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Simulating bi-directional pedestrian flow in a cellular automaton model considering the body-turning behavior
Accepted Manuscript Simulating bi-directional pedestrian flow in a cellular automaton model considering the body-turning behavior Cheng-Jie Jin, Rui Jiang, Jun-Lin Yin, Li-Yun Dong, Dawei Li PII: DOI: Reference:
S0378-4371(17)30433-8 http://dx.doi.org/10.1016/j.physa.2017.04.117 PHYSA 18219
To appear in:
Physica A
Received date: 10 December 2016 Revised date: 21 March 2017 Please cite this article as: C.-J. Jin, R. Jiang, J.-L. Yin, L.-Y. Dong, D. Li, Simulating bi-directional pedestrian flow in a cellular automaton model considering the body-turning behavior, Physica A (2017), http://dx.doi.org/10.1016/j.physa.2017.04.117 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
The body-turning behavior in bi-directional pedestrian flow experiments is studied. This behavior is introduced into one cellular automaton model. The occupied area of each pedestrian is set as a rectangle of 0.4m*0.2m. The results from experiments and simulations are compared by quantitative analysis. The effect of new rules is significant in two different simulation scenarios.
*Manuscript Click here to view linked References
Simulating bi-directional pedestrian flow in a cellular automaton model considering the body-turning behavior Cheng-Jie Jin 1,2 , Rui Jiang 3, Jun-Lin Yin 1,2, Li-Yun Dong 4, Dawei Li 1,2 1 Jiangsu Key Laboratory of Urban ITS, Southeast University of China, Nanjing, Jiangsu, 210096, People's Republic of China 2 Jiangsu Province Collaborative Innovation Center of Modern Urban Traffic Technologies, Nanjing, Jiangsu, 210096, People's Republic of China 3 MOE Key Laboratory for Urban Transportation Complex Systems Theory and Technology, Beijing Jiaotong University, Beijing, 100044, People’s Republic of China 4 Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai, 200072, People's Republic of China Abstract: In the experiments of bi-directional pedestrian flow, it is often observed that pedestrians turn their bodies and change from walking straight to walking sideways, in order to mitigate or avoid the conflicts with opposite walking ones. When these conflicts disappear, pedestrians restore and walk straight again. In the turning states, the forward velocities of pedestrians are not affected. In order to simulate this body-turning behavior, we use a cellular automaton (CA) model named ITP model, which has been proposed before. But the occupied area of one pedestrian is set as 0.4m*0.2m. After the introduction of new rules of turnings and restorations, the pedestrians become more intelligent and flexible during the lane formation process, and some improvements of the fundamental diagram of pedestrian flow can be found. The simulation results of two different scenarios under open boundary conditions are also presented, and compared with the experimental data. It is shown that the new model performs much better than the original model in various tests, which further confirms the validity of the new rules. We think this approach is one useful contribution to the pedestrian flow modeling. Keywords: pedestrian flow; bi-directional flow; experiment; cellular automaton; turning. 1. Introduction The study of pedestrian flow has a very long history [1]. Many different approaches are used for the modeling of pedestrian flow, including the macroscopic ones in which the flow-density and velocity-density relationship is analyzed, and the microscopic ones in which the movement of each pedestrian is simulated and presented. The latter can be further classified into several types, and in this paper we concentrate on one simple type: cellular automaton (CA) model. In CA models the time and space are both discretized, which makes the computer simulation fast and easy to run [2]. In recent years, CA models become popular in many different fields, including the study of pedestrian movements. Particularly, there are many studies concerning the pedestrian flows in a narrow corridor [3-14], which is a common situation in real life. For this situation, the bi-directional flow is more interesting due to the complexity: some phenomena of self-organization often attract researchers' attention, including the automatic lane formation, etc.
But in these previous CA simulations of bi-directional pedestrian flow, there are still some disadvantages. Although many different modeling methods and techniques are used, in the simulations the jams usually appear at lower densities, and the critical density, critical flow and jam density may be much smaller than that in the empirical data [15-19]. It seems that the rules of pedestrian movements are usually not realistic enough. Besides, in some pedestrian flow experiments [20], we observed some interesting phenomena, in which pedestrians occupy smaller area than ordinary situations, and they can turn the bodies to pass the high-density area without waiting. This behavior is very flexible, and can help to significantly enhance the capacity of bi-directional flow. However, until now the modeling of this behavior is not sufficient. The main reason is that in many previous studies, the area occupied by one pedestrian is set as a square (usually it is 0.4m*0.4m). Thus no matter it is turned or not, it has no influence on the other pedestrians or the surrounding environment. In the recent years, some researchers try to use rectangle to represent the pedestrian. For example, in Ref.[8] the turning behaviors in bi-directional flow are simulated with the name of "sidle effects", and in Ref. [21] these phenomena in pedestrian evacuations are discussed. Both of them are lattice gas models, in which the moving direction of each pedestrian is determined by the calculation of transition probabilities. However, some problems still exist, e.g., the occupied area of one pedestrian is set as 0.6m*0.3m in Ref.[8] and 0.45m*0.3m in Ref.[21]. Both of them are much larger than the real results in the crowded situation. Besides, the simulation results in bi-directional flow [8] are not compared with the empirical or experimental data, so the validity is not clear. In order to investigate the essence of pedestrian movements, we organized some pedestrian flow experiments, and we also find some similar phenomena: pedestrians occupy smaller area, and they try to turn and restore frequently, so that they can pass others quickly and easily. In order to simulate this situation, we use the new configuration in which one pedestrian occupies the area of 0.4m*0.2m, and add the necessary rules of turnings and restorations into the pedestrian model named ITP model [22]. The effects of new rules are obvious: both the simulation results under periodic boundary conditions and open boundary conditions show that the new model performs much better than the previous model, and the pedestrians become more intelligent than before. The quantitative analysis of both experimental results and simulation results can help us to learn more about the mechanisms of bi-directional flow. Thus we think our attempts are great help for the pedestrian flow modeling. This paper is organized as follows. The data of the pedestrian bi-directional flow experiments is analyzed and discussed in Section 2. The rules of the previous ITP model are briefly introduced in Section 3. The new rules of turnings and restorations, and the corresponding simulation results under periodic boundary conditions are shown in Section 4. The simulation results under open boundary conditions with two different scenarios are presented, and compared with experimental results in Section 5. The conclusion is given in Section 6. 2. Experimental data analysis The experiments of bi-directional pedestrian flow were conducted in a narrow corridor, which is located in the campus of Shanghai University. The length and the width of the corridor is about 8m and 1.6m. 30 runs were recorded by a camera equipped above the corridor. In each run,
20-40 students are used, and in 18 runs the numbers of the pedestrians in two direction are the same. During the experiment, pedestrians wear different caps to represent their different moving directions. Although this experiment is simple, we find some very important phenomena: in all the runs, no deadlock occurs. This is because pedestrians are very flexible and they have many methods to pass. This procedure seems very easy for the pedestrians, and it does not need much time. We find there are two factors which can be crucial to avoid deadlocks: (1) Smaller occupied area: at small densities, it is easy to understand that pedestrians want to have large rooms in order to be comfortable. This is why the occupied area of one pedestrian is usually set as a square of 0.4*0.4m in most previous studies of modeling. But when the density is high, we find they don't like waiting at all. They will try to move as quickly as possible, and will not be afraid of frequent body contacts. This strong tendency leads to the fact that their occupied areas are usually much smaller than the normal situation. Although the width of a pedestrian is not easy to be changed, the length seems easier to be compressed. Smaller length will make the occupied area looks like a narrow rectangle, rather than a normal square. Thus we think the area can be considered as 0.4m*0.2m in the modeling, which will be discussed later. (2) Frequent body-turning behaviors: in order to avoid conflicts, sometimes pedestrians can turn their bodies. They can change from walking straight to walking sideways, in order to pass the others. This intelligent behavior can help to form the pedestrian lanes very quickly, especially when the pedestrians from two opposite directions meet each others. The turning states of some pedestrians can be maintained for a long time, until they finally pass. And then, they can restore if there is enough room. These findings can be clearly shown in some typical examples. Here we use one software named Tracker (http://physlets.org/tracker/) to extract the information of pedestrians' movement in the video, including the positions, velocities, turning angles and shoulder widths, etc. The time step for all the data collection is 1s. Before the extraction, the length and the width of corridor (8m and 1.6m) are used for calibrating the video. To ensure the accuracy, all the experimental data presented in this section is collected by hand. In this paper, the pedestrians with different moving directions are marked by dark blue and yellow, which can give enough contrast in black-and-white mode. And in this section, all the pedestrians in the sketches are represented by ellipses. Here we take 20° as one critical value, and the turning angles which are smaller than 20° will not be considered in all the sketches and statistics. Now the sketches of the Example 1 are shown in Fig.1. All the pedestrians start walking from the two ends of the corridor, and we call this configuration Scenario A in the following sections. When the two groups just meet, no obvious phenomena can be observed in Fig.1(a). After a short while, 4 lanes are primarily formed in Fig.1(b). There are 9 pedestrians who are turning their bodies, and the maximum turning angle is 72°. In Fig.1(c) the 4 lanes become more clear, but there are still 9 pedestrians who are turning, and the maximum angle is 77°. In Fig.1(d), when some pedestrians have reached the end, 3 pedestrians are still moving forward and in turning states.
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(d) Fig.1. The typical sketches of Example 1. It belongs to Scenario A in the experiment. (a) T=3s; (b) T=5s; (c) T=7s; (d) T=9s. With some other initial configuration, although the procedure is different, the smaller occupied area and body-turning behaviors still can be found. Another typical situation is that at the beginning, 16 red pedestrians and 16 blue pedestrians stand very close. The sketches of the Example 2 are shown in Fig.3, and we call this configuration Scenario B in the following sections. In order to be compared with Fig.1, we choose the same moments to show the sketches. Due to the short distance, in Fig.3(a) the area is more crowded than that in Scenario A, and 15 pedestrians do not face their original moving directions. Especially, two clusters are temporarily formed, as marked by black curves. In these clusters 4 and 9 pedestrians are involved, and all of them are turning. The maximum angle even reaches 83°. After a short time, 4 obvious lanes are automatically formed in Fig.3(b), but there are still 13 pedestrians who are in turning states, and the maximum angle is still 75°. Finally, in Fig.3(c)(d) the temporary congestion disperses, and all the pedestrians gradually restore.
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(d) Fig.2. The typical sketches of Example 2. It belongs to Scenario B in the experiment. (a) T=3s; (b) T=5s; (c) T=7s; (d) T=9s. Then we check some time series data. The averaged velocities of pedestrians are shown in Fig.3, in which the Example 1 and 2 are compared. It should be noted that due to the limit of the camera scope, in Example 1 the velocities of some pedestrians cannot be recorded, so are all the other runs of Scenario A in the experiment. For example, in Fig.1(a) when T=3s, only 8 blue ones can be seen, and the last blue one (the 15th one) does not enter the camera scope until T=6s. Nevertheless, we can find something useful: (1) The basic trends are easy to understand. For Scenario A, the velocities reach the peak at the beginning, but for Scenario B, the velocities could only gradually increase. After the short-time congestion, both of them can increase again. (2) It is interesting that when the two groups of pedestrians meet each others, the results of two examples become similar, no matter when there exists one short-time congestion (T=4~6s) or when the congestion disperses (T>=7s). Starting from T=6s, when all the movements of blue ones are recorded in the camera, the averaged velocities become nearly quantitatively the same. It seems that the initial states of pedestrian flow do not have much influence on the results. (3) There are some obvious fluctuations of velocities when the congestion disperses (T=8s). This is mainly due to the features of open boundary conditions, rather than the properties of pedestrians: when some pedestrians with higher velocities reach the end of corridor, they will be
removed, and only some others with lower velocities are left for the statistics in the next time step. After a short while, they also can become fast.
Fig.3. The averaged velocities of pedestrians in two examples. The averaged velocities of the first 5 seconds in Example 1 are represented by different symbols, since they are not the results of all the pedestrians. The proportions of turned pedestrians are shown in Fig.4. The basic trends of the two examples are similar: during 3 seconds (T=3~5s in Example 1 and T=5~7s in Example 2), many pedestrians choose to turn. After this "peak period", most of them restore soon. This "peak period" is also the period when short-time congestion occurs in the experiment. During this period, the maximum proportion of turned pedestrians in Example 1 is about 33%, while that in Example 2 is about 50%. It also implies the center area in Example 2 is more crowded.
Fig.4. The proportions of turned pedestrians in two examples. The averaged velocities of turned and unturned pedestrians are show in Fig.5. It is very clear that in Example 1, the velocities of turned and unturned ones are nearly the same at every second. In Example 2, when T=3~4s, the velocities of turned ones are even larger. These results are very important, which shows the turning behaviors do not affect the forward movement of pedestrians at all. On the contrary, it becomes one great help for the solution of congestions.
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(b) Fig.5. The averaged velocities of turned and unturned pedestrians in two examples. (a) Example 1; (b) Example 2. Next, we focus on the microscopic information about pedestrians' turning behaviors. The distributions of the shoulder widths are shown in Fig.6. The results in both examples have the appearances which are similar to normal distribution. The peak is at 0.37m, which is also the averaged widths in the two examples. In China the averaged shoulder widths for male and female between 18-25 years old are 0.375m and 0.351m. Since in our experiment most pedestrians are male, the measured values are close to the statistical results.
Fig.6. The distributions of the shoulder widths in two examples. In Fig. 7 and 8, all the data are collected at each second when turning behaviors are found in
the experimental video. As shown in Fig.4 and Fig.5, in Example 1 the collecting time is T=4~9s, while for Example 2 it is T=1~7s. The distributions of the turning angles are shown in Fig.7. The differences between two examples can be seen: in Example 1 there are more smaller angles (A=20°~30°), while in Example 2 more results can be found at A=30°~70°. This shows not only the different levels of crowdedness in two examples, but also the great flexibility of pedestrians' movement.
Fig.7. The distributions of the turning angles in two examples. Note that in the experiment, pedestrians can choose to turn left or turn right, according to their own thoughts. This process is also interesting, and we show some statistics about the turning directions in Tab.1. Some new results can be found: (1) The number of right-turning pedestrians is much greater than that of left-turning pedestrians in two examples. It is not strange, since in China the basic traffic rules make people try to walk on the right side. (2) It is very interesting that some statistical results of right-turning pedestrians in two different examples are nearly the same, including the averaged velocities and averaged turning angles. We think it implies the universal tendency of pedestrians' right-turning behaviors (3) On the contrary, the statistical results of left-turning pedestrians in two examples are completely different. In Example 1 the velocities are higher and the angles are smaller, while in Example 2 the velocities are lower and the angles are similar to that of right-turning pedestrians. The reasons for this phenomenon still need to be investigated in the future, and we think one possible explanation is that in Example 1 the sample is not large enough. Table.1. The statistics for the pedestrians' turning directions Example
Turning direction
Number
Averaged velocity (m/s)
Averaged angle (°)
1 2 1 2
Left Left Right Right
15 24 23 45
0.69 0.43 0.60 0.59
32 41 41 41
And then, the relationship between each pedestrian's velocities and turning angles is shown in Fig.8. The scatter of data points in both figures further confirms that the turning angles have no influence on the velocities. Besides, the differences between two examples also can be seen. In
Fig.8(a) all the larger turning angles result from right-turning behaviors, which shows the general preferences of many pedestrians. On the contrary, in Fig.8(b) the maximum angle results from left-turning behaviors, which shows left-turning is also one possible option for some pedestrians when there is not enough room.
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(b) Fig.8. The relationship between each pedestrian's velocities and turning angles in two examples. (a) Example 1; (b) Example 2. Finally, we also investigate some statistics for all the 18 runs in our experiment, e.g., the averaged durations. In the runs of Scenario A, the total pedestrian numbers are 20, 30, 40. while in the runs of Scenario B, the total numbers are 20, 24, 28, ...,40. For each number two runs are performed, and the durations shown in Fig.9 are the averaged value of the two runs. It is clear that the times used for two scenarios are close to each other, which also coincides with the conclusion (2) we made above. We think the small difference between them is mainly due to the different walking distances: in Scenario A everyone needs to walk further.
Fig.9. the averaged durations of two scenarios in the experiment. 3. The rules of the basic model Here we briefly recall the rules of the ITP model, which has been discussed in our previous papers [22]. It is an improved version of the "two-process" model [23-25], in which the forward movement and the lateral movement of pedestrians are split and handled at two different sub-steps. The time step is set to 1s. In order to describe the pedestrian movements more accurately, the size of one cell is set to 0.1m*0.1m. In the previous paper [22], each pedestrian occupies the area of 0.4m*0.4m, which is a traditional configuration used in many other studies. But as we mentioned in Section 2, when the pedestrians are passing through others in the bi-directional flow, the occupied area could be considered as a rectangle of 0.4m*0.2m. Thus in all the following simulations, we will adopt this configuration, and we call this model as "basic model". The schematic illustration of the pedestrian's movement is shown in Fig.10, and the variables involved are introduced in Tab.2. The parameters used in the basic model are presented in Tab.3, in which the values and related explanations are also given.
Fig.10. An example of the surrounding environment of a pedestrian X. Table.2. The variables used in the basic model Name /
Definition and explanation The front gap to the nearest pedestrian. In Fig.7, since The lateral gap to the nearest pedestrian on the left/right side. The front velocity of the pedestrian.
.
The lateral velocity of the pedestrian. The "conflict parameter". We call the nearest pedestrian in the moving direction as "key one". If the moving directions of the pedestrian and the "key one" are the same, C=0; otherwise C=1. For example, in Fig.7, pedestrian B is the "key one" of pedestrian X and C=0. Besides, if , the one who has larger C value will be considered as the “key one”.
C
Table.3. The parameters used in the basic model Name and value =2.4m/s
=0.8m/s
R=0.1m/s2
S=5m
P=0.1
Definition and explanation The maximum value of . In Example 1, the maximum front velocity of all the pedestrians is about 2.3m/s at the beginning (when T=2s). In Example 2, the maximum value is only about 1.7m/s at the end (when T=7s). Therefore, we set the value as 2.4m/s in the simulations, since it is better to be an integer multiple of 0.4m. The maximum value of . In Example 1, the maximum lateral velocity is about 0.66m/s, and in Example 2 the value is about 0.63m/s. Therefore, we set the maximum value as 0.8m/s in the simulations, since it is better to be an integer multiple of 0.4m. The pedestrian-dependent preferred front velocity. In the experimental data, the averaged velocity of free walking is about 1.5m/s. Therefore, in order to present the variety of pedestrians' preferences, is assumed to be uniformly distributed over the interval [1.0, 2.0]m/s in the simulations. Here only the discrete value of 1.0, 1.1, 1.2..., 2.0 can be chosen. An random acceleration/deceleration value used in the update of forward movement. It represents the random fluctuations of velocities, and this idea is similar to that used in vehicular traffic flow models. It cannot be measured in this experiment, and we set it as 0.1m/s2. One critical gap which controls the actions of pedestrians. When , the pedestrian starts to consider the possibility of lateral movement. When the new rules of turning and restoration are added, for this situation the pedestrian also considers the possibility of turning. In this paper we set it as 5m in the simulations. It will be further discussed in Section 4, since it is crucial to the bi-directional flow. The randomization probability used for the velocity update. We set it as 0.1, which is the common value used in many CA models.
The flow chart of the rules of basic model is shown in Fig.11. The further explanations of some important steps are marked by numbers (1)(2)(3)(4), and presented as follows:
Fig.11. The flow chart of the basic model rules. (1) If one of following is satisfied, the lateral movement is needed: ① C=1 and . It implies potential conflict in the future. ② C=0 and . This rule is similar to the model rule in multilane vehicular traffic flow, which means that the forward movement is hindered by the "key one". (2) Sometimes new conflicts may occur, when there are many lateral moves, e.g., two adjacent pedestrians want to occupy the same cell. For this situation, one of them has to abandon the plan. (3) For the lateral movement, something should be noted: ① Suppose the conflict parameters and front gaps before(after) the lateral move are C1(C2) and ( ), respectively. If one of following is satisfied, it becomes "better": a.C2=0 and C1=1. It means the pedestrian can avoid potential conflicts by lateral move. b.C1=C2 and . It implies the moving condition is not significantly changed, but the front gap becomes larger. ② The exhaustive range of lateral velocity is: 0.1, 0.2, ..., min ( , ). Find the smallest velocity which can help the pedestrian become "better", and record the value. ③ A pedestrian tries to move to right firstly; if failed, then it tries to move to left. If it tries to move to left and ② is fulfilled, there is still 50% chance to give up the lateral movement. ④ One cooperative strategy between different pedestrians is used: for a pair of pedestrians (A and B) with opposing directions, if they are the '"key one" of each other, the potential plan should be particularly considered. Specifically, if A has the potential plan of moving left (right), B will not try to move right (left). (4) For the forward movement, there are three different situations to be considered:
① If
,
The deterministic calculation is similar to that in the Blue-Adler model [23-25]. It includes the consideration of the potential movement of the opposite pedestrian. The random deceleration rule is similar to that in many vehicular traffic flow models. ② If
and
,
③ If
and
,
The latter two situations also consider the random fluctuations of the front velocity. 4. The periodic boundary simulation with turnings and restorations As we discussed in Section 2, the rules of frequent turnings and restorations are necessary for the simulation of bi-directional flow. When these rules are added into the basic model, we get the "new model". The flow chart of the rules of new model is shown in Fig.12. It is clear that the new rules (framed by dashed lines) can be considered as one component whose structure is similar to the basic rules. The further explanations of some important steps are marked by numbers (5)(6)(7) (8), and presented as follows: (5) For the turning rules, we set 4 necessary conditions: Condition A: (There is enough gap before the pedestrian for the turnings). Condition B: C=1 (The "key one" has opposite moving direction). Condition C: (The pedestrian cannot move freely in the next time step). Condition D: (The gap is not large enough). It is clear that A has to be fulfilled, and B, C, D can be chosen. Actually, there could be many possible options for the turning rules, and it is difficult to determine which one is the best. Here we just choose one simple and reasonable rule: A and (B or C) and D. For the restoration rules, we also have 4 necessary conditions: Condition E: and (There is enough gap on the left and right of the pedestrian for the restorations). Condition F: C=0 (The "key one" has the same moving direction). Condition G: (The pedestrian can move freely in the next time step). Condition H: (The gap is large enough). It can be found that Condition EFGH are similar to the opposite situation of Condition ABCD, but there are still some minor differences. For example, if the critical value in Condition H is also set as S, it becomes meaningless, since in the high-density situation it is impossible for all the pedestrians to fulfill it. It is also difficult to determine the best restoration rules, and here we choose one simple combination: E and F and H. Since there is always , Condition G can be ignored here. (6) Sometimes new conflicts may occur, when there are many turnings or restorations, e.g., two adjacent pedestrians want to occupy the same cell. For this situation, one of them has to abandon the plan. This is similar to (2).
Fig.12. The flow chart of the new model rules. The rules of turning and restoration are framed by dashed lines. (7) For the operations of turnings and restorations, the schematic illustration is shown in Fig.13. Here we consider the process of "0.4m*0.2m→0.2m*0.4m" as "turning", and the process of "0.2m*0.4m→0.4m*0.2m" as "restoration". There might be different ways to do so, and we only choose the ones which are easy to perform in the simulation. We use the way in Fig.13(a) for the turnings, which implies that for the safety, there should be enough gap before the pedestrian. Here the critical value is 0.2m. For the restorations, we use the way in Fig.13(b). The reason why we choose it is that everybody wants to move forward, rather than backward. This method implies
that for the safety, there should be enough gaps both on the left and right of the pedestrian. The critical value is 0.1m. Besides, for the special situation when the pedestrian is near the boundary of the walking area (e.g., the wall), we use the way in Fig.13(c). Here the boundary is on the left side of the pedestrian, and the situation when it is on the right side is similar.
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(c) Fig.13. The illustrations of pedestrian turnings and restorations when the occupied area is a rectangle. The cyan cells represent the occupied area, and the white cells represent the empty area. We use (a) for the turnings and use (b) (c) for the restorations. (8) After turnings, the lateral movement of pedestrians will be affected. In the rules of the basic model, we set the as 0.8m/s, which is two times of the pedestrian's width. We keep this idea unchanged, and in the new model, the after turnings will be 0.4m/s, which is two times of the new width. Besides, as shown in Fig.5, the forward movement of pedestrians will not be affected after turnings. To testify the simulation results of pedestrian flow models, fundamental diagrams are very important [20, 26-28]. In our simulations, the fundamental diagrams are obtained under the periodic boundary conditions. Initially, the left-going (blue squares in the snapshots) and right-going pedestrians (red squares in the snapshots) are distributed randomly on the square lattice of L*W. Here we set L =200cells =20m, W =40cells =4m. In the following diagrams, the velocity v is the averaged value of the front velocity (the lateral velocity is not considered). We run 3600 time steps to remove the transient state. Then the averaged velocities of all the
pedestrians are recorded during another 1800 time steps. The flow J is obtained by J = ρ*v, in which ρ is the pedestrian density. The results presented in the figures are all averaged over 50 simulations. The simulation results with different rules are shown in Fig.14. It is clear that the use of turning rules can significantly enhance the flow and velocity in the fundamental diagram, and the use of restoration rules can reduce them a little. Due to the complexity of bi-directional flow, the results of two empirical datasets [18-19] are quite different. The determination of fundamental diagram of pedestrian flow needs the help of experiments under periodic boundary conditions, which are out of the scope of this paper, and here we just do some simple comparisons. It is clear that the result of new model is much better than the basic model, since the critical flow, critical density and jam density are all closer to the empirical facts.
Fig.14. The empirical and simulated fundamental diagrams of bi-directional flow, when various rules are used. The points are the two typical sets of empirical data [18-19], which are obtained from Ref. [20]. They can be downloaded at http://www.asim.uni-wuppertal.de/en/ database/data-from-literature/fundamental-diagrams.html. As mentioned in Section 3, the parameter S is crucial to the turning behaviors in bi-directional flow. Thus here we investigate its influence on the fundamental diagrams, as shown in Fig.15. It is clear that larger S leads to larger flows, since there is enough room and time for pedestrians to turn. If we make S even larger, e.g., S=10m or 20m, the simulation results will be the same as that of S=5m. This is not difficult to understand: at high densities (e.g., ), the situation when is seldom found in the simulation, which makes large S meaningless. On the contrary, smaller S makes turning behaviors more difficult, and the minimum S which is meaningful should be 0.2m (otherwise turning is impossible in our model). Nevertheless, in Fig.15 we present the result when S=0 for comparison, since the new model with S=0 degenerates to the basic model.
Fig. 15. The influence of S on the fundamental diagrams of the new model. 5. The open boundary simulation of different scenarios In order to test the microscopic performance of the new rules, we also do some open boundary simulations. One typical run of Scenario A with the new model is shown in Fig.16, which is similar to the experiment in Fig.1. Here the conditions are different from that in periodic boundary simulations. The length and width are set as 8m and 1.6m, which are the same as that in the experiment. In each direction 16 pedestrians (4 rows * 4 columns) try to go through the corridor, and the total number is N=32. The initial positions of all the pedestrians are the two ends of the corridor. At the beginning, if the cells at the ends are occupied by the front ones, the pedestrian is not shown in the simulation and its velocity is set as 0. Until the cells are empty, it will be shown and start moving according to the model rules. In Fig.16(a), they start to move from the two ends of the corridor. When they meet the others with opposite moving directions, many of them turn their bodies to give way to the others, as shown in Fig.16(b). They need some time to form the lanes, as shown in Fig.16(c). Finally, the pedestrians can restore when they have enough room (see Fig. 16(d)).
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(d) Fig.16. The typical patterns of Scenario A under open boundary conditions, when N=32 and the new model is used. (a)-(d) correspond to different time steps. (a) T=3s. (b) T=6s. (c) T=9s. (d) T=13s. And then, we can compare Fig.1 and Fig.16. Actually they are qualitatively similar: the appearances of Fig.1(a)(b)(c)(d) are similar to that of Fig.16(a)(b)(c)(d). But on the quantitative level, there are still some differences. For example, the simulation needs more time to run, e.g., the time of Fig.1(d) is T=9s, but in Fig.16(d) it is T=13s. Besides, in Fig.16 the pedestrian lanes are not so clear as shown in Fig.1. Nevertheless, we think our work is a good beginning, since it is much better than the result of basic model. It is easy to understand that the pedestrians cannot easily go through such a narrow corridor without turnings and restorations, as shown in Fig.17. Due to the effect of lateral moves, some of them can pass. But some others fail, and finally, the deadlock appears, which cannot be solved at all.
Fig.17. The typical pattern of Scenario A under open boundary conditions, when N=32 and the basic model is used. This snapshot is the final result, and the deadlock can be seen clearly. Then we consider the Scenario B, which can be compared to the experiment in Fig.2. The other configurations are the same, but the pedestrians stand very close at the beginning. Here we set the gap between the pedestrians with the same moving direction is 0.6m, and the initial gap between the two groups is 1.2m. One typical run of Scenario B with the new model is shown in Fig.18, and the total number is also N=32. Here the moments we choose to show are exactly the same as that in Fig.16. In Fig.18(a), all the pedestrians start to turn. In Fig.18(b), some pedestrians have passed, while in the short-time congestion, some others struggle to find their ways. And in Fig.18(c), the temporary
clusters gradually dissolve with the help of turning behaviors . Finally, all the pedestrians successfully pass and restore in Fig.18(d).
(a)
(b)
(c)
(d) Fig.18. The typical patterns of Scenario B under open boundary conditions, when N=32 and the new model is used. (a)-(d) correspond to different time steps. (a) T=3s. (b) T=6s. (c) T=9s. (d) T=13s. The comparisons between Fig.2 and Fig.18 have some similar results: the appearances of Fig.2(a)(b)(c)(d) are qualitatively similar to that of Fig.18(a)(b)(c)(d). But on the quantitative level, the differences between the durations can be seen. For example, since the pedestrians in the simulations are not so "intelligent" as real ones, the clusters in Fig.18 (b)(c) may maintain for longer times. If we compare with the basic model, we find the new rules of turnings and restorations perform better in the test of Scenario B. The basic model performs very bad, as shown in Fig.19. Since the room around the pedestrians are not enough large, they cannot have any chances to do lateral moves. Therefore, they could only form such a regular deadlock, and no one can pass. To our knowledge, for many previous cellular automaton models, this deadlock is also inevitable in the test of Scenario B. This comparison further confirms the significance of our new rules.
Fig.19. The typical pattern of Scenario B under open boundary conditions, when N=32 and the basic model is used. This snapshot is the final result, and the inevitable deadlock can be seen clearly. Then we concentrate on the statistical results. Actually, it is not a good choice to directly compare the averaged velocities in experimental results and simulation results, since in the Example 1 not all the pedestrians' movements are recorded by the camera until T=6s, as mentioned in Section 2. Nevertheless, we show the results in Fig.20, and some important features can be found: (1) In both scenarios, the minimum averaged velocities are about 0.4m/s in the experiment, which is nearly the same as the values in our simulations. Besides, the times for reaching this minimum velocity are just the same: in Scenario A they are T=6s, and in Scenario B they are T=4s. It means the transition from free flow to congestion in the bi-directional flow is well reproduced by our model (T=3~6s). (2) The main differences between the experimental results and simulation results are the transition from congestion back to free flow. In the experiment it only needs very short time, and the averaged velocity can quickly recover; while in the simulation the time is longer, and the averaged velocity gradually increases. (3) Due to the different initial configurations, the situations in the starting 2~3s are a little different. For example, in Scenario B, we need to give each pedestrian some room to turn, so we make them stand neatly, and they can obtain the maximum velocities at the first second. On the contrary, in the experiments the pedestrians stand at random positions, so they could only gradually accelerate, and the maximum velocities are reached later. (4) Some other phenomena are similar. For example, in the simulations we also find the fluctuations of velocities when the congestion disperses (T=13s in Scenario A and T=14s in Scenario B), although they are smaller than the results in the experiment.
(a)
(b) Fig.20. The averaged velocities of pedestrians in experimental results and simulation results. (a) Scenario A. (b) Scenario B. Then we compare the results considering the turning behaviors. In CA models, due to simplicity, many parameters are fixed: the turning angle is always 90° or 0°, and the widths of pedestrians are always 0.4m. Thus only the proportion of turned pedestrians can be used for comparisons. As shown in Fig.21(a) and (b), although the basic trends are qualitatively similar, the quantitative differences between experimental results and simulation results can be found. The effect of turning in simulation results is not so good as real ones: in two scenarios more pedestrians turn in the simulations, but they still use more time to pass the others.
(a)
(b) Fig.21. The proportions of turned pedestrians in experimental results and simulation results. (a) Scenario A. (b) Scenario B. For further comparisons, we also investigate some other statistical results of Scenario A and
B, and compare them with the experimental results. The deadlock probabilities of various situations are shown in Fig.22. Since the length of the simulation area is just 8m, the maximum pedestrian number in Scenario B is 40 (5 rows in each direction). As we mentioned in Section 2, since there is no deadlock in all the runs, the deadlock probabilities in the experiments are always 0. Although the differences between simulation results and experimental results exist, it is clear that the new model performs much better than the basic model. For example, in Scenario A, when there are 32 pedestrians, the deadlock probability of new model is just 9%, while that of basic model is as high as 72%. For the new model, the pedestrian number for the complete deadlock is 80, while for the basic model it is just 48. In Scenario B the differences are even larger: for the basic model the deadlock probabilities are always 100%, while for the new model the result is similar to that of Scenario A.
Fig.22. The deadlock probabilities of different situations. Both Scenario A and B are considered. Finally, the durations under different conditions are shown in Fig.23. The durations in the simulation results are the averaged values of the cases in which deadlock does not occur. The experimental results when N=24, 32, 40 (corresponding to 3, 4, 5 rows) are also presented here for comparisons. In Scenario A we only make the run of 30 pedestrians, and we think it is possible to use its value to represent the result of 32 pedestrians in Fig.23, since the difference should be little. Besides, since the deadlock probability of basic model in Scenario B is always 100%, for this situation the valid durations do not exist, and only three simulated curves are presented. The trend is similar to that in Fig.22: although the gap between simulation results and experimental data exists, the new model performs better than the basic model, and the results of Scenario B are a little smaller than that of Scenario A.
Fig.23. The averaged durations of different situations. Both Scenario A and B are considered. 6. Conclusion In this paper we discuss the new CA model for bi-directional pedestrian flow. In the experiments, we find some complex body-turning behaviors of pedestrians when they are passing through others, which can help to form the automatic lanes and avoid deadlocks. Many of them prefer to turn right, and in the turning states, the velocities of pedestrians are not affected. In order to simulate these phenomena, we use a new configuration in which each pedestrian occupies an area of 0.4m*0.2m. Then we design some new rules for the turnings and restorations, based on the basic model we proposed before. The effects of the new model can be found not only in the fundamental diagrams under periodic boundary conditions, but also in the simulation results of two different scenarios under open boundary conditions. The results show that the simulation results of new model qualitatively coincide with the experimental data. By the comparisons with the basic model, we find the improvement of the statistical results (including deadlock probabilities and averaged durations) is significant. Therefore, we think these results can be useful contributions to the field of pedestrian flow modeling. Nevertheless, there still exist some problems. As shown in Section 5, for both Scenario A and B, there are some quantitative differences between the experimental results and simulation results. For example, in the real life pedestrians need shorter time to form lanes, and the clusters can dissolve more quickly. Sometimes deadlock can occur with small probabilities in our simulations, but in the experiments it never occurs. Besides, in our simulation with high densities, when lane formation phenomenon can be found and no deadlock occurs, most of the pedestrians (sometimes even all of them) turn the bodies and keep this state for long time. But in the experimental data, only some of them temporarily turn and they can restore soon. These differences imply there are still some disadvantages in the current model. We think the problems come down to the fact that some basic configuration in CA model is oversimplified. For example, the turning angles are always 90° or 0° in the simulations, while in the real life they have many different choices. However, when time and space are both discrete, it is very difficult to accurately simulate these complex behaviors. Thus how to solve this problem in CA model, and how to make pedestrians more flexible is an important task, which needs to be investigated in the future.
Acknowledgements We thank the anonymous referees for fruitful suggestions. This work was funded by the National Natural Science Foundation of China (No. 11302022, 11422221, 51608115, 71371175, 71621001), and the Natural Science Foundation of Jiangsu Province (No. BK20150613, BK20150619). REFERENCES [1] E. Papadimitriou, G. Yannis, J. Golias, A critical assessment of pedestrian behaviour models. Transp. Res. F 12 (2009) 242–255. [2] S. Wolfram, Statistical Mechanics of Cellular Automata, Rev.Mod.Phys.55 (1983) 601. [3] M. Muramatsu, T. Irie, T. Nagatani, Jamming transition in pedestrian counter flow, Physica A 267 (1999) 487-498. [4] Y. Tajima, K. Takimoto, T. Nagatani, Pattern formation and jamming transition in pedestrian counter flow, Physica A 313 (2002) 709-723. [5] C. Burstedde, K. Klauck, A. Schadschneider, J. Zittartz, Simulation of pedestrian dynamics using a two-dimensional cellular automaton, Physica A 295 (2001) 507–525. [6] J. Li, L. Yang, D. Zhao, Simulation of bi-direction pedestrian movement in corridor, Physica A 354 (2005) 619–628. [7] W. G. Weng, T. Chen, H. Y. Yuan, W. C. Fan, Cellular automaton simulation of pedestrian counter flow with different walk velocities, Phys. Rev. E 74 (2006) 036102. [8] M. Fukamachi, T. Nagatani, Sidle effect on pedestrian counter flow, Physica A 377 (2007) 269-278. [9] H. Yue, H. Hao, X. Chen, C. Shao, Simulation of pedestrian flow on square lattice based on cellular automata model．Physica A 384 (2007) 567-588. [10] H. Yue, H. Guan, J. Zhang, C. Shao, Study on bi-direction pedestrian flow using cellular automata simulation, Physica A 389 (2010) 527-539. [11] S. Sarmady, F. Haron, A. Z. Talib, A cellular automata model for circular movements of pedestrians during Tawaf, Simulat. Model. Pract. Theory 19 (2011) 969–985. [12] P. Zhang, X. X. Jian, S. C. Wong, K. Choi, Potential field cellular automata model for pedestrian flow, Phys. Rev. E 85 (2012) 021119. [13] S. Nowak, A. Schadschneider, Quantitative analysis of pedestrian counterflow in a cellular automaton model, Phys. Rev. E 85 (2012) 066128. [14] J. Zhang, W. Klingsch, A. Schadschneider, A. Seyfried, Transitions in pedestrian fundamental diagrams of straight corridors and T-junctions, J. Stat. Mech. (2012) P06004. [15] B. D. Hankin, R. A. Wright, Passenger flow in subways, Oper. Res. Q. 9 (1958) 81. [16] M. Mori, H. Tsukaguchi, A new Method for Evaluation of Level of Service in Pedestrian Facilities, Transp. Res. A 21 (1987) 223-234. [17] D. Helbing, A. Johansson, H. Al-Abideen, Dynamics of crowd disasters: An empirical study, Phys. Rev. E 75 (2007) 046109. [18] S.J. Older, Movement of Pedestrians on Footways in Shopping Streets, Traffic Eng. Control 10 (1968) 160. [19] F. D. Navin, R. J. Wheeler, Pedestrian flow characteristics, Traffic Eng. 39 (1969) 30. [20] J. Zhang, W. Klingsch, A. Schadschneider, A. Seyfried, Ordering in bidirectional pedestrian
flows and its influence on the fundamental diagram, J. Stat. Mech. (2012) P02002. [21] X. Xu, W. Song, Staircase evacuation modeling and its comparison with an egress drill, Building and Environment 44 (2009) 1039–1046. [22] C.J. Jin, W. Wang, R. Jiang, L.Y. Dong, Simulating pedestrian flow by an improved two-process cellular automaton model, International Journal of Modern Physics C 28 (2017) 1750016. [23] V.J. Blue, J.L. Adler, Emergent fundamental pedestrian flows from cellular automata microsimulation, Transp. Res. Rec. 1644 (1998) 29-36. [24] V.J. Blue, J.L. Adler, Cellular automata microsimulation of bi-directional pedestrian flows, Transp. Res. Rec. 1678 (1999) 135-141. [25] V.J. Blue, J.L. Adler, Cellular automata micro-simulation for modeling bi-directional pedestrian walkways, Transp. Res. B 35 (2001) 293–312. [26] A. Seyfried, B. Steffen, W. Klingsch, M. Boltes, The fundamental diagram of pedestrian movement revisited, J. Stat. Mech. (2005) P10002. [27] A. Schadschneider, A. Seyfried, Validation of CA models of pedestrian dynamics with fundamental diagrams, Cybernetics and Systems: An International Journal 40 (2009) 367-389. [28] G. Flötteröd, G. Lämmel, Bidirectional pedestrian fundamental diagram, Transp. Res. B 71 (2015) 194–212.
S0378-4371(17)30433-8 http://dx.doi.org/10.1016/j.physa.2017.04.117 PHYSA 18219
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Physica A
Received date: 10 December 2016 Revised date: 21 March 2017 Please cite this article as: C.-J. Jin, R. Jiang, J.-L. Yin, L.-Y. Dong, D. Li, Simulating bi-directional pedestrian flow in a cellular automaton model considering the body-turning behavior, Physica A (2017), http://dx.doi.org/10.1016/j.physa.2017.04.117 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
The body-turning behavior in bi-directional pedestrian flow experiments is studied. This behavior is introduced into one cellular automaton model. The occupied area of each pedestrian is set as a rectangle of 0.4m*0.2m. The results from experiments and simulations are compared by quantitative analysis. The effect of new rules is significant in two different simulation scenarios.
*Manuscript Click here to view linked References
Simulating bi-directional pedestrian flow in a cellular automaton model considering the body-turning behavior Cheng-Jie Jin 1,2 , Rui Jiang 3, Jun-Lin Yin 1,2, Li-Yun Dong 4, Dawei Li 1,2 1 Jiangsu Key Laboratory of Urban ITS, Southeast University of China, Nanjing, Jiangsu, 210096, People's Republic of China 2 Jiangsu Province Collaborative Innovation Center of Modern Urban Traffic Technologies, Nanjing, Jiangsu, 210096, People's Republic of China 3 MOE Key Laboratory for Urban Transportation Complex Systems Theory and Technology, Beijing Jiaotong University, Beijing, 100044, People’s Republic of China 4 Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai, 200072, People's Republic of China Abstract: In the experiments of bi-directional pedestrian flow, it is often observed that pedestrians turn their bodies and change from walking straight to walking sideways, in order to mitigate or avoid the conflicts with opposite walking ones. When these conflicts disappear, pedestrians restore and walk straight again. In the turning states, the forward velocities of pedestrians are not affected. In order to simulate this body-turning behavior, we use a cellular automaton (CA) model named ITP model, which has been proposed before. But the occupied area of one pedestrian is set as 0.4m*0.2m. After the introduction of new rules of turnings and restorations, the pedestrians become more intelligent and flexible during the lane formation process, and some improvements of the fundamental diagram of pedestrian flow can be found. The simulation results of two different scenarios under open boundary conditions are also presented, and compared with the experimental data. It is shown that the new model performs much better than the original model in various tests, which further confirms the validity of the new rules. We think this approach is one useful contribution to the pedestrian flow modeling. Keywords: pedestrian flow; bi-directional flow; experiment; cellular automaton; turning. 1. Introduction The study of pedestrian flow has a very long history [1]. Many different approaches are used for the modeling of pedestrian flow, including the macroscopic ones in which the flow-density and velocity-density relationship is analyzed, and the microscopic ones in which the movement of each pedestrian is simulated and presented. The latter can be further classified into several types, and in this paper we concentrate on one simple type: cellular automaton (CA) model. In CA models the time and space are both discretized, which makes the computer simulation fast and easy to run [2]. In recent years, CA models become popular in many different fields, including the study of pedestrian movements. Particularly, there are many studies concerning the pedestrian flows in a narrow corridor [3-14], which is a common situation in real life. For this situation, the bi-directional flow is more interesting due to the complexity: some phenomena of self-organization often attract researchers' attention, including the automatic lane formation, etc.
But in these previous CA simulations of bi-directional pedestrian flow, there are still some disadvantages. Although many different modeling methods and techniques are used, in the simulations the jams usually appear at lower densities, and the critical density, critical flow and jam density may be much smaller than that in the empirical data [15-19]. It seems that the rules of pedestrian movements are usually not realistic enough. Besides, in some pedestrian flow experiments [20], we observed some interesting phenomena, in which pedestrians occupy smaller area than ordinary situations, and they can turn the bodies to pass the high-density area without waiting. This behavior is very flexible, and can help to significantly enhance the capacity of bi-directional flow. However, until now the modeling of this behavior is not sufficient. The main reason is that in many previous studies, the area occupied by one pedestrian is set as a square (usually it is 0.4m*0.4m). Thus no matter it is turned or not, it has no influence on the other pedestrians or the surrounding environment. In the recent years, some researchers try to use rectangle to represent the pedestrian. For example, in Ref.[8] the turning behaviors in bi-directional flow are simulated with the name of "sidle effects", and in Ref. [21] these phenomena in pedestrian evacuations are discussed. Both of them are lattice gas models, in which the moving direction of each pedestrian is determined by the calculation of transition probabilities. However, some problems still exist, e.g., the occupied area of one pedestrian is set as 0.6m*0.3m in Ref.[8] and 0.45m*0.3m in Ref.[21]. Both of them are much larger than the real results in the crowded situation. Besides, the simulation results in bi-directional flow [8] are not compared with the empirical or experimental data, so the validity is not clear. In order to investigate the essence of pedestrian movements, we organized some pedestrian flow experiments, and we also find some similar phenomena: pedestrians occupy smaller area, and they try to turn and restore frequently, so that they can pass others quickly and easily. In order to simulate this situation, we use the new configuration in which one pedestrian occupies the area of 0.4m*0.2m, and add the necessary rules of turnings and restorations into the pedestrian model named ITP model [22]. The effects of new rules are obvious: both the simulation results under periodic boundary conditions and open boundary conditions show that the new model performs much better than the previous model, and the pedestrians become more intelligent than before. The quantitative analysis of both experimental results and simulation results can help us to learn more about the mechanisms of bi-directional flow. Thus we think our attempts are great help for the pedestrian flow modeling. This paper is organized as follows. The data of the pedestrian bi-directional flow experiments is analyzed and discussed in Section 2. The rules of the previous ITP model are briefly introduced in Section 3. The new rules of turnings and restorations, and the corresponding simulation results under periodic boundary conditions are shown in Section 4. The simulation results under open boundary conditions with two different scenarios are presented, and compared with experimental results in Section 5. The conclusion is given in Section 6. 2. Experimental data analysis The experiments of bi-directional pedestrian flow were conducted in a narrow corridor, which is located in the campus of Shanghai University. The length and the width of the corridor is about 8m and 1.6m. 30 runs were recorded by a camera equipped above the corridor. In each run,
20-40 students are used, and in 18 runs the numbers of the pedestrians in two direction are the same. During the experiment, pedestrians wear different caps to represent their different moving directions. Although this experiment is simple, we find some very important phenomena: in all the runs, no deadlock occurs. This is because pedestrians are very flexible and they have many methods to pass. This procedure seems very easy for the pedestrians, and it does not need much time. We find there are two factors which can be crucial to avoid deadlocks: (1) Smaller occupied area: at small densities, it is easy to understand that pedestrians want to have large rooms in order to be comfortable. This is why the occupied area of one pedestrian is usually set as a square of 0.4*0.4m in most previous studies of modeling. But when the density is high, we find they don't like waiting at all. They will try to move as quickly as possible, and will not be afraid of frequent body contacts. This strong tendency leads to the fact that their occupied areas are usually much smaller than the normal situation. Although the width of a pedestrian is not easy to be changed, the length seems easier to be compressed. Smaller length will make the occupied area looks like a narrow rectangle, rather than a normal square. Thus we think the area can be considered as 0.4m*0.2m in the modeling, which will be discussed later. (2) Frequent body-turning behaviors: in order to avoid conflicts, sometimes pedestrians can turn their bodies. They can change from walking straight to walking sideways, in order to pass the others. This intelligent behavior can help to form the pedestrian lanes very quickly, especially when the pedestrians from two opposite directions meet each others. The turning states of some pedestrians can be maintained for a long time, until they finally pass. And then, they can restore if there is enough room. These findings can be clearly shown in some typical examples. Here we use one software named Tracker (http://physlets.org/tracker/) to extract the information of pedestrians' movement in the video, including the positions, velocities, turning angles and shoulder widths, etc. The time step for all the data collection is 1s. Before the extraction, the length and the width of corridor (8m and 1.6m) are used for calibrating the video. To ensure the accuracy, all the experimental data presented in this section is collected by hand. In this paper, the pedestrians with different moving directions are marked by dark blue and yellow, which can give enough contrast in black-and-white mode. And in this section, all the pedestrians in the sketches are represented by ellipses. Here we take 20° as one critical value, and the turning angles which are smaller than 20° will not be considered in all the sketches and statistics. Now the sketches of the Example 1 are shown in Fig.1. All the pedestrians start walking from the two ends of the corridor, and we call this configuration Scenario A in the following sections. When the two groups just meet, no obvious phenomena can be observed in Fig.1(a). After a short while, 4 lanes are primarily formed in Fig.1(b). There are 9 pedestrians who are turning their bodies, and the maximum turning angle is 72°. In Fig.1(c) the 4 lanes become more clear, but there are still 9 pedestrians who are turning, and the maximum angle is 77°. In Fig.1(d), when some pedestrians have reached the end, 3 pedestrians are still moving forward and in turning states.
(a)
(b)
(c)
(d) Fig.1. The typical sketches of Example 1. It belongs to Scenario A in the experiment. (a) T=3s; (b) T=5s; (c) T=7s; (d) T=9s. With some other initial configuration, although the procedure is different, the smaller occupied area and body-turning behaviors still can be found. Another typical situation is that at the beginning, 16 red pedestrians and 16 blue pedestrians stand very close. The sketches of the Example 2 are shown in Fig.3, and we call this configuration Scenario B in the following sections. In order to be compared with Fig.1, we choose the same moments to show the sketches. Due to the short distance, in Fig.3(a) the area is more crowded than that in Scenario A, and 15 pedestrians do not face their original moving directions. Especially, two clusters are temporarily formed, as marked by black curves. In these clusters 4 and 9 pedestrians are involved, and all of them are turning. The maximum angle even reaches 83°. After a short time, 4 obvious lanes are automatically formed in Fig.3(b), but there are still 13 pedestrians who are in turning states, and the maximum angle is still 75°. Finally, in Fig.3(c)(d) the temporary congestion disperses, and all the pedestrians gradually restore.
(a)
(b)
(c)
(d) Fig.2. The typical sketches of Example 2. It belongs to Scenario B in the experiment. (a) T=3s; (b) T=5s; (c) T=7s; (d) T=9s. Then we check some time series data. The averaged velocities of pedestrians are shown in Fig.3, in which the Example 1 and 2 are compared. It should be noted that due to the limit of the camera scope, in Example 1 the velocities of some pedestrians cannot be recorded, so are all the other runs of Scenario A in the experiment. For example, in Fig.1(a) when T=3s, only 8 blue ones can be seen, and the last blue one (the 15th one) does not enter the camera scope until T=6s. Nevertheless, we can find something useful: (1) The basic trends are easy to understand. For Scenario A, the velocities reach the peak at the beginning, but for Scenario B, the velocities could only gradually increase. After the short-time congestion, both of them can increase again. (2) It is interesting that when the two groups of pedestrians meet each others, the results of two examples become similar, no matter when there exists one short-time congestion (T=4~6s) or when the congestion disperses (T>=7s). Starting from T=6s, when all the movements of blue ones are recorded in the camera, the averaged velocities become nearly quantitatively the same. It seems that the initial states of pedestrian flow do not have much influence on the results. (3) There are some obvious fluctuations of velocities when the congestion disperses (T=8s). This is mainly due to the features of open boundary conditions, rather than the properties of pedestrians: when some pedestrians with higher velocities reach the end of corridor, they will be
removed, and only some others with lower velocities are left for the statistics in the next time step. After a short while, they also can become fast.
Fig.3. The averaged velocities of pedestrians in two examples. The averaged velocities of the first 5 seconds in Example 1 are represented by different symbols, since they are not the results of all the pedestrians. The proportions of turned pedestrians are shown in Fig.4. The basic trends of the two examples are similar: during 3 seconds (T=3~5s in Example 1 and T=5~7s in Example 2), many pedestrians choose to turn. After this "peak period", most of them restore soon. This "peak period" is also the period when short-time congestion occurs in the experiment. During this period, the maximum proportion of turned pedestrians in Example 1 is about 33%, while that in Example 2 is about 50%. It also implies the center area in Example 2 is more crowded.
Fig.4. The proportions of turned pedestrians in two examples. The averaged velocities of turned and unturned pedestrians are show in Fig.5. It is very clear that in Example 1, the velocities of turned and unturned ones are nearly the same at every second. In Example 2, when T=3~4s, the velocities of turned ones are even larger. These results are very important, which shows the turning behaviors do not affect the forward movement of pedestrians at all. On the contrary, it becomes one great help for the solution of congestions.
(a)
(b) Fig.5. The averaged velocities of turned and unturned pedestrians in two examples. (a) Example 1; (b) Example 2. Next, we focus on the microscopic information about pedestrians' turning behaviors. The distributions of the shoulder widths are shown in Fig.6. The results in both examples have the appearances which are similar to normal distribution. The peak is at 0.37m, which is also the averaged widths in the two examples. In China the averaged shoulder widths for male and female between 18-25 years old are 0.375m and 0.351m. Since in our experiment most pedestrians are male, the measured values are close to the statistical results.
Fig.6. The distributions of the shoulder widths in two examples. In Fig. 7 and 8, all the data are collected at each second when turning behaviors are found in
the experimental video. As shown in Fig.4 and Fig.5, in Example 1 the collecting time is T=4~9s, while for Example 2 it is T=1~7s. The distributions of the turning angles are shown in Fig.7. The differences between two examples can be seen: in Example 1 there are more smaller angles (A=20°~30°), while in Example 2 more results can be found at A=30°~70°. This shows not only the different levels of crowdedness in two examples, but also the great flexibility of pedestrians' movement.
Fig.7. The distributions of the turning angles in two examples. Note that in the experiment, pedestrians can choose to turn left or turn right, according to their own thoughts. This process is also interesting, and we show some statistics about the turning directions in Tab.1. Some new results can be found: (1) The number of right-turning pedestrians is much greater than that of left-turning pedestrians in two examples. It is not strange, since in China the basic traffic rules make people try to walk on the right side. (2) It is very interesting that some statistical results of right-turning pedestrians in two different examples are nearly the same, including the averaged velocities and averaged turning angles. We think it implies the universal tendency of pedestrians' right-turning behaviors (3) On the contrary, the statistical results of left-turning pedestrians in two examples are completely different. In Example 1 the velocities are higher and the angles are smaller, while in Example 2 the velocities are lower and the angles are similar to that of right-turning pedestrians. The reasons for this phenomenon still need to be investigated in the future, and we think one possible explanation is that in Example 1 the sample is not large enough. Table.1. The statistics for the pedestrians' turning directions Example
Turning direction
Number
Averaged velocity (m/s)
Averaged angle (°)
1 2 1 2
Left Left Right Right
15 24 23 45
0.69 0.43 0.60 0.59
32 41 41 41
And then, the relationship between each pedestrian's velocities and turning angles is shown in Fig.8. The scatter of data points in both figures further confirms that the turning angles have no influence on the velocities. Besides, the differences between two examples also can be seen. In
Fig.8(a) all the larger turning angles result from right-turning behaviors, which shows the general preferences of many pedestrians. On the contrary, in Fig.8(b) the maximum angle results from left-turning behaviors, which shows left-turning is also one possible option for some pedestrians when there is not enough room.
(a)
(b) Fig.8. The relationship between each pedestrian's velocities and turning angles in two examples. (a) Example 1; (b) Example 2. Finally, we also investigate some statistics for all the 18 runs in our experiment, e.g., the averaged durations. In the runs of Scenario A, the total pedestrian numbers are 20, 30, 40. while in the runs of Scenario B, the total numbers are 20, 24, 28, ...,40. For each number two runs are performed, and the durations shown in Fig.9 are the averaged value of the two runs. It is clear that the times used for two scenarios are close to each other, which also coincides with the conclusion (2) we made above. We think the small difference between them is mainly due to the different walking distances: in Scenario A everyone needs to walk further.
Fig.9. the averaged durations of two scenarios in the experiment. 3. The rules of the basic model Here we briefly recall the rules of the ITP model, which has been discussed in our previous papers [22]. It is an improved version of the "two-process" model [23-25], in which the forward movement and the lateral movement of pedestrians are split and handled at two different sub-steps. The time step is set to 1s. In order to describe the pedestrian movements more accurately, the size of one cell is set to 0.1m*0.1m. In the previous paper [22], each pedestrian occupies the area of 0.4m*0.4m, which is a traditional configuration used in many other studies. But as we mentioned in Section 2, when the pedestrians are passing through others in the bi-directional flow, the occupied area could be considered as a rectangle of 0.4m*0.2m. Thus in all the following simulations, we will adopt this configuration, and we call this model as "basic model". The schematic illustration of the pedestrian's movement is shown in Fig.10, and the variables involved are introduced in Tab.2. The parameters used in the basic model are presented in Tab.3, in which the values and related explanations are also given.
Fig.10. An example of the surrounding environment of a pedestrian X. Table.2. The variables used in the basic model Name /
Definition and explanation The front gap to the nearest pedestrian. In Fig.7, since The lateral gap to the nearest pedestrian on the left/right side. The front velocity of the pedestrian.
.
The lateral velocity of the pedestrian. The "conflict parameter". We call the nearest pedestrian in the moving direction as "key one". If the moving directions of the pedestrian and the "key one" are the same, C=0; otherwise C=1. For example, in Fig.7, pedestrian B is the "key one" of pedestrian X and C=0. Besides, if , the one who has larger C value will be considered as the “key one”.
C
Table.3. The parameters used in the basic model Name and value =2.4m/s
=0.8m/s
R=0.1m/s2
S=5m
P=0.1
Definition and explanation The maximum value of . In Example 1, the maximum front velocity of all the pedestrians is about 2.3m/s at the beginning (when T=2s). In Example 2, the maximum value is only about 1.7m/s at the end (when T=7s). Therefore, we set the value as 2.4m/s in the simulations, since it is better to be an integer multiple of 0.4m. The maximum value of . In Example 1, the maximum lateral velocity is about 0.66m/s, and in Example 2 the value is about 0.63m/s. Therefore, we set the maximum value as 0.8m/s in the simulations, since it is better to be an integer multiple of 0.4m. The pedestrian-dependent preferred front velocity. In the experimental data, the averaged velocity of free walking is about 1.5m/s. Therefore, in order to present the variety of pedestrians' preferences, is assumed to be uniformly distributed over the interval [1.0, 2.0]m/s in the simulations. Here only the discrete value of 1.0, 1.1, 1.2..., 2.0 can be chosen. An random acceleration/deceleration value used in the update of forward movement. It represents the random fluctuations of velocities, and this idea is similar to that used in vehicular traffic flow models. It cannot be measured in this experiment, and we set it as 0.1m/s2. One critical gap which controls the actions of pedestrians. When , the pedestrian starts to consider the possibility of lateral movement. When the new rules of turning and restoration are added, for this situation the pedestrian also considers the possibility of turning. In this paper we set it as 5m in the simulations. It will be further discussed in Section 4, since it is crucial to the bi-directional flow. The randomization probability used for the velocity update. We set it as 0.1, which is the common value used in many CA models.
The flow chart of the rules of basic model is shown in Fig.11. The further explanations of some important steps are marked by numbers (1)(2)(3)(4), and presented as follows:
Fig.11. The flow chart of the basic model rules. (1) If one of following is satisfied, the lateral movement is needed: ① C=1 and . It implies potential conflict in the future. ② C=0 and . This rule is similar to the model rule in multilane vehicular traffic flow, which means that the forward movement is hindered by the "key one". (2) Sometimes new conflicts may occur, when there are many lateral moves, e.g., two adjacent pedestrians want to occupy the same cell. For this situation, one of them has to abandon the plan. (3) For the lateral movement, something should be noted: ① Suppose the conflict parameters and front gaps before(after) the lateral move are C1(C2) and ( ), respectively. If one of following is satisfied, it becomes "better": a.C2=0 and C1=1. It means the pedestrian can avoid potential conflicts by lateral move. b.C1=C2 and . It implies the moving condition is not significantly changed, but the front gap becomes larger. ② The exhaustive range of lateral velocity is: 0.1, 0.2, ..., min ( , ). Find the smallest velocity which can help the pedestrian become "better", and record the value. ③ A pedestrian tries to move to right firstly; if failed, then it tries to move to left. If it tries to move to left and ② is fulfilled, there is still 50% chance to give up the lateral movement. ④ One cooperative strategy between different pedestrians is used: for a pair of pedestrians (A and B) with opposing directions, if they are the '"key one" of each other, the potential plan should be particularly considered. Specifically, if A has the potential plan of moving left (right), B will not try to move right (left). (4) For the forward movement, there are three different situations to be considered:
① If
,
The deterministic calculation is similar to that in the Blue-Adler model [23-25]. It includes the consideration of the potential movement of the opposite pedestrian. The random deceleration rule is similar to that in many vehicular traffic flow models. ② If
and
,
③ If
and
,
The latter two situations also consider the random fluctuations of the front velocity. 4. The periodic boundary simulation with turnings and restorations As we discussed in Section 2, the rules of frequent turnings and restorations are necessary for the simulation of bi-directional flow. When these rules are added into the basic model, we get the "new model". The flow chart of the rules of new model is shown in Fig.12. It is clear that the new rules (framed by dashed lines) can be considered as one component whose structure is similar to the basic rules. The further explanations of some important steps are marked by numbers (5)(6)(7) (8), and presented as follows: (5) For the turning rules, we set 4 necessary conditions: Condition A: (There is enough gap before the pedestrian for the turnings). Condition B: C=1 (The "key one" has opposite moving direction). Condition C: (The pedestrian cannot move freely in the next time step). Condition D: (The gap is not large enough). It is clear that A has to be fulfilled, and B, C, D can be chosen. Actually, there could be many possible options for the turning rules, and it is difficult to determine which one is the best. Here we just choose one simple and reasonable rule: A and (B or C) and D. For the restoration rules, we also have 4 necessary conditions: Condition E: and (There is enough gap on the left and right of the pedestrian for the restorations). Condition F: C=0 (The "key one" has the same moving direction). Condition G: (The pedestrian can move freely in the next time step). Condition H: (The gap is large enough). It can be found that Condition EFGH are similar to the opposite situation of Condition ABCD, but there are still some minor differences. For example, if the critical value in Condition H is also set as S, it becomes meaningless, since in the high-density situation it is impossible for all the pedestrians to fulfill it. It is also difficult to determine the best restoration rules, and here we choose one simple combination: E and F and H. Since there is always , Condition G can be ignored here. (6) Sometimes new conflicts may occur, when there are many turnings or restorations, e.g., two adjacent pedestrians want to occupy the same cell. For this situation, one of them has to abandon the plan. This is similar to (2).
Fig.12. The flow chart of the new model rules. The rules of turning and restoration are framed by dashed lines. (7) For the operations of turnings and restorations, the schematic illustration is shown in Fig.13. Here we consider the process of "0.4m*0.2m→0.2m*0.4m" as "turning", and the process of "0.2m*0.4m→0.4m*0.2m" as "restoration". There might be different ways to do so, and we only choose the ones which are easy to perform in the simulation. We use the way in Fig.13(a) for the turnings, which implies that for the safety, there should be enough gap before the pedestrian. Here the critical value is 0.2m. For the restorations, we use the way in Fig.13(b). The reason why we choose it is that everybody wants to move forward, rather than backward. This method implies
that for the safety, there should be enough gaps both on the left and right of the pedestrian. The critical value is 0.1m. Besides, for the special situation when the pedestrian is near the boundary of the walking area (e.g., the wall), we use the way in Fig.13(c). Here the boundary is on the left side of the pedestrian, and the situation when it is on the right side is similar.
(a)
(b)
(c) Fig.13. The illustrations of pedestrian turnings and restorations when the occupied area is a rectangle. The cyan cells represent the occupied area, and the white cells represent the empty area. We use (a) for the turnings and use (b) (c) for the restorations. (8) After turnings, the lateral movement of pedestrians will be affected. In the rules of the basic model, we set the as 0.8m/s, which is two times of the pedestrian's width. We keep this idea unchanged, and in the new model, the after turnings will be 0.4m/s, which is two times of the new width. Besides, as shown in Fig.5, the forward movement of pedestrians will not be affected after turnings. To testify the simulation results of pedestrian flow models, fundamental diagrams are very important [20, 26-28]. In our simulations, the fundamental diagrams are obtained under the periodic boundary conditions. Initially, the left-going (blue squares in the snapshots) and right-going pedestrians (red squares in the snapshots) are distributed randomly on the square lattice of L*W. Here we set L =200cells =20m, W =40cells =4m. In the following diagrams, the velocity v is the averaged value of the front velocity (the lateral velocity is not considered). We run 3600 time steps to remove the transient state. Then the averaged velocities of all the
pedestrians are recorded during another 1800 time steps. The flow J is obtained by J = ρ*v, in which ρ is the pedestrian density. The results presented in the figures are all averaged over 50 simulations. The simulation results with different rules are shown in Fig.14. It is clear that the use of turning rules can significantly enhance the flow and velocity in the fundamental diagram, and the use of restoration rules can reduce them a little. Due to the complexity of bi-directional flow, the results of two empirical datasets [18-19] are quite different. The determination of fundamental diagram of pedestrian flow needs the help of experiments under periodic boundary conditions, which are out of the scope of this paper, and here we just do some simple comparisons. It is clear that the result of new model is much better than the basic model, since the critical flow, critical density and jam density are all closer to the empirical facts.
Fig.14. The empirical and simulated fundamental diagrams of bi-directional flow, when various rules are used. The points are the two typical sets of empirical data [18-19], which are obtained from Ref. [20]. They can be downloaded at http://www.asim.uni-wuppertal.de/en/ database/data-from-literature/fundamental-diagrams.html. As mentioned in Section 3, the parameter S is crucial to the turning behaviors in bi-directional flow. Thus here we investigate its influence on the fundamental diagrams, as shown in Fig.15. It is clear that larger S leads to larger flows, since there is enough room and time for pedestrians to turn. If we make S even larger, e.g., S=10m or 20m, the simulation results will be the same as that of S=5m. This is not difficult to understand: at high densities (e.g., ), the situation when is seldom found in the simulation, which makes large S meaningless. On the contrary, smaller S makes turning behaviors more difficult, and the minimum S which is meaningful should be 0.2m (otherwise turning is impossible in our model). Nevertheless, in Fig.15 we present the result when S=0 for comparison, since the new model with S=0 degenerates to the basic model.
Fig. 15. The influence of S on the fundamental diagrams of the new model. 5. The open boundary simulation of different scenarios In order to test the microscopic performance of the new rules, we also do some open boundary simulations. One typical run of Scenario A with the new model is shown in Fig.16, which is similar to the experiment in Fig.1. Here the conditions are different from that in periodic boundary simulations. The length and width are set as 8m and 1.6m, which are the same as that in the experiment. In each direction 16 pedestrians (4 rows * 4 columns) try to go through the corridor, and the total number is N=32. The initial positions of all the pedestrians are the two ends of the corridor. At the beginning, if the cells at the ends are occupied by the front ones, the pedestrian is not shown in the simulation and its velocity is set as 0. Until the cells are empty, it will be shown and start moving according to the model rules. In Fig.16(a), they start to move from the two ends of the corridor. When they meet the others with opposite moving directions, many of them turn their bodies to give way to the others, as shown in Fig.16(b). They need some time to form the lanes, as shown in Fig.16(c). Finally, the pedestrians can restore when they have enough room (see Fig. 16(d)).
(a)
(b)
(c)
(d) Fig.16. The typical patterns of Scenario A under open boundary conditions, when N=32 and the new model is used. (a)-(d) correspond to different time steps. (a) T=3s. (b) T=6s. (c) T=9s. (d) T=13s. And then, we can compare Fig.1 and Fig.16. Actually they are qualitatively similar: the appearances of Fig.1(a)(b)(c)(d) are similar to that of Fig.16(a)(b)(c)(d). But on the quantitative level, there are still some differences. For example, the simulation needs more time to run, e.g., the time of Fig.1(d) is T=9s, but in Fig.16(d) it is T=13s. Besides, in Fig.16 the pedestrian lanes are not so clear as shown in Fig.1. Nevertheless, we think our work is a good beginning, since it is much better than the result of basic model. It is easy to understand that the pedestrians cannot easily go through such a narrow corridor without turnings and restorations, as shown in Fig.17. Due to the effect of lateral moves, some of them can pass. But some others fail, and finally, the deadlock appears, which cannot be solved at all.
Fig.17. The typical pattern of Scenario A under open boundary conditions, when N=32 and the basic model is used. This snapshot is the final result, and the deadlock can be seen clearly. Then we consider the Scenario B, which can be compared to the experiment in Fig.2. The other configurations are the same, but the pedestrians stand very close at the beginning. Here we set the gap between the pedestrians with the same moving direction is 0.6m, and the initial gap between the two groups is 1.2m. One typical run of Scenario B with the new model is shown in Fig.18, and the total number is also N=32. Here the moments we choose to show are exactly the same as that in Fig.16. In Fig.18(a), all the pedestrians start to turn. In Fig.18(b), some pedestrians have passed, while in the short-time congestion, some others struggle to find their ways. And in Fig.18(c), the temporary
clusters gradually dissolve with the help of turning behaviors . Finally, all the pedestrians successfully pass and restore in Fig.18(d).
(a)
(b)
(c)
(d) Fig.18. The typical patterns of Scenario B under open boundary conditions, when N=32 and the new model is used. (a)-(d) correspond to different time steps. (a) T=3s. (b) T=6s. (c) T=9s. (d) T=13s. The comparisons between Fig.2 and Fig.18 have some similar results: the appearances of Fig.2(a)(b)(c)(d) are qualitatively similar to that of Fig.18(a)(b)(c)(d). But on the quantitative level, the differences between the durations can be seen. For example, since the pedestrians in the simulations are not so "intelligent" as real ones, the clusters in Fig.18 (b)(c) may maintain for longer times. If we compare with the basic model, we find the new rules of turnings and restorations perform better in the test of Scenario B. The basic model performs very bad, as shown in Fig.19. Since the room around the pedestrians are not enough large, they cannot have any chances to do lateral moves. Therefore, they could only form such a regular deadlock, and no one can pass. To our knowledge, for many previous cellular automaton models, this deadlock is also inevitable in the test of Scenario B. This comparison further confirms the significance of our new rules.
Fig.19. The typical pattern of Scenario B under open boundary conditions, when N=32 and the basic model is used. This snapshot is the final result, and the inevitable deadlock can be seen clearly. Then we concentrate on the statistical results. Actually, it is not a good choice to directly compare the averaged velocities in experimental results and simulation results, since in the Example 1 not all the pedestrians' movements are recorded by the camera until T=6s, as mentioned in Section 2. Nevertheless, we show the results in Fig.20, and some important features can be found: (1) In both scenarios, the minimum averaged velocities are about 0.4m/s in the experiment, which is nearly the same as the values in our simulations. Besides, the times for reaching this minimum velocity are just the same: in Scenario A they are T=6s, and in Scenario B they are T=4s. It means the transition from free flow to congestion in the bi-directional flow is well reproduced by our model (T=3~6s). (2) The main differences between the experimental results and simulation results are the transition from congestion back to free flow. In the experiment it only needs very short time, and the averaged velocity can quickly recover; while in the simulation the time is longer, and the averaged velocity gradually increases. (3) Due to the different initial configurations, the situations in the starting 2~3s are a little different. For example, in Scenario B, we need to give each pedestrian some room to turn, so we make them stand neatly, and they can obtain the maximum velocities at the first second. On the contrary, in the experiments the pedestrians stand at random positions, so they could only gradually accelerate, and the maximum velocities are reached later. (4) Some other phenomena are similar. For example, in the simulations we also find the fluctuations of velocities when the congestion disperses (T=13s in Scenario A and T=14s in Scenario B), although they are smaller than the results in the experiment.
(a)
(b) Fig.20. The averaged velocities of pedestrians in experimental results and simulation results. (a) Scenario A. (b) Scenario B. Then we compare the results considering the turning behaviors. In CA models, due to simplicity, many parameters are fixed: the turning angle is always 90° or 0°, and the widths of pedestrians are always 0.4m. Thus only the proportion of turned pedestrians can be used for comparisons. As shown in Fig.21(a) and (b), although the basic trends are qualitatively similar, the quantitative differences between experimental results and simulation results can be found. The effect of turning in simulation results is not so good as real ones: in two scenarios more pedestrians turn in the simulations, but they still use more time to pass the others.
(a)
(b) Fig.21. The proportions of turned pedestrians in experimental results and simulation results. (a) Scenario A. (b) Scenario B. For further comparisons, we also investigate some other statistical results of Scenario A and
B, and compare them with the experimental results. The deadlock probabilities of various situations are shown in Fig.22. Since the length of the simulation area is just 8m, the maximum pedestrian number in Scenario B is 40 (5 rows in each direction). As we mentioned in Section 2, since there is no deadlock in all the runs, the deadlock probabilities in the experiments are always 0. Although the differences between simulation results and experimental results exist, it is clear that the new model performs much better than the basic model. For example, in Scenario A, when there are 32 pedestrians, the deadlock probability of new model is just 9%, while that of basic model is as high as 72%. For the new model, the pedestrian number for the complete deadlock is 80, while for the basic model it is just 48. In Scenario B the differences are even larger: for the basic model the deadlock probabilities are always 100%, while for the new model the result is similar to that of Scenario A.
Fig.22. The deadlock probabilities of different situations. Both Scenario A and B are considered. Finally, the durations under different conditions are shown in Fig.23. The durations in the simulation results are the averaged values of the cases in which deadlock does not occur. The experimental results when N=24, 32, 40 (corresponding to 3, 4, 5 rows) are also presented here for comparisons. In Scenario A we only make the run of 30 pedestrians, and we think it is possible to use its value to represent the result of 32 pedestrians in Fig.23, since the difference should be little. Besides, since the deadlock probability of basic model in Scenario B is always 100%, for this situation the valid durations do not exist, and only three simulated curves are presented. The trend is similar to that in Fig.22: although the gap between simulation results and experimental data exists, the new model performs better than the basic model, and the results of Scenario B are a little smaller than that of Scenario A.
Fig.23. The averaged durations of different situations. Both Scenario A and B are considered. 6. Conclusion In this paper we discuss the new CA model for bi-directional pedestrian flow. In the experiments, we find some complex body-turning behaviors of pedestrians when they are passing through others, which can help to form the automatic lanes and avoid deadlocks. Many of them prefer to turn right, and in the turning states, the velocities of pedestrians are not affected. In order to simulate these phenomena, we use a new configuration in which each pedestrian occupies an area of 0.4m*0.2m. Then we design some new rules for the turnings and restorations, based on the basic model we proposed before. The effects of the new model can be found not only in the fundamental diagrams under periodic boundary conditions, but also in the simulation results of two different scenarios under open boundary conditions. The results show that the simulation results of new model qualitatively coincide with the experimental data. By the comparisons with the basic model, we find the improvement of the statistical results (including deadlock probabilities and averaged durations) is significant. Therefore, we think these results can be useful contributions to the field of pedestrian flow modeling. Nevertheless, there still exist some problems. As shown in Section 5, for both Scenario A and B, there are some quantitative differences between the experimental results and simulation results. For example, in the real life pedestrians need shorter time to form lanes, and the clusters can dissolve more quickly. Sometimes deadlock can occur with small probabilities in our simulations, but in the experiments it never occurs. Besides, in our simulation with high densities, when lane formation phenomenon can be found and no deadlock occurs, most of the pedestrians (sometimes even all of them) turn the bodies and keep this state for long time. But in the experimental data, only some of them temporarily turn and they can restore soon. These differences imply there are still some disadvantages in the current model. We think the problems come down to the fact that some basic configuration in CA model is oversimplified. For example, the turning angles are always 90° or 0° in the simulations, while in the real life they have many different choices. However, when time and space are both discrete, it is very difficult to accurately simulate these complex behaviors. Thus how to solve this problem in CA model, and how to make pedestrians more flexible is an important task, which needs to be investigated in the future.
Acknowledgements We thank the anonymous referees for fruitful suggestions. This work was funded by the National Natural Science Foundation of China (No. 11302022, 11422221, 51608115, 71371175, 71621001), and the Natural Science Foundation of Jiangsu Province (No. BK20150613, BK20150619). REFERENCES [1] E. Papadimitriou, G. Yannis, J. Golias, A critical assessment of pedestrian behaviour models. Transp. Res. F 12 (2009) 242–255. [2] S. Wolfram, Statistical Mechanics of Cellular Automata, Rev.Mod.Phys.55 (1983) 601. [3] M. Muramatsu, T. Irie, T. Nagatani, Jamming transition in pedestrian counter flow, Physica A 267 (1999) 487-498. [4] Y. Tajima, K. Takimoto, T. Nagatani, Pattern formation and jamming transition in pedestrian counter flow, Physica A 313 (2002) 709-723. [5] C. Burstedde, K. Klauck, A. Schadschneider, J. Zittartz, Simulation of pedestrian dynamics using a two-dimensional cellular automaton, Physica A 295 (2001) 507–525. [6] J. Li, L. Yang, D. Zhao, Simulation of bi-direction pedestrian movement in corridor, Physica A 354 (2005) 619–628. [7] W. G. Weng, T. Chen, H. Y. Yuan, W. C. Fan, Cellular automaton simulation of pedestrian counter flow with different walk velocities, Phys. Rev. E 74 (2006) 036102. [8] M. Fukamachi, T. Nagatani, Sidle effect on pedestrian counter flow, Physica A 377 (2007) 269-278. [9] H. Yue, H. Hao, X. Chen, C. Shao, Simulation of pedestrian flow on square lattice based on cellular automata model．Physica A 384 (2007) 567-588. [10] H. Yue, H. Guan, J. Zhang, C. Shao, Study on bi-direction pedestrian flow using cellular automata simulation, Physica A 389 (2010) 527-539. [11] S. Sarmady, F. Haron, A. Z. Talib, A cellular automata model for circular movements of pedestrians during Tawaf, Simulat. Model. Pract. Theory 19 (2011) 969–985. [12] P. Zhang, X. X. Jian, S. C. Wong, K. Choi, Potential field cellular automata model for pedestrian flow, Phys. Rev. E 85 (2012) 021119. [13] S. Nowak, A. Schadschneider, Quantitative analysis of pedestrian counterflow in a cellular automaton model, Phys. Rev. E 85 (2012) 066128. [14] J. Zhang, W. Klingsch, A. Schadschneider, A. Seyfried, Transitions in pedestrian fundamental diagrams of straight corridors and T-junctions, J. Stat. Mech. (2012) P06004. [15] B. D. Hankin, R. A. Wright, Passenger flow in subways, Oper. Res. Q. 9 (1958) 81. [16] M. Mori, H. Tsukaguchi, A new Method for Evaluation of Level of Service in Pedestrian Facilities, Transp. Res. A 21 (1987) 223-234. [17] D. Helbing, A. Johansson, H. Al-Abideen, Dynamics of crowd disasters: An empirical study, Phys. Rev. E 75 (2007) 046109. [18] S.J. Older, Movement of Pedestrians on Footways in Shopping Streets, Traffic Eng. Control 10 (1968) 160. [19] F. D. Navin, R. J. Wheeler, Pedestrian flow characteristics, Traffic Eng. 39 (1969) 30. [20] J. Zhang, W. Klingsch, A. Schadschneider, A. Seyfried, Ordering in bidirectional pedestrian
flows and its influence on the fundamental diagram, J. Stat. Mech. (2012) P02002. [21] X. Xu, W. Song, Staircase evacuation modeling and its comparison with an egress drill, Building and Environment 44 (2009) 1039–1046. [22] C.J. Jin, W. Wang, R. Jiang, L.Y. Dong, Simulating pedestrian flow by an improved two-process cellular automaton model, International Journal of Modern Physics C 28 (2017) 1750016. [23] V.J. Blue, J.L. Adler, Emergent fundamental pedestrian flows from cellular automata microsimulation, Transp. Res. Rec. 1644 (1998) 29-36. [24] V.J. Blue, J.L. Adler, Cellular automata microsimulation of bi-directional pedestrian flows, Transp. Res. Rec. 1678 (1999) 135-141. [25] V.J. Blue, J.L. Adler, Cellular automata micro-simulation for modeling bi-directional pedestrian walkways, Transp. Res. B 35 (2001) 293–312. [26] A. Seyfried, B. Steffen, W. Klingsch, M. Boltes, The fundamental diagram of pedestrian movement revisited, J. Stat. Mech. (2005) P10002. [27] A. Schadschneider, A. Seyfried, Validation of CA models of pedestrian dynamics with fundamental diagrams, Cybernetics and Systems: An International Journal 40 (2009) 367-389. [28] G. Flötteröd, G. Lämmel, Bidirectional pedestrian fundamental diagram, Transp. Res. B 71 (2015) 194–212.