Statistical Analysis of the Pedestrian Flow

Copyright © IFAC Control in Transportation Systems, Tokyo,Japan,2003

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STATISTICAL ANALYSIS OF THE PEDESTRIAN FLOW Takeshi Iizuka

Department of Physics, Faculty of Sciences, Ehime University, Matsuyama, 790-8577 Japan

Abstract: We consider a numerical model of pedestrian movement in a counter flow footway using a cellar automaton model. The enter rate of a pedestrian into the footway is the control parameter in our study. The probability of the panic occurrence is calculated statistically. If the width of footway is large, we find critical behavior of the panic probability which is similar to the phase transition obtained from statistical mechanics. We also consider the asymmetric case where the number of pedestrian moving one direction is different form that of the couter direction. Copyright © 2003 IFAC

Keywords: pedestrian flow, cellar automaton model, panic probability

1. INTRODUCTION

Section §4 is devoted to the case that the number of pedestrians are different from that of couterwalking pedestrians.

Pedestrians flow is one of the interesting parts in the field of the traffic engineering. Actually pedestrian behavior in public place such as architectural space and street crossing has been analyzed in many situations. For designing public facilities, quantitative analysis of pedestrian flow is important(Feurtey 2000).

2. CELLAR AUTOMATON MODEL Here we introduce a model which simulates the two-way pedestrian flow. The footway is assumed to be a two-dimensional square lattice.

Pedestrian counter flow(Muramatsu et al 1999) is remarkable situation as compared with the car traffic flow. In particular, panic is one of interesting phenomenon of the pedestrian flow dynamics. Shocking accident on July, 21, 2001 in the footbridge, Akashi, Japan is nothing but panic of the pedestrians. To avoid such accident, analysis of two-directional counter flow in facilities such as footway seems very important problem.

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The traffic flow is modeled in some kinds manners. The celler automaton model is one of popular method to investigate traffic motion, which is applicable to the two-dimensional case(Biham 1992).

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Fig. 1. Cellar automaton model to the two-way pedestrian move. The walkway is a rectangle and each pedestrian is located in a square lattice site.

This paper also employs a simple celler automaton model which will be explained in the next section. In §3, panic occurrence are analyzed statistically.

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Each lattice site is occupied by one person at most. There are two groups of pedestrian, one of which tends to move one direction and the other opposite direction. Figure 1 shows a configuration of the footway, where solid and blank circles indicate the pedestrians. A solid circle tends to move rightward (on paper) and a blank circle leftward. We apply the following two celler automaton rules for the motion of the pedestrians. One of them (rule A) impose that if a pedestrian is likely to head-on collide with another pedestrian, he tend to avoid right side. In the other rule (rule A), he chose his direction randomly. Detail of the rules are followings.

5) Front, left-front and right-front sites are occu-

pied the pedestrian cannot move.

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right-front sites are vacant, the pedestrian choses a site randomly for the next step.

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2) If the front site and left-front site are occupied and right-front site is vacant, the pedestrian steps in it.

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The above rules are for a motion of a pedestrians. In our model, all pedestrians do not move simultaneously to avoid long range correlation in the automaton rule. Therefore, we must fix the turn of pedestrians in appling the rules.

3) If the front site and right-front site are occupied

and left-front site is vacant, the pedestrian steps in it.

Same as rule A.

Ordering the pedestrians

In one time step, every pedestrians applied once. 7) Solid circle pedestrians (see fig. 1)have priority to be applied. 8) Hindward pedestrians (left on the paper) have priority to be applied. 9) Order of the solid circles in the same column are fixed randomly. 10) Next, the rules are applied to blank circles. The ordering of them are determined in the same method as solid ones.

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4) If front site is occupied and both left-front and right-front sites are vacant, the pedestrian steps in right-front site.

Rules 1)-5) and application order 6)-10), consist of the cellar automaton model for one time step. Rule 7) is introduced for convenience. Actually

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solid circle need not have the initial priority. Since the simulation are done many time steps, initial" difference" should not affect the long time behavior of the pedestrians. On the other hand, rule 8) is important. As a simple case, consider single directional(rightward) flow and a congestion cluster in it. As seen from fig.2, if the hindward pedestrians have the priority to be applied the motion, only the front pedestrian can move in one time step. In case of the forward priority, all pedestrians can move simultaneously in one time step. In real footpath, the pedestrians tend to keep interval when walking. Therefore, the hindward priority is better choice in our simulation. Hideward priority

The solid circles obey the cellar automaton rules. Figure 3 show Flaw-RATE diagram in case of rule A. Almost same result is obtained for rule B. Here, RATE is treated as a control parameter and Flaw is defined by,

Flaw _ # of pedestrians escaped from the exits -

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In this paper total time step for one simulation is set 1000. For low RATE values, the Flaw grows in linear with it. The curve is bended around RATE = 0.3, and become flat for RATE> 0.4. The fundamental diagram of the car traffic flows usually have the maximum value. Contrarily our model restricts the pedestrian's density due to the entrance rule 12). That is the reason why the Flow does not decline even in the high RATE values.

Flow 1000 Forward priority

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Fig. 2. Difference between hindward and forward priority. The former case seems realistic. In the exit or entrance of the walkway we treat the pedestrians in the following manner.

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11) Pedestrians in the exit of the footway can always escape next time. 12) After I-step movement, new pedestrians step into the footway. If one site in the entrance is vacant, new one can occupy it with probability RATE, which plays a role as a control parameter.

11 Fig. 3. One-directional flow. If RATE is larger than about 0.4, the diagrams becomes flat.

We note that the side wall is considered to be occupied by a pedestrian. In our analysis all sites are assumed to be vacant as the initial condition. RATE and width and length of the footway (Length, Width EN) are the parameters of our simulation.

Let us come back to the two-directional case. In low RATE area, the motion of pedestrians very smooth, while it getting larger, pedestrian often stop (see rule 5)). If RATE is large enough, collisions between couter directional pedestrians occur frequently and we observe a freeze of the flow. It is nothing but the panic. Typical configuration of the paic is shown in fig.4.

3. PEDESTRIAN FLOW

Our main aim is statistical behaviour of the pedestrian flow. We take the ensemble average of the number of pedestrians who path thorough the footway (Flaw).

Before analyzing two-directional case, we consider one-directional pedestrian motions. Here we do not induct the blank circles into the footway.

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is wider, the difference becomes clear, however quantitative difference is not observed in our simulation. Therefore in the following we mainly apply the rule A only. Figure 6 shows RATE - Flow diagram in which the panic cases are taken into the ensemble.

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Fig. 4. A configuration of the panic. The twoway pedestrians flow freezes when RATE is large enough. The figure shows singlepanic wall, whereas numerical simulations sometimes show the panic with multi-walls.

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We see a maximum point around RATE = 0.98. This gives an optimal flow, above which panic may occur. In our automaton model, the velocity of the pedestrians is 0 or 1, and the their density is limited due to the entrance rule.

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The probability that panic occures seems interesting quantity for the footway designing. In the next section, we comprehensively treat the 'panic probability'.

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We show the RATE - Flow diagram, in case the width and length is 10 and 100 respectively and the ensemble number is set 50. As seen from the figure 5, the flow of the pedestrians grows linear with the RATE, though the curve bents when it is larger.

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RATE 4. PANIC PROBABILITY

Fig. 5. Average of total number of pedestrians who path through the footway for rule A and B. Clear difference is not seen. If RATE> 0.098, panic can occur, and the panic probability is almost 100% at RATE = 0.12. The ensemble are taken only from non-panic case.

As seen before, if Length = 100, Width = 10, and 0.098 < RATE < 0.12, the panic possibly occures. Ensembles of simulations gives the probability of the panic occurrence. In this section we pay attention on the panic probability applying rule A. Ensemble number is set 200.

If RATE is over 0.98, the panic begins to occur and finally, for RATE = 0.12 or less, the panic occures 100 %. In the figure, the ensemble are chosen only form non-panic case.

Figure ?? show the panic probability for a fixed width (=10). Three kinds of length are chosen (Length = 10,40,160). We see that that panic probability becomes larger with the length of the pathway. We should also note that the critical RATE above which panic probability becomes finite are almost common. (about 0.1). Anyhow the larger length becomes, the steeper curve of the diagram becomes.

Results of rule A and B are given in the figure, though we do not observe quantitative difference between them. Even if rule A and B are essentially different, the dynamical behavior of them are quite similar. If the width of the footpath

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sembles the phase transition of macroscopic matter. According to the statistical mechanics, if the system size becomes larger, a statistical quantity (order parameter) changes critically at a point of the control parameter. Here we can regard as the panic probability as the the order parameter.

Panic probabilty

Width=10 Length=16

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5. ASYMMETRIC CASE

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So far the enter rate (RATE) of the two groups is assumed common. In real situation, the flow usually asymmetric. Therefore, we need consider such case in which RATE of the solid circles and blank circles are different.

RATE

Fig. 7. Panic probability for a fixed width (=10) The larger length becomes, the steeper curve of the diagram becomes. Figure 8 show the panic probability for a fixed length (=200). We change the Width from 10 to 640. As Width becomes larger, the transitions of panic probability in the figure becomes steeper.

Flow 1500

We expect phase transition like phenomenon for a large width. Remark that the saturation RATE on which the panic probability is 100% are common (- 0.12). This is comparable with fig.7.

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RATE Fig. 8. Panic probability for a fixed length (=200). If width is large enough, the diagram exhibits phase transition like behavior. Two results represented by figs.8 and ??, suggest that if the size of footpath is large enough, transition of the panic probability is steep. This re-

Similarly to the 'symmetric' case the panic occures above RATE - 0.15 Panic probability is shown in fig.lO.

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Fig. 10. Panic probability in asymmetric case. The ensemble number(= 50) is not large enough. Then the curve seems ugly.

6. SUMMERY AND DISCUSSIONS We have considered pedestrian couter flow in a footway. If the rate of pedestrians entrance (RATE) is large enough, panic phenomena occur. Employing RATE as the control parameter, we have statistically calculated panic probability which is regardes as a order parameter. If the size of the footpath becomes larger, the transition of the panic probability becomes steeper, which resembles the behavior of an order parameter in statistical mechanics. In applying our model to real situation, we have to set values of lattice size l and time of one step C:..t. Here we set l the psychological distance in intimate relation ship l "" 0.5m (Feurtey 2000) and C:..t = l/walking speed. If we choose walking speed 1.5 m/s , C:..t = 1/3(8). Using those values we can calculate the real panic rate for a given width, length and number of pedestrians entering into the footway.

REFERENCES O.Biham, A.A.Middleton and D.Levine, Phys Rev. A46(1992)R6124. F .Feurtey, Simulating the Collison Avoidance Behavior of Pedestrians. Master thesis -the University of Tokyo, dep. Electronic Eng. (2000). M.Muramatsu, T.Nagatani, Physica A267(1999)487498.

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