January 2012 Keywords: Team-moving Pedestrian movement Cellular automation Corridor capacity Marginal utility of team-moving
abstract We propose a cellular automation model to simulate team-moving behavior in bidirectional pedestrian flow. The moving rules for double-pedestrian teaming include the constraint that pedestrians remain on adjacent cells. Phase transition, critical density — team number, velocity–density, and flow–density relationships are key component parts of the analysis. Simulations show that team-moving produces significant corridor capacity effects, and effects highly depend on the type of teaming behavior. In daily life, pedestrians prefer traverse teaming; under this bias, as teaming pedestrians increase in number, critical density reduces; that means traverse teaming will weaken the capacity of the corridor. The effect of traverse team-moving is nonlinear, and capacity will continually reduce as the team numbers increase; however, reduction rate will decay. We call this phenomenon, ‘‘the marginal utility of team-moving.’’ © 2012 Elsevier B.V. All rights reserved.
1. Introduction Cellular automation is widely used to simulate pedestrian movements. By setting simple local rules to each individual, an approximation of actual pedestrian behavior can be created, and collective pedestrian behavior emerges as an outgrowth interaction of the micro-simulation rules set [1–10]. This is very helpful in studying pedestrian movement law, as well as for analyzing and designing transportation facilities. In most current studies of pedestrian dynamics simulation, the main idea is taking the isolated individuals as the modeling objective. Related research work includes the Two Process Model [1], Lattice Gas Model [2–4], Floors Model [5–10], Pre-fixed Probabilities Model [11], Dynamic Parameters Model [12] and the Multi-grid Model [13,14]. In these studies usually the pedestrian flow is partitioned into three types according to the moving directions, which are bi-direction flow, laterallyinterfered flow and evacuation flow. Bi-direction flow restricts the moving direction to forward and backwards [15], and laterally-interfered flow restricts the pedestrian move laterally [16], while evacuation flow has little restriction on flow direction [17,18]. Among them the bi-direction flow model uses simple rules to reveal complex collective moving phenomena, such as ‘‘quick is low’’ [19,20], self-organized, jam formulation, and has been considerately used in the studying of pedestrian movement. Among the researches for the bi-directional model, Fang, et al. presented a set of four-direction moving rules for CA local moving [15], which allowed each pedestrian four latent positions for next step moving. Though the rules are simple, they can reveal most pedestrian dynamics, and the ‘‘back-step strategy’’ can reduce the jamming conditions significantly. Based on Fang’s work, Yue et al. modified the moving rules to eight-direction [21], and discussed the pedestrian dynamics under
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Corresponding author at: School of Traffic and Transportation, Beijing Jiaotong University, Beijing, 100044, China. E-mail address: [email protected] (Z.Y. Wang).
0378-4371/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2011.12.066
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the conditions of direction split and right-side walking habit. Similar work includes Refs. [22–25]. Unfortunately, most of the work did not consider the team-moving phenomenon, though it is very common in reality. For example, observations at Beijing’sWangFuJing Business Street, demonstrate more than 70% of pedestrians were in a team-moving state. In other public places, like parks and theaters, it is quite similar. Definition 1. Team-moving means that pedestrians join to form a team; in moving they keep close and keep together. Pedestrians in the same team tend to remain close to each other to avoid obstacles; it is quite different with isolated individual movement. The collective dynamics are different. Yang et al. discussed the kin behavior in evacuation flow [26]; the main work was using different kin behavior coefficients to generate the team-moving phenomenon. That study lacked the quantitative measurement of team-moving effects. Moussaïd et al. observed the team-moving phenomenon in public parks under natural conditions [27], and used a ‘‘social force model’’ to show that the transformation depends on two factors – communication – and crowd density. Unfortunately, their work did not measure the effect of different teaming manners. Summarily speaking, pedestrian dynamics studies of the team-moving phenomenon have been largely neglected. Interactions of individuals together with the collective dynamics under different teaming manners require additional deep study. A new bi-directional model with team-moving behavior is proposed here, which describes the individuals moving forwards and avoiding obstacles under team-moving condition. In this model, the probability of position transition reflects the pedestrian bias to different teaming manners. The simulation results show in the quantitative way how team-moving affects the collective dynamics, such as phase transition, velocity–density curve and flow–density curve. It is found that the effect is primarily decided by the bias to teaming manner. The collective dynamics under traverse teaming is quite different with that under lengthways teaming, while diagonal teaming will also play an important role to effect the collective dynamics. It is also found that there is some marginal utility in the effect of team-moving to the corridor capacity, the teaming effect will decay with the increase of team number. 2. Bi-direction model with team-moving 2.1. Space partition and moving rules In the model, pedestrian moving space is portioned into W × W grids in the plane. Every grid is a cell. Pedestrians move on the grids, and at each time step, each cell must be occupied by one pedestrian or must be empty. Under the doublepedestrian team-moving condition, each team occupies two cells. The size of a cell corresponds to 40 × 40 cm2 , and this the typical space occupied by a pedestrian in a dense crowd [28]. There are two type of moving directions. One is moving forward, the other is moving backwards; back stepping is inhibited. According to the manner of teaming, there are ten types of individuals in the system; five of them move forward and the other five backward. Figs. 1–5 show at k step the positions of the five forward moving types and their possible transition positions at k + 1 step. Fig. 1 shows the conditions for isolated individuals, at step k its state is T10 (k), it has 6 possible positions at step k + 1, those are T11 (k + 1), T12 (k + 1), . . . , T16 (k + 1) respectively. The possibility of transition to i those 6 positions are D11 /N1 , D21 /N1 , . . . , D61 /N1 respectively, and D11 , D21 , . . . , D61 are nonnegative numbers, N1 = i=1 D1 . Figs. 2–5 show conditions for teaming individuals. Taking Fig. 2 as an example, at step k the state is signed by T10 (k), by enumerating it has 14 possible positions at step k + 1, those are signed by T21 (k + 1), T22 (k + 1), . . . , T214 (k + 1) respectively.
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i The possibility of transition to those 14 positions are D2 /N2 , with D2 ≥ 0, j = 1, 2, . . . , 14 respectively, and N2 = i =1 D 2 . The position transition condition of the five backwards moving individuals is the center symmetry to Figs. 1–5, signed by 0 (k). The possibility values represent different moving and teaming biases. T60 (k), T70 (k), . . . , T10 The boundaries of upper and bottom are periodic, that means when the forward-moving pedestrian reaches the upper boundary, it will move back from the bottom boundary, and when the backwards-moving pedestrian arrives at the bottom it will return back from the upper boundary. The left and right boundaries are close. The total number of each type of individual is constant.
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2.2. Updating rules Five updating rules exist in the model. (1)At the initial step, all the isolated individuals and team-moving individuals are randomly distributed on the W × W grids. (2)The update is consequential. Every individual is assigned a number randomly, and at each time step the individual positions are updated according by the numbers. Isolated individuals each have a number, and each team has a number. (3) Transition positions are decided by Figs. 1–5, and are computed randomly by the possibility. When the potential positions are blocked by other pedestrians or by a wall, the corresponding possibility should be modified to 0. For example, 13 when at step k some traverse teaming individuals meet an obstacle as Fig. 6-a, D22 , D32 , D42 , D11 2 , D2 should be set to 0. When 8 13 1 11 the left wall is besides them as Fig. 6-b, D2 , D2 , D2 , D2 should be set to 0.
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Fig. 1. Possible transition positions for isolated forward moving individuals.
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Fig. 2. Possible transition positions for forward traverse teaming individuals. A1 A1
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Fig. 3. Possible transition positions for forward lengthways teaming individuals.
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Fig. 4. Possible transition positions for forward diagonal teaming individuals.
(4) The position transition should be consistent to team-moving habit in common sense, otherwise the corresponding possibility should be set to 0. (5) Team-moving habit definitions: (a) when one pedestrian is ahead of another, the front is the leader and the other is the follower. The follower should not violate the leader’s moving purpose; (b) when one moves to the left, the other one should not move to the right. So from these two team-moving habits in common practice, the following possibility values 9 13 16 5 9 15 9 10 15 14 11 12 14 1 12 14 should be set 0, D13 2 , D2 , D3 , D3 , D3 , D4 , D4 , D4 , D4 , D4 , D4 , D5 , D5 , D5 , D5 , D5 , D5 . 3. Simulation results N is the total number of pedestrians. M is the total number of teams. The total density p is defined as the value of N divided by the total number of cells, W × W . The velocity of pedestrians moving in one time-step is defined as the value
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Fig. 5. Possible transition positions for forward reverse-diagonal teaming individuals.
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Fig. 6. Transition possibility values modification: (a) front cell is occupied; (b) left side is boundary.
of the number of pedestrians moving ahead divided by the total number of walkers, N. The flow of pedestrians is the sum number of up walkers moving through the upper boundary plus down walkers moving through the bottom boundary in one time unit. ⟨v⟩ and ⟨f ⟩ are computed according to the mean values of velocity and flow. For each simulation, altogether 1500 steps were carried out. The first 1000 steps were used to start the simulation and were then discarded. The values of ⟨v⟩ and ⟨f ⟩ are computed according to the last 500 steps. 3.1. Parameter configuration The transition position possibility represents different moving and teaming bias. There are four kinds of teaming manners, traverse teaming (Fig. 2, T20 (k)), lengthways teaming (Fig. 3, T30 (k)), diagonal teaming (Fig. 4, T40 (k)), and reverse-diagonal teaming (Fig. 5, T50 (k)). Traverse teaming is favored because it is helpful for communication [27]. The police and army when patrolling often adopt lengthways teaming. In daily life, pedestrians will assume mixed teaming behaviors of the four types. For studying the effect of each manner, we decompose them as isolated behaviors, and then later recompose some of them. (1) Configuration 1 (C1): Traverse teaming is prior, diagonal teaming and reverse-diagonal teaming are allowed, and lengthways teaming is forbidden. D1 = D6 = [100 10000 100 1 1 1], D2 = D7 = [100 0 100 100000000 100 0 100 10000 10000 10000 1000000 1000000 0 0], D3 = D8 = [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0], D4 = D9 = [1000000 0 0 1000000 0 0 0 1 0 1000 0 100 0 0], D5 = D10 = [0 0 1000000 0 1000000 0 0 1 0 0 100 0 100 0 0]; In C1, traverse teaming is the biggest possibility, and in order to focus on the effect of traverse teaming, the possibilities of transition to lengthways teaming are set to 0. (2) Configuration 2 (C2): Traverse teaming is allowed, lengthways teaming, diagonal teaming and reverse-diagonal teaming are forbidden. D1 = D6 = [100 10000 100 1 1 1], D2 = D7 = [0 0 0 10000 0 0 0 1 1 1 100 100 0 0], D3 = D8 = [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0], D4 = D9 = [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0], D5 = D10 = [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]; C2 has an extreme bias to traverse teaming; it forbids all other teaming manners.
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Fig. 7. Effect on phase transition (a. N = 170, M = 0, b. N = 170, M = 20, c. N = 170, M = 40).
(3) Configuration 3 (C3): Lengthways teaming is prior, traverse is forbidden, diagonal teaming and reverse-diagonal teaming are allowed. D1 = D6 = [100 10000 100 1 1 1], D2 = D7 = [0 0 0 0 0 0 0 0 0 0 0 0 0 0], D3 = D8 = [0 100 100000000 100 0 1 1 10000 0 10000 1000000 0 0 10000 1000000 0], D4 = D9 = [0 100000000 100000000 0 0 1000000 1000000 1 0 100 0 0 10000 0 0], D5 = D10 = [0 100000000 0 100000000 0 1000000 1000000 1 0 0 10000 0 100 0 0]; In C3, lengthways teaming is given the biggest possibility, and in order to focus on the effect of lengthways teaming, the possibilities of transition to traverse teaming are set to 0. (4) Configuration 4 (C4): Lengthways teaming is allowed, traverse teaming, diagonal teaming and reverse-diagonal teaming are forbidden. D1 = D6 = [100 10000 100 1 1 1], D2 = D7 = [0 0 0 0 0 0 0 0 0 0 0 0 0 0], D3 = D8 = [0 0 10000 0 0 0 0 1 0 1 100 0 0 1 100 0], D4 = D9 = [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0], D5 = D10 = [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]; C4 has an extreme bias to lengthways teaming; it forbids all other teaming manners. C1–C4 are designed, refer to Refs. [11,15,19]. In these references whenever there is no obstacle in the front, pedestrians will move forward, because forward moving has the largest gain. When the front cell is occupied, a pedestrian will take avoidance actions. This indicates pedestrians will take the transition corresponding to the largest gain. Suggested configuration rules: 1. Possibility values should be consistent with the demanding bias, such as in C1, traverse teaming has higher priority. 2. Under the same teaming manner, the transitions corresponding to higher gain has higher priority. 3. Shared teaming manner and gain motivations increase the transitions in which the two pedestrians move forward as a team. 4. The possibility of one priority level is 100 times greater than its adjacent one-degree-lower priority level. 3.2. Effect on phase transition Experiment 1. Use C1 as the possibility configuration and set W = 20, N = 170. Change M from smaller values to bigger values, and observe the phase transition course. Consider Fig. 7. Red triples represent forward moving pedestrians, and green triples stand for backwards moving pedestrians. Two triples with a line means a team. When there is no team-moving (N = 170, M = 0), because of high density, the crowd begins to self-organize (Fig. 7-a). When there are a small amount of teams (N = 170, M = 20), as shown by Fig. 7-b, the self-organized straps change from narrow to wide. In order to overcome blockage, the self-organized straps change from narrow to wide. When there are more teams, the system will enter a jam state (N = 170, M = 40), as shown by Fig. 7-c. As Refs. [15,21–25,29] report that without team-moving there is a critical density value in the system. When the density is below its critical value, the pedestrians will move freely; when the density exceeds that value, the system will become jammed. This phase transition is quite similar to the phase transition shown by Fig. 7. This illuminates that the effect of merely adding teams is equal to adding the pedestrian number, they all create more blocks in the crowd. In experiment when M = 0 and M = 20, N = 170 is smaller than the critical density, while when M = 40, N = 170 is bigger than the critical density, this means adding teams will make the critical density decrease.
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Fig. 8. Relationship of critical density — team number when W = 20.
When using the configurations C2–C4 to repeat this experiment, the phenomenon of critical density point drift also occurs. Experiment 2 was designed to quantify the relationship of M with critical density under different configurations. 3.3. Effect on critical density Experiment 2. Set W = 20, and use C1–C4 respectively. Under each configuration, the critical point was recorded corresponding to different team numbers. As an example, take C1—first fix M = 0, and set N = 0, change the value of N to continually increase. The phase transition course was observed and critical density recorded. Then re-fix M = 10 and repeat. Finally, change M to larger numbers. Results are shown by Fig. 8. By Fig. 8, the below phenomena can be found. (1) In the experiments of C2 and C4, adding team numbers will decrease critical density; in other words, team-moving increases jamming. This means both the traverse teaming and lengthways teaming will create blocks in a crowd, and reduce corridor capacity. The curve of C2 is below the curve of C4, this means traverse-teaming makes more blocks than lengthways teaming. (2) The curve of C2 is below the curve of C1. This means for the same team numbers M the critical density of C1 is significantly higher than that of C2. This is because in C1, diagonal-teaming and reverse-diagonal-teaming are allowed. The effects of diagonal teaming and reverse-diagonal-teaming are like a lubricant. According to aerodynamics, diagonal teaming and reverse-diagonal-teaming will reduce blocks significantly. (3) Considering C3, adding team numbers will enlarge critical density. This is because under C3, an efficient avoid mode is formed for teaming pedestrians. In C3, the lengthways teaming is a priority, so the blocks are not as big as seen in C1 or C2. In lengthways teaming, one pedestrian is a leader, and the other is a follower. When they meet an obstacle, the leader will take avoidance action first; this action will change teaming manner to diagonal or reverse-diagonal, so the block will be reduced further. Very little space is needed to change teaming manner from diagonal and reverse-diagonal to lengthways; and it is easy to revert lengthways teaming. This avoid-mode has high efficiency. In C4, the diagonal and reverse-diagonal motions are forbidden, so the avoid efficiency is lower than that of C3. Summarily speaking the effect of team-moving on corridor capacity is mainly decided by bias to teaming-manner. When pedestrians are apt to lengthways teaming, diagonal teaming and reverse diagonal teaming, the negative effect on capacity is small, and sometimes it will even enlarge the capacity. While when pedestrians are apt to traverse teaming the capacity will reduce significantly and the jam is happens easily. 3.4. Effect on velocity–density curve and flow–density-curve Experiment 3. Use C1 configuration and set W = 20. Draw the velocity–density curves and flow–density-curves at the conditions that M = 0, 10, 20, . . . , 70 respectively. See Fig. 9.
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Fig. 9. Effect on velocity–density curve and flow–density curve when W = 20.
Fig. 10. Reduced capacity-added team number.
(1) Fig. 9 shows that the velocity–density curves (and flow–density) under different M values have the same varying tendency. There are critical density points in the curves. Below the critical density point, the velocity varies gently, and stays near v = 1.0 level, which means pedestrians can move freely; flow increases with the density, proportionally. When the density exceeds the critical point, both the velocity and flow will drop sharply. While the critical density points corresponding to different M are different. Fig. 9 shows that by increasing M, the critical density will decrease continually. The critical density is just the density corresponding to different M in the green solid line of Fig. 8 (the results under C1). Therefore, Fig. 9 is just another description of Fig. 8 when taking C1 as the configuration.
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Fig. 11. Team number of diagonal and reverse-diagonal teams.
Fig. 12. Relationship of critical density — team number when W = 30.
(2) Fig. 9 also shows when the density is small, team-moving has little effect on velocity and flow (Fig. 9-a and b, the curves when N ≤ 100). While from the range of N = 100 to the critical density, to the same density when M increases the velocity will decrease, and to each curve with the increase of N, the velocity will decrease, too. This shows that both the ways of merely increasing M and N will create blocks in crowd, and will decrease velocity. To flow, Fig. 9 shows a similar phenomenon. (3) Figs. 8 and 9 have a corresponding relationship. For the same M, the densities in the green solid line of Fig. 8 (the results under C1) equals critical density in Fig. 9. It is easy to see that for other configurations (C2–C4), the velocity–density curves and flow–density curves under different M are similar to Fig. 9, except that the critical densities for each configuration are different. 3.5. Marginal utility of effect on corridor capacity There is an interesting phenomenon seen in Fig. 9. The distance of critical points of the adjacent two lines will decrease when M increases. For example, the critical points distance of M = 10 and M = 20 is smaller than that of M = 0 and M = 10. The critical points distance of M = 20 and M = 30 is smaller than that of M = 10 and M = 20. This means that the effect of increasing M to corridor capacity is nonlinear. When adding the first 10 teams to the system, the capacity will
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Fig. 13. Course of evolution with N = 750 (a. M = 0, t = 250 step b. M = 0, t = 300 step c. M = 0, t = 400 step d. M = 20, t = 50 step e. M = 20, t = 100 step f. M = 20, t = 150 step g. M = 40, t = 50 step h. M = 40, t = 100 step i. M = 40, t = 150 step).
drop fast, while when adding the second 10 teams to the system, the capacity will not drop as fast. Fig. 10 is derived from Fig. 9; it shows that by adding each 10 teams to the system, capacity drops, while increasing M will reduce the effect on corridor capacity. This is quite similar to the ‘‘marginal utility’’ concept in economics. ‘‘Marginal utility’’ here – in this discussion – is hypothesized as follows: After C1 is used, increasing M will create more blocks in a crowd, so the capacity will drop when M increases. In C1, diagonal and reverse-diagonal teaming is allowed, so when the blocks increase, more pedestrians will take diagonal and reverse-diagonal teaming manners. Diagonal and reverse-diagonal teaming will reduce blocks significantly and slow the decrease of capacity. There is a negative feedback loop in the intersection; this is consistent with the concept of marginal utility in economics. To prove this hypothesis the below experiment was designed: Experiment 4. Use the configuration C1 and set W = 20. Fix N = 130, N = 150, N = 170 respectively. For each fixed N recording the number of diagonal and reverse-diagonal teams. More than 40 independent simulations were conducted for each set of parameters, and the mean values are provided in Fig. 11. It can be found that for each fixed N, the teams taking diagonal and reverse-diagonal manner will increase with the increase of M. While the curve of N = 170 is above that of N = 150 and the curve of N = 150 is above that of N = 130. Adding more teams and adding more pedestrians will stimulate more diagonal and reverse-diagonal teams. 3.6. Other phenomena analysis (a) Effect of space size Experiment 5. Set W = 30. Use configurations C1–C4 respectively to repeat Experiment 2. See Fig. 12.
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By comparing Fig. 12 with Fig. 8, when taking different space size, the curves of critical density — team numbers are quite similar. This shows that when W = 30, the results and laws discussed in 3.2–3.5 are still valid. (b) Effect on evolution course with time Experiment 6. Set W = 40, N = 750, use C1. Carry out 3 simulations with M = 0, M = 20, M = 40 respectively, and recording the revolution course with times for each simulation. In the three simulations, the initial pedestrian distributions are the same. For the simulation of M = 20, 20 randomly paired teams are used. In the simulation of M = 40 there are 40 teams. See Fig. 13. It can be found that the jam is first occurring locally (Fig. 13-(a), (d) and (g)), later it will diffuse to the whole system. Then a complete jam will form. When there is no team-moving (M = 0), more time passes from an initial condition to a local jam (Fig. 13-(a), (b)). With team-moving, much less time is required to form a jam (Fig. 13-(e), and (h)). This is another validation that team-moving facilitates jamming. It can also be found that when there is no team-moving or when there are only small amounts of team-moving, there is only one local jam point in the formation of complete jam (Fig. 13-(a), (d)). With increased team-moving, there may be more than one local jam point (Fig. 13-(g))). One jam point may enlarge incrementally, and others will join this jam point or disappear. When there are multi-jam points, it will spend a lot of time to form complete jam. 4. Conclusion A cellular automation model is established to simulate the pedestrian team-moving behavior, and the rules for doublepedestrian team-moving are defined. Effects of team-moving are discussed considering velocity, flow, corridor capacity and phase transitions. Simulation results show the significant effect of team-moving. The effect is further influenced by bias to teaming manners. When pedestrians join in traverse teaming, team-moving will create blocks in a crowd, and the critical density will decrease and the corridor capacity will weaken. When pedestrians utilize other teaming manners, such as lengthways, diagonal or reverse-diagonal teaming, fewer and smaller blockages result. When pedestrians take the mixed teaming behaviors including lengthways, diagonal, and reverse-diagonal, the corridor capacity will increase as the team number increases. The marginal utility of team-moving is also noted. It is reasonable to believe that this model has potential practical applications in future pedestrian dynamics studies. Acknowledgments We would like to thank the anonymous referees for their insightful suggestions. This work is funded by the Fundamental Research Funds of Ministry of Education of China (2011JBM064, 2011JBM058), National Natural Science Foundation of China (70901007), Doctoral Fund of Ministry of Education of China (20090009120014). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
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