A dynamic cellular automaton model for evacuation process with obstacles

(1)

A dynamic cellular automaton model for evacuation process with obstacles

R. Alizadeh

Department of Mathematics, Shahed University, P.O. Box 18151-159, Tehran, Iran

a r t i c l e i n f o

Article history:

Received 24 January 2010

Received in revised form 14 August 2010 Accepted 3 September 2010

Keywords:

Cellular automaton Evacuation process Pedestrian dynamics

a b s t r a c t

A dynamic cellular automaton (CA) model is proposed to simulate the evacuation process in the rooms with obstacles. Besides the basic parameters such as human psychology, placement of the doors, doors width, position of the obstacles and light of the environment, distribution of the crowd plays an important role in this model. Applying our model, simulation of the evacuation process for a restaurant and a classroom are presented. Also effects of pedestrians distribution, doors position and doors width on the evacuation time are discussed and the obtained results are compared with several static models.

1. Introduction

There has been a great interest in studying CA models in order to simulate various physical and biological processes (for some examples seeJeon and Yoo, 2008; Vanem and Skjong, 2006; Pelec- hano and Malkawi, 2008; Georgoudas et al., 2007; Rinaldi et al., 2007; Encinas et al., 2007; Herr and Kvan, 2007). Besides the social forces (Helbing and Molnar, 1995; Helbing et al., 2000) and lattice gas (Helbing et al., 2003; Tajima et al., 2001; Isobe et al., 2004) models, CA models are used successfully to simulate evacuation process. CA approach could be appropriate to describe pedestrian dynamics in complex situations because of its simplicity, flexibility and efficiency.

Cellular automata for pedestrian dynamics have been proposed in Fukui and Ishibashi (1996, 1999), Muramatsu et al. (1999), Muramatsu and Nagatani (2000a,b), and Klzüpfel et al. (2000).

These models can be considered as generalizations of the Biham–

Middleton–Levine model for city traffic (Biham et al., 1992). Most of these works have focussed on the occurrence of a jamming transition as the density of pedestrians is increased.Burstedde et al.

(2001)proposed a two-dimensional cellular model with exclusion statistics and parallel dynamics which could be used in a large room without obstacle and with reduced visibility, e.g. due to fail- ure of lights or smoke. In this model, Long-range interactions between the pedestrians are mediated by a so-called floor field which modified the transition rates to neighboring cells and it could be modified by the motion of the pedestrians. Indeed, the model used an idea similar to chemotaxis (seeJacob (1997)for a review), but with pedestrians following a virtual rather than a

chemical trace. They showed that the introducing such a floor field is sufficient to model collective effects and self-organization phe- nomena (seeHelbing, 2001; Helbing et al. (2002)for an overview and a comprehensive list of references) in pedestrian dynamics.

Kirchner and Schadschneider (2002)characterized different classes of behavior exhibited by the model inBurstedde et al. (2001). Con- sidering the simple human judgment and introduction ofback step- ping,Fang et al. (2003)established a set of rules for bi-direction pedestrian movements and proposed a new CA model. Using this model, they found that the critical pedestrian density increases as the probability of back stepping increases at the same system size.Daoliang et al. (2005)proposed a two-dimensional CA model to investigate about optimal widths, positions and separations of exits in a large room without obstacles.Song et al. (2005)introduced a new CA model entitled cellular automata with forces essentials (CAFE). In this model, the interactions in evacuation are classified into three types: attraction, repulsion and friction.

Song et al. (2006)improved CAFE model and compared the arching, clogging and faster-is-slower behaviors as well as the evacuation time, in detail with those of the original social force model introduced byHelbing et al. (2000).Yamamotoa et al. (2007)proposed the real-coded cellular automata (RCA) as a new numerical model for pedestrian dynamics in a room without obstacles. It is based on the real-coded lattice gas (RLG), which has been developed for fluid simulation (Hashimoto et al., 2000).Zhao et al. (2008)applied a CA random model to occupant evacuation considering the influence of human psychology and behavior in a large room without obstacles.

Using this model, they concluded that the exit width must be big- ger than a certain value to ensure a dilute state of evacuation and the optimal value of the exit separationf, is independent of the exit width, but is related to the total width of the buildingD:f0.3D.

doi:10.1016/j.ssci.2010.09.006

⇑ Tel.: +98 021 51212631.

E-mail address:[email protected]

Contents lists available atScienceDirect

Safety Science

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / s s c i

(2)

simulate the process of evacuation of pedestrians in a room with fixed obstacles.

Most of the mentioned models partition floor into rectangular cells and assign a weight to each cell in every time step, depending on human psychology and behaviors, placement of the doors, doors width, position of the obstacles, light of the environment and etc.

The set of these cells together with their weights is called floor filed of the model. There exist two kinds of the floor fields, static and dynamic. Static floor fields do not evolve with the time and do not change by the presence of the pedestrians. In a static floor field, considering placement of the doors, doors width and position of obstacles, a fix weight is assigned to each cell. In order to deter- mine the static weight of the cells, a metric should be used. Nish- inari et al. (2004) metrics are well known metrics on cellular environments. Euclidian metric is suitable for the rooms with no obstacles, but Manhattan and Dijkstra metrics are used for the rooms that contain obstacles. To calculate Dijkstra metric a visibility graph is constructed and using Dijkstra algorithm the shortest path is specified. In comparison to Manhattan metric, pedestrians movements are more realistic with Dijkstra metric. However con- structing a static floor field with Dijkstra metric may consume a considerable time in large environments.Varas et al. (2007)introduced a metric which works as well as Dijkstra metric but is easier and faster to compute. To obtain this metric, they used a simple recursive process.

Dynamic floor fields are modified by changing the time and the presence of pedestrians. In a dynamic floor field, regarding static floor field, human psychology and behavior, distribution of pedestrians and some other factors, a weight is assigned to each cell in every time step. Finally, movement of the pedestrians is determined by the weight of the cells and rules of pedestrians interaction.

Most of the introduced models are based on the assumption that pedestrians are uniformly distributed and in some models distribution of the crowd is only discussed in the rooms without obstacles and around the exits.

In our model, distribution of the crowd plays an important role and affects the weight of the cells in every time step (see Section 3). We use the static floor field constructed inVaras et al. (2007).

Also the pedestrians movements are based on the rules inHelbing and Molnar (1995), Burstedde et al. (2001), Von and Burks (1966), Wolfram (1986), Wolfram (1994), Yang et al. (2003).

The aim of our paper is to consider the effects of the obstacles and crowd distribution in evacuation process, to provide the occupants safety in enclosed environments, avoiding and/or reducing the number of fatalities (and/or the number of injuries). Also research objective is to identify information that might be useful in building designing to assess buildings and their ability to provide sufficient time for the occupants to evacuate safely in the event of a emergency.

The structure of the remainder of this paper is as follows:

First, in Section2, we describe the set of the rules which govern the motion of the pedestrians in the model. In order to consider the distribution of the pedestrians during the evacuation process, a new dynamic CA model is proposed in Section3. Then, in Section 4, we apply the model to a restaurant and a classroom and discuss about the effects of the pedestrians distribution, doors position, doors width and mean velocity of the pedestrians on the evacuation time. Finally, in Sections5 and6, we summarize the results and point out to future research.

2. Preliminaries

Varas et al. (2007) constructed a bidimensional CA model to simulate the evacuation process of pedestrians in a room with good visibility and fixed obstacles. In this model, the obtained floor

field is similar to the Manhattan metric, in the sense that the floor field at each cell is the minimum path length from an exit door to the cell. The difference lies in the fact that diagonal movements are allowed. The floor field has been assigned as follows:

(1) The room is divided into a rectangular grid. The exit door is assigned a value 0. Then

(2) all adjacent cells to the previous one are assigned a value according to the following rules:

(2.1) If a cell has valueN, then adjacent cells in the vertical or horizontal directions are assigned a valueN+ 1 and adjacent cells in diagonal directions are assigned a valueN+ 1.5. This is a simple attempt to represent the fact that the distance between two diagonally adjacent cells is larger than in horizontal or vertical directions.

(2.2) If there are conflicts in the assignment of a value to a cell, because it is adjacent to cells with different floor fields, then the minimum possible value is assigned to the cell in conflict.

(3) Then the third layer of the cells is calculated, which is all cells adjacent to the second layer, and not in the first layer.

(4) The process is repeated until all cells are evaluated.

(5) Walls are also considered when defining the grid. Cells belonging to walls are given very high values of the floor field. This ensures that pedestrians will never attempt to occupy one of those cells.

In our model, local rules for pedestrian movements and interactions are based on the rules inHelbing and Molnar (1995), Burstedde et al. (2001), Von and Burks (1966), Wol- fram (1986, 1994), Yang et al. (2003)as follows:

(6) Each occupant chooses one of the adjacent cells at next time step depending on their weights (here the cell with smaller weight is chosen).

(7) For each cell where more than one occupant wants to enter, randomly assign it to one of them with a certain probability;

other occupants still stay where they are. Here, we consider every viewer has the same probability.

(8) In order to avoid a deterministic model, a random slowdown rule is introduced: Give each person who has decided to move a probability to stay. Here, an experiential value, 5%, is used.

Note that the weight of a cell in the constructed floor field, depends on the position of the cell, exits and obstacles. Since these parameters are constant and do not change with the time, the obtained floor field is independent of the time and so is static. We need a floor field that updates itself during the time with respect

Fig. 1.A 1828 floor with 91 pedestrians which all of them are located in the right side of the door A.

(3)

to the crowd distribution. For example, considerFig. 1, a room with two exits A and B, in which pedestrians are not distributed uniformly. Here we have 91 pedestrians which all of them are located in the right side of A. The yellow colored occupants are about 5% of the pedestrians in the rule 8.

Fig. 2displays the obtained floor field ofFig. 1, using the rules 1–5. Applying this floor filed and using the rules 6–8, simulations show that all pedestrians will move towards A and none of them pass through the door B. In a real situation this is not acceptable.

In the next section, we will establish another floor field in which the distribution of the pedestrians is considered.

3. Model description

In order to consider the distribution of pedestrians during the evacuation process, we construct a dynamic floor field. In this model for determining the weight of a cellxwith respect to a door Ain theith step ‘W^A_iðxÞ’, we consider two following parameters:

(i) W^A_staticðxÞ: The distance from x to A with respect to the applied metric.

(ii) T^A_iðxÞ: The number of persons who are nearer thanxtoAin theith step.

We set W^A_iðxÞ ¼W^A_staticðxÞ

|fflfflfflfflfflffl{zfflfflfflfflfflffl}

static weight

þ

a

_T^A_i_ðxÞ

|fflffl{zfflffl}

dynamic weight

; ð3:1Þ

where

TÂ_iðxÞ ¼jPÂ_iðxÞj þ¹₂jEÂ_iðxÞj dA

;

Vi¼ fyjyis occupied by a person in theith stepg;

PÂ_iðxÞ ¼ fyjWÂ_staticðyÞ<WÂ_staticðxÞandy2Vig;

EÂ_iðxÞ ¼ fyjWÂ_staticðyÞ ¼WÂ_staticðxÞandy2Vig;

anddAis the width of the doorA. HereP^A_iðxÞ

andE^A_iðxÞ

denote the number of elements ofPÂ_iðxÞandEÂ_iðxÞ, respectively. AlsoWÂ_staticðÞ could be Euclidian, Manhattan, Dijkstra or any other proper metric.

In our simulations we use the metric introduced in the previous section. Also

a

is a constant that depends on the pedestrians and environment conditions. We call

a

_,the coefficient of crowd avoid- ance. Regarding(3.1), larger

a

increases the effect ofT^A_i (distribution efficiency) and hence implies less interest in moving towards the crowded doors. For

a

= 0, the equality W^A_iðxÞ ¼W^A_staticðxÞ holds and the dynamic model is same as the static model.

Finally the weight of a cellxin the stepiP1, is displayed by Wi(x) and is calculated by the following formula

WiðxÞ ¼minfW^A_iðxÞjAis a door in the roomg;

We will refer from now on the model introduced in the previous section and the new proposed model, the static and dynamic model respectively.

Fig. 3displays the floor field ofFig. 1respecting the dynamic model for

a

= 1 in the first step.

Figs. 4 and 5indicate snapshots of the simulation given inFig. 1 after 10 time steps according to the static and our dynamic model respectively. The evacuation time forFig. 1with respect to the static and dynamic model is 190 and 110 time steps respectively.

Since the new model is dynamic, it is possible that some of the pedestrians select a door firstly, but after several time steps, regarding the present situation, change their idea and move towards another door. To see this, considerFig. 6which is a snapshot of simulation after 41 time steps. As we see in this figure, three of the pedestrians move from the door A towards the door B.

4. Simulation results

Determining the proper locations of the exits in order to reduce the evacuation time is interesting in the field of evacuation process. In most of the models, crowd distribution is not considered and it is supposed that the crowd is uniformly distributed in a large room without any obstacle, but this parameter is important in Fig. 2.The floor field ofFig. 1, obtained by the rules of the static model.

R. Alizadeh / Safety Science 49 (2011) 315–323

(4)

specifying doors locations. In this section, we discuss about the effects of crowd distribution on evacuation process. For this purpose we give two examples. In these examples rooms are described by a two-dimensional grid. Each cell can be empty, occupied by an obstacle or by a pedestrian and its size corresponds to 0.4 m by 0.4 m, the typical surface occupied by a person in a dense situation (Burstedde et al., 2001). A single pedestrian (not interacting with others) moves with a velocity of one cell per timestep, i.e., 0.4 m per timestep. The mean velocity of a pedestrian is about 1 m/s, which yieldsD_t_’0.4 s. Also the width of the doors or obstacles dimensions can be calculated by the cells which they contain. For example, inFig. 7, two types of dining tables with dimensions of 1.2 m0.8 m and 0.8 m0.8 m are available.

Considering the randomicity in our simulations, each datum is the mean value of 10 times independent calculations.

Example 1.In this example, we consider a restaurant with 18 dining tables and 112 persons. Dining tables play the role of obstacles (Fig. 7).

Fig. 3.The floor field ofFig. 1with respect to the dynamic model, fora= 1 in the first step.

Fig. 4.Snapshot of the simulation given inFig. 1after 10 times steps according to the static model.

Fig. 5.Snapshot of the simulation given inFig. 1after 10 times steps according to the dynamic model.

Fig. 6.Snapshot of the simulation given inFig. 1after 41 time steps according to the dynamic model. Three of the pedestrians move towards the door B.

(5)

The obtained floor field ofFig. 7with respect to the dynamic model, for

a

= 1, in the first step is given inFig. 8.

Here we change the location of two exits along the cells labeled from 1 to 81 and study variations of the evacuation time.Fig. 9a and b shows the contour graph which provides the relation between the location of the doors and evacuation time for the static (dynamic) model.

As we see, for both cases, in the red parts, evacuation time is high and in the dark blue parts, evacuation time is low. The evacuation time of other parts lies between these ranges. The contours are symmetric with respect to the liney=x, because swapping the locations of two doors does not change the evacuation time. Also

the liney=xis located in the red part, since in this case, two doors coincide and in fact we have only one door.

Comparing these contours, we get the following facts:

(i) Evacuation time in the dynamic floor field is efficiently less than the evacuation time in the static floor field.

(ii) The area of the dark blue regions efficiently increases in the dynamic case. Hence there are further suitable places for the doors in the case of dynamic model.

(iii) Dark blue regions in the static case almost remain dark blue in the dynamic model. Hence usually suitable positions for the doors in the static case remain suitable in the dynamic case.

Fig. 7.A restaurant with 18 dining tables and 112 persons.

Fig. 8.The obtained floor field ofFig. 7with respect to the dynamic model, fora= 1 in the first step.

(6)

(iv) At the first step, if distribution is in such a way that the number of pedestrians who select the doors A and B are approximately equal, then the evacuation time decreases. To see this, letNbe the number of pedestrians and set

N^A_static¼ jfxjW^B_staticðxÞ6W^A_staticðxÞgj;

N^A₁¼ jfxjW^B₁ðxÞ6W^A₁ðxÞgj;

D_static_¼₁^minfN

A

static;NNÂ_staticg maxfNÂ_static;NNÂ_staticg and D₁_¼₁^minfN

A

1;NNÂ1g maxfNÂ₁;NNÂ₁g:

D_static andD₁are positive real numbers lie between 0 and 1.

Clearly ifD_static_andD₁are close to zero, then in both models the persons who intend to pass through the doors A and B are approximately equal firstly. Also in both models, ifD_static_andD₁_{are close} to one. Most of the pedestrians decide to pass through one of the doors and a small number of them decide to pass through the other, at the first step.

Fig. 10a and b displays the relation betweenD_staticandD₁with respect to doors positions respectively. As we see if a point is located in the dark blue¹position inFig. 10a and b, then it is also located in dark blue portion inFig. 9a and b. The converse is not true in dynamic case, because according toFigs. 9band 10b, it is possible that most of the pedestrians select a specified door at the first step, but the evacuation time remains low.

If pedestrians are distributed uniformly or homogeneously, then the static and dynamic models behave in the same way. For example inFig. 11the evacuation time for different doors locations in the static and dynamic models are similar (Fig. 12).

To investigate the relation between the width of the doors and evacuation time, consider two doors A and B situated in the cells numbered 23 and 55 located in the dark blue part i.e. for which the evacuation time is in low ranges. We increase the width of the doors symmetrically in both left and right directions. In the static model, as shown inFig. 13a, increasing width of the doors will decrease the evacuation time. Also increasing the width of the wider door has small effects on reduction of the evacuation time. In the dynamic model (Fig. 13b), increasing the width of the wider door has more effects on decreasing the evacuation time. Note that in this figure the width of the doors, is expressed based on the number of cells that they have occupied. For example, if the width of a door is equal to 10, then its width will be 4 m.

Example 2.In this example we investigate the effect of

a

variations on evacuation time. Consider a classroom with 10 tables, 30 students and 2 single exit as inFig. 14.Fig. 15shows the relation between

a

and evacuation time steps, where

a

_changes

from 0 to 2. As we see, increasing

a

implies decreasing the evacuation time. For

a

= 0, the obtained floor field is static and in fact the distribution is not considered. In this case all of the students select door A and pass through it. For

a

_{= 0.2, the}

coefficient of crowd avoidance is not large enough to effect on evacuation time. By increasing

a

, more students select the door B and this decreases the evacuation time.

Fig. 9.Contours which provide the relations between the location of the doors and evacuation time step, for the static (a) and the dynamic (b) model ofFig. 1.

Fig. 10.The contours which provide the variation ofD_static(Fig. 10a) andD₁(Fig. 10b) with respect to doors positions.

1 For interpretation of color in Figs. 9 and 10, the reader is referred to the web version of this article.

(7)

Fig. 11.The pedestrians are distributed uniformly in the restaurant.

Fig. 12.contours which provide the relation between the location of the doors and evacuation time step for the static (left contour) and the dynamic (right contour) floor field ofFig. 11.

Fig. 13.The contours which provide the variation of evacuation time step in the static floor field (a) and the dynamic floor field (b) with respect to doors width inFig. 7.

(8)

If we consider attraction, repulsion and friction in human behaviors with the parameters stated in Song et al. (2006) and

a

= 1, then the relation between evacuation time and mean velocity of pedestrians inFig. 14, could be seen inFig. 16. As we see, when the desired velocity is small, increasing velocity will decrease the evacuation time. However, trying to move faster above a certain level leads to longer evacuation time, because clogging becomes more severe (faster-is-slower behavior).

Now letJ1/ms andVm/s be outflow (the number of pedestrians leaving the room per second per meter of door width) and velocity

of pedestrians respectively. The relationship between efficiency of leaving, which is demonstrated with the variableJ=V1=m² and velocity of pedestrians is displayed inFig. 17. As we see, like as (Song et al., 2006), desired velocities higher than about 1.3 m/s, reduce the efficiency of leaving.

5. Recommendations for further work

Certainly, there are several features which can be improved in the model. For instance, it is possible that during the evacuation process, some pedestrians jump over the dining tables. For considering such a factor, parameters such as physical ability and age should be considered. Also, obstacles in the model can only occupy an integer number of cells. This is not true in general, but could be improved by using a finer discretization (Kirchner et al., 2004). An- other important point is that chairs inFigs. 7 and 14, have been not considered. In fact, chairs could be considered as a movable obstacle, and this is an interesting modification to the model. Finally, the pedestrians in this model (and other stated models) are almost uniformly modeled. However, in the real world, the crowd evacuation is a complex system composed of different pedestrians and environments. These pedestrians have various psychological states and physiological characteristics and in an evacuation process, they interact and are affected differently by the environments around them. Therefore, in further research, pedestrians should be considered in the form of heterogeneous individuals or groups.

Fig. 14.A class with 10 tables, 30 students and two single exit doors.

Fig. 15.The relation betweenaand evacuation time step inFig. 14.

Fig. 16.relation between the mean velocity of the pedestrians and evacuation time inFig. 14.

Fig. 17.OutflowJdivided by velocity for different velocities inFig. 14.

(9)

6. Conclusions

In this paper, a dynamic CA model was proposed to simulate the evacuation process with obstacles. The model has been designed in such a way that persons in every step consider position of obstacles, exits and distribution of the crowd and make the best decision for exit. Our simulations show that: (i) The evacuation time in the dynamic case is efficiently less than the static case. (ii) Proper positions for doors in the static model remain suitable in the dynamic case. (iii) In both models, if the number of pedestrians who select different doors in the first step are approximately equal, then evacuation time decreases. Converse is not true in the dynamic model, i.e. it is possible that most of the pedestrian select a specified door in the first step, but the evacuation time remains low. (iv) In comparison with the static model, increasing the width of the wider door has more effects on decreasing the evacuation time in the dynamic model.

References

Biham, O., Middleton, A.A., Levine, D., 1992. Self-organization and a dynamical transition in traffic-flow models. Physical Review A 46 (10), R6124–R6127.

Burstedde, C., Klauck, K., Schadschneider, A., Zittartz, J., 2001. Simulation of pedestrian dynamics using a two-dimensional cellular automaton. Physica A 295, 507–525.

Daoliang, Z., Lizhong, Y., Jian, L., 2005. Exit dynamics of occupant evacuation in an emergency. Physica A 363, 501–511.

Encinas, L.H., White, S.H., del Rey, A.M., Sánchez, G.R., 2007. Modelling forest fire spread using hexagonal cellular automata. Applied Mathematical Modelling 31, 1213–1227.

Fang, W., Yang, L., Fan, W., 2003. Simulation of bi-direction pedestrian movement using a cellular automata model. Physica A 321, 633–640.

Fukui, M., Ishibashi, Y., 1996. Traffic flow in 1D cellular automaton model including cars moving with high speed. Journal of the Physical Society of Japan 65, 1868–

1870.

Fukui, M., Ishibashi, Y., 1999. Self-organized phase transitions in cellular automaton models for pedestrians. Journal of the Physical Society of Japan 68, 2861–2863.

Georgoudas, I.G., Sirakoulis, G.C., Andreadis, I., 2007. Modelling earthquake activity features using cellular automata. Mathematical and Computer Modelling 46, 124–137.

Hashimoto, Y., Chen, Y., Ohashi, H., 2000. Immiscible real-coded lattice gas.

Computer Physics Communications 129, 56–62.

Helbing, D., 2001. Traffic and related self-driven many-particle systems. Reviews of Modern Physics 73, 1067–1141.

Helbing, D., Molnar, P., 1995. Social force model for pedestrian dynamics. Physical Review E 51, 4282–4286.

Helbing, D., Farkas, I., Vicsek, T., 2000. Simulating dynamical features of escape panic. Nature 407, 487–490.

Helbing, D., Farkás, I.J., Molnár, P., Vicsek, T., 2002. Simulation of pedestrian crowds in normal and evacuation situations. In: Schreckenberg, M., Sharma, S.D. (Eds.), Pedestrian and Evacuation Dynamics. Springer, Berlin, Germany, pp. 21–58.

Helbing, D., Isobe, M., Nagatani, T., Takimoto, K., 2003. Lattice gas simulation of experimentally studied evacuation dynamics. Physical Review E 67, 067101.

Herr, C.M., Kvan, T., 2007. Adapting cellular automata to support the architectural design process. Automation in Construction 16, 61–69.

Isobe, M., Adachi, T., Nagatani, T., 2004. Experiment and simulation of pedestrian counter flow. Physica A 336, 638–650.

Jacob, E.B., 1997. From snowflake formation to growth of bacterial colonies, Part II:

cooperative formation of complex colonial patterns. Contemporary Physics 38, 205–241.

Jeon, J.C., Yoo, K.Y., 2008. Montgomery exponent architecture based on programmable cellular automata. Mathematics and Computers in Simulation 79, 1189–1196.

Kirchner, A., Schadschneider, A., 2002. Simulation of evacuation processes using a bionics-inspired cellular automaton model for pedestrian dynamics. Physica A 312, 260–276.

Kirchner, A., Klüpfel, H., Nishinari, K., Schadschneider, A., Schreckenberg, M., 2004.

Discretization effects and the influence of walking speed in cellular automata models for pedestrian dynamics. Journal of Statistical Mechanics 10, 10011–

10032.

Klzüpfel, H., Meyer-König, T., Wahle, J., Schreckenberg, M., 2000. In: Bandini, S., Worsch, T. (Eds.), Theory and Practical Issues on Cellular Automata. Springer, Berlin.

Muramatsu, M., Nagatani, T., 2000. Jamming transition in two-dimensional pedestrian traffic. Physica A 275, 281–291.

Muramatsu, M., Nagatani, T., 2000. Jamming transition of pedestrian traffic at a crossing with open boundaries. Physica A 286, 377–390.

Muramatsu, M., Irie, T., Nagatani, T., 1999. Jamming transition in pedestrian counter flow. Physica A 267, 487–498.

Nishinari, K., Kirchner, A., Namazi, A., Schadschneider, A., 2004. Extended floor field CA model for evacuation dynamics. In: IEICE Trans. Inf. & Syst., vol. E87-D, pp.

726–732.

Pelechano, N., Malkawi, A., 2008. Evacuation simulation models: challenges in modeling high rise building evacuation with cellular automata approaches.

Automation in Construction 17, 377–385.

Rinaldi, P.R., Dalponte, D.D., Vénere, M.J., Clausse, A., 2007. Cellular automata algorithm for simulation of surface flows in large plains. Simulation Modelling Practice and Theory 15, 315–327.

Song, W.G., Yu, Y.F., Fan, W.C., Zhang, H.P., 2005. A cellular automata evacuation model considering friction and repulsion. Science in China Series E 48 (4), 403–

413.

Song, W.G., Yu, Y.F., Wang, B.H., Fan, W.C., 2006. Evacuation behaviors at exit in CA model with force essentials: a comparison with social force model. Physica A 371, 658–666.

Tajima, Y., Takimoto, K., Nagatani, T., 2001. Scaling of pedestrian channel flow with a bottleneck. Physica A 294, 257–268.

Vanem, E., Skjong, R., 2006. Designing for safety in passenger ships utilizing advanced evacuation analyses – a risk based approach. Safety Science 44, 111–

135.

Varas, A., Cornejoa, M.D., Mainemera, D., Toledob, B., Rogana, J., Muñoza, V., Valdiviaa, J.A., 2007. Cellular automaton model for evacuation process with obstacles. Physica A 382, 631–642.

Von, N.J., Burks, A.W., 1966. Theory of Self-reproducing Automata. University of Illinois Press, Urbana.

Wolfram, S., 1986. Theory and Applications of Cellular Automata. World Scientific, Singapore.

Wolfram, S., 1994. Cellular Automata and Complexity. Addison-Wesley Publishing Company, Reading, MA.

Yamamotoa, K., Kokuboa, S., Nishinarib, K., 2007. Simulation for pedestrian dynamics by real-coded cellular automata (RCA). Physica A 379, 654–660.

Yang, L.Z., Fang, W.F., Li, J., Huang, R.Q., Fan, W.C., 2003. Cellular automata pedestrian movement considering human behavior. Chinese Science Bulletin 48 (16), 1695–1699.

Zhao, D.L., Li, J., Zhu, Y., Zou, L., 2008. The application of a two-dimensional cellular automata random model to the performance-based design of building exit.

Building and Environment 43 (4), 518–522.