Cellular Automaton Simulation of Evacuation Process in Story

Transcription

1 Commun. Theor. Phys. (Beijing, China) 49 (2008) pp c Chinese Physical Society Vol. 49, No. 1, January 15, 2008 Cellular Automaton Simulation of Evacuation Process in Story ZHENG Rong-Sen, 1 QIU Bing, 2 DENG Min-Yi, 3, KONG Ling-Jiang, 3 and LIU Mu-Ren 3 1 Department of Physics and Information Science, Yulin Normal College, Yulin , China 2 Department of Information Material Science and Engineering, Guilin University of Electronic Technology, Guilin , China 3 College of Physics and Electronic Engineering, Guangxi Normal University, Guilin , China (Received January 18, 2007) Abstract Computer simulations on the evacuation process in a story are launched with cellular automaton in this article. The story is composed of five rooms and one corridor. Influence of various parameters on the evacuation process is investigated. It shows that the width of the door of rooms has little influence but the width of the corridor and the maximum velocity of the pedestrian have great influence on the time for evacuation. The relation between evacuation time and the width of corridor is found as t c W It is also found that appropriate shape of the room is helpful to evacuation. PACS numbers: Fh, k, m Key words: cellular automaton, pedestrian flow, evacuation 1 Introduction It is important to know about the characteristics of pedestrian flow. The jam of pedestrians could cause many accidents, and brings large loss to our lives. In recent years, accidents caused by the jam and trample of crowd have happened frequently. So, great attention has been paid to the problem of evacuation, which has become a pop problem in the studies on the pedestrian flow. Up to now, the models commonly used to simulate the evacuation process on computer are the social force model, [1] the lattice-gas model, [2 5] and the cellular automaton model. [6 9] In most researches, what people are most interested in are the time for evacuation and the states of crowd in evacuation process. However, all these are closely in connection with the surroundings around people. Muramatsu et al. presented a lattice-gas model of biased-random walkers to investigate the pedestrian counter flow in a channel. [2] Dynamical jamming transition from the freely moving state to the stop state was reproduced by computer simulation. After that, the model was widely used to simulate the pedestrian flow in all kinds of channels, such as channel with a bottleneck, [3] T-shape channel, [4] merging channel without bottleneck, [5] and so on. On the other hand, Fukui et al. presented a cellular automaton model to study the pedestrian counter flow in an underpass. [6] The computer simulation shows that complex behaviors such as self-organizations and jamming transitions would occur. Blue also presented a cellular automaton model of pedestrian flow, [7] which is based on the multi-lane cellular automaton model of traffic flow. The fundamental diagram of pedestrian flow in a one-direction channel was given, and relations between the shape of fundamental diagram and various parameters were discussed. Some modifications were put to the lattice-gas model of biased-random walkers by Tajima et al. [10] The new model was used to simulate the evacuation from a hall with only one exit. It was shown that some transients occur during the evolution of crowd flow. And the scale behavior between the transition time and width of exit was found. This model was also used by Takimoto et al. [11] to investigate the time for evacuating the hall. It was found that initial position of people has great effect on the vacuation time. Nagai et al. studied the evacuation from a dark room. [12] Effect of exit configuration on the evacuation process was discussed in conditions that people are all invisible. Compared with the data obtained in experiment, the result of simulation is rather reasonable. It is important to the real lives to study the evacuation process from the story. We have ever used the latticegas model of biased-random walker [13] and the multi-speed cellular automaton model of pedestrian flow [14] to simulate the evacuation process in a corridor. It was found that the width of door has little influence but the width of corridor and the maximum velocity of people have great influence on the evacuation time. Using the reformative cellular automaton model, we studied the evacuation process from a big room. [15] However, it shows that the evacuation time is sensitive to the width of door. The story is commonly composed of rooms and a corridor. Rooms are joined to the corridor at the doors. During the evacuation, people in rooms and in the corridor always evolve as a whole. In this article, computer simulation on the evacuation process in a story is launched with cellular automaton. The influence of various parameters on the pedestrian flow is discussed. 2 Model Suppose that the story is composed of five rooms and one corridor. All the rooms are collocated along the same side of the corridor. Each room is represented by the square of B A sites, and has only one door facing to The project supported by National Natural Science Foundation of China under Grant Nos and and the Natural Science Foundation of Guangxi Province of China under Grant No dengminyi97@tom.com

2 No. 1 Cellular Automaton Simulation of Evacuation Process in Story 167 the corridor. The width of each door is d. The corridor is represented by the square of W L, where W is the width and L = 5 A is the length of the corridor. The final exit lies at the end of the corridor on the right. At the beginning, all the people are well-distributed in all the rooms. Each person occupies only one site in the system. No two or more persons are allowed to stay in one site at the same time step. The initial density of people in rooms is ρ = 0.3. From beginning, everybody moves towards the exit simultaneously as fast as he can. The movement in rooms is different from that in the corridor. Once a walker steps out of a room, he joins the pedestrian flow in corridor. When a walker gets to the exit, he would be got rid of from the system. Considering that people in rooms have to change their directions frequently according to their localities, and they could not move as quickly as people in corridor, we use a one-speed cellular automaton model to simulate their movement. It means that every one in rooms can move no more than one site distance in a single time step. However, people in corridor seldom change their directions, and could move in a relatively high speed. So, we use a multi-speed cellular automaton to simulate their movement, which means that people can move over several sites distance in a single time step, so as it would not go beyond V max. In every time step, all people change their place simultaneously. The rules of evolvement are described as follows. 3 Rules for People in Rooms (i) In the area which faces directly to the exit, the preferred direction is up-wards, and the sub-preferred directions are left-wards and right-wards. That is, if there is nobody in his way, the walker moves up-wards to the nearest site, otherwise, he could turn to the right site or the left site if there is more space in front of it. (ii) Since the doors are all located on the top left corner of the rooms, in the area which faces indirectly to the exit, the preferred directions are left-wards and up-wards. If there is nobody in his way, the walker goes left to the nearest site in probability of D x = x x 0 /( x x 0 + y y 0 ), and goes up to the nearest site in probability of D y = y y 0 /( x x 0 + y y 0 ). (x,y) is the coordinates of the walker, and (x 0,y 0 ) is the coordinates of the edge of the door. If D x > D y, the sub-preferred direction is down-wards, and if D x < D y, the sub-preferred direction is right-wards. 4 Rules for People in Corridor The corridor is supposed to be made up of W parallel lanes, which have the same length of L. (i) According to his surrounding, the person in corridor finds out an optimal lane from the current lane and the lanes nearby. The optimal lane is the one along which he could move in the highest speed in the coming time step. (ii) Every person moves to his optimal lanes simultaneously. (iii) Every person moves towards the exit along the lane as fast as he can. The maximum speed of people is V max = 3, which means that a person can move at most 3 sites distance in a single time step. The walker who gets to the exit would be removed from the system. Considering that all people update in parallel, collision is unavoidable. When collision happens among several walkers, one of them would be chosen randomly to occupy the site. The density is defined as p = N/[L (W+B)], where N is the number of walkers in the story. We define the mean velocity as v = N m /N, where N m is the total number of sites that all people move over in a single time step. The mean flow rate is defined as J = vp = N m /[L (W +B)]. 5 Computer Simulation and Results Setting the width of room A = 20, the length of room B = 30, the width of corridor W = 5, and the width of door d = 4, we put down the density p, velocity v, and the mean flow rate J at each time step. And we note down the time for evacuation t c of each process of evacuation. The process of evacuation is simulated 100 times by the computer. The result is averaged over them. Fig. 1 Plots of velocity V (a), density p (b), mean flow rate J (c) against time step t.

3 168 ZHENG Rong-Sen, QIU Bing, DENG Min-Yi, KONG Ling-Jiang, and LIU Mu-Ren Vol. 49 Figure 1 is the plot of velocity, density, and mean flow rate against time step. Figure 2 is the spatial configuration of people at different time step. At the beginning, most people are in the rooms, and most of them are movable. So, the velocity approaches 1. The density keeps its value at the beginning because nobody has left the system. Figure 2(a) shows the distribution of people at that moment. Fig. 2 Spatial configuration of people at different time step. (a) t = 0; (b) t = 60; (c) t = 160. Before long, more and more people accumulate in front of the doors, and velocity decreases rapidly. On the other hand, when more and more people enter the corridor, they move towards the exit at an even faster speed. And then, velocity would stop decreasing at about 0.5 and keep its value in a comparatively long period. Along with people going out of the system, the density also decreases uniformly. Figure 2(b) shows the distribution of people at that moment. Afterwards, the accumulations in front of the doors fade away gradually, velocity increases again. Because people in corridor can move at a comparatively high speed, the velocity would go up quickly to about 2.3, which is only a little lower than V max. In the end, the mean flow rate decreases suddenly because that most people have already evacuate the story. It is shown in Fig. 1 that the density keeps decreasing uniformly during almost the whole evacuation process. Therefore, the pedestrian flow seems to flow out of the story equably, no matter how protean it is in the story. Furthermore, it is shown in Fig. 2(c) that in different rooms, the time for people to evacuate the room is different, though all the people are well-distributed in each room at the beginning. People in the room that is the farthest from the exit could evacuate the room firstly, because they can enter the corridor successfully without being blocked by the pedestrian flow from the room behind. However, it is always difficult for people in the other rooms to enter the corridor successfully because that they would often be blocked by the pedestrian flow from the rooms behind. Moreover, the pedestrians who have entered the corridor could be blocked by the pedestrian flow from the rooms before. The farther the room is from the exit, the more seriously the pedestrians from it would be blocked. The blocking of pedestrians in

4 No. 1 Cellular Automaton Simulation of Evacuation Process in Story 169 corridor would also prevent people in rooms from going into the corridor. Therefore, in all rooms except the farthest one, the time for people to evacuate rooms would extend with the increase of the distance from the room to the exit. The influence of various parameters on the time for evacuation is investigated next. Fig. 3 The plot of mean J against t under different W. Fig. 4 The log-log plot of t c and W for different V max. Fig. 5 The plot of J against t under different V max. Fig. 6 The plots of t c against V max. Fig. 7 The plot of J against t under different width of door d. Figure 3 is the plot of mean flow rate J against time step t under different width of corridor W. It shows that the time for evacuation t c decreases distinctly with the increases of W. And the value of J in the steady phase Fig. 8 The plot of J against t under different shape of room. would also increase when the corridor is widened. Therefore, to widen the corridor is effective for improving the evacuation in story. Figure 4 is the log-log plot of t c and W for different

5 170 ZHENG Rong-Sen, QIU Bing, DENG Min-Yi, KONG Ling-Jiang, and LIU Mu-Ren Vol. 49 V max. It is found in the figure that t c scales with W as t c W Figure 5 is the plot of J against t under different V max. It is shown that the influence of V max on the value of J in the steady phase is not as distinct as that of W. However, V max also has great influence on t c. Apparently the increase of V max could shorten t c. Figure 6 is the plot of t c against V max. Figure 7 is the plot of J against t under different width of the door, d. It is shown that the width of door has little influence on both t c and the value of J in the steady phase. Therefore, to widen the width of the door is useless to improve the evacuation in story, though it was found to be effective in the system of a sole big room [15]. Finally, we change the shape of the rooms. Given the width of the room A = 30, the length of the room B = 20, and the length of corridor being lengthened accordingly, which is now L = A 5 = 150, figure 8 is the plot of J against t, compared with that of the former shape. It is seen that the time for evacuation is almost the same, but the value of J in steady phases has increased. So, the appropriate shape of room is also helpful to the evacuation process in story. 6 Conclusion Using the cellular automaton model of pedestrian flow, we have simulated the evolution of pedestrian escaping flow in a story. The characteristic of escaping pedestrian flow in story is also analyzed. Some conclusions are obtained. First, in different rooms, time for people to evacuate the room is different. People in the room that is the farthest from the exit could evacuate the room first. In the other rooms, the time would extend with the increase of the distance of the room from the exit. Second, it was found that the width of the door d has little influence but the width of the corridor W and the maximum velocity of pedestrian V max have great influence on the evacuation time. So, to widen the width of the corridor could accelerate the evacuation process. But to widen the width of door is useless. Third, the appropriate shape of room is also helpful to the evacuation process. References [1] D. Helbing, Phys. Rev. E 51 (1995) [2] M. Muramatsu, T. Irie, and T. Nagatani, Physica A 267 (1999) 487. [3] Y. Tajima, K. Takimoto, and T. Nagatani, Physica A 294 (2001) 257. [4] Y. Tajima and T. Nagatani, Physica A 303 (2002) 239. [5] T. Nagatani, Physica A 307 (2002) 505. [6] M. Fukui and Y. Ishibashi, J. Phys. Soc. Jpn. 68 (1999) [7] V.J. Blue and J.L. Adler, Trans. Res. Rec (1998) 29. [8] V.J. Blue and J.L. Adler, in Proceedings of the 79 th Transportation Research Board, Transportation Research Board, Washington D.C. (2000). [9] V.J. Blue and J.L. Adler, Trans. Resh. Part B 35 (2001) 293. [10] Y. Tajima and T. Nagatani, Physica A 292 (2001) 545. [11] K. Takimoto and T. Nagatani, Physica A 320 (2003) 611. [12] R. Nagai, T. Nagatani, M. Isobe, and T. Adachi, Physica A 343 (2004) 712. [13] B. Qiu, H.L. Tani, L.J. Kong, and M.R. Liu, Chin. Phys. 13 (2004) 990. [14] B. Qiu, H.L. Tani, C.Y. Zhang, L.J. Kong, and M.R. Liu, Int. J. Mod. Phys. C 16 (2005) 225. [15] Tan Hui-Li, Qiu Bing, and Liu Mu-Ren, J. Guangxi Normal University 22 (2004) 1.