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Doktorarbeit / Dissertation, 2003
160 Seiten
1 Introduction
1.1 Why Simulating Crowd Motion and Evacuation Processes?
1.2 Models for Social Systems
1.3 Evacuation Assessment and how to Improve
1.4 The Perspective of Physics
2 Modeling Pedestrian and Crowd Dynamics - Methodology
2.1 General Concepts
2.2 Movement Dynamics
2.3 Representation of Space: Discrete vs. Continuous
2.4 Population and Behavior
2.5 Empirical Data: Literature Review
2.6 Velocity Distribution and Dependance on Group Size
3 A Two-dimensional Cellular Automaton Model for Crowd Motion
3.1 Description of the Model
3.2 Distance Keeping and Paths for v max > 1
3.3 Cell Size and Discretization
3.4 Walking Direction and Orientation Based on a Potential
3.5 Transition Probabilities
3.6 Comparison of the Different Update Types
3.7 Model Extensions to Include Further Aspects of Crowd Motion
3.8 Relation to Other Lattice Based Models
4 Evacuation Simulations: Implementation and Validation
4.1 The Implementation of the Model Into a Simulation
4.2 Simulation Programs - Overview
4.3 Validation of Simulation Results by Comparison with Evacuation Exercises
5 Evacuation Analysis for Passenger Ships
5.1 Why the Case of a Ship is the Most Complex
5.2 The Procedure: Assembly and Evacuation Phase
5.3 Regulations Concerning the Safety of Ships
5.4 Ship Motion and Further Influences
5.5 Results from Full Scale Tests and Simulations
6 Summary and Conclusion
6.1 Summary
6.2 Open Questions
6.3 Conclusions
Lebenslauf (Curriculum Vitae)
Danksagung (Acknowledgements)
List of Publications
List of Figures
List of Tables
Glossary
Bibliography
Index
The movement of crowds is a field of research that attracts increasing interest. This is due to three major reasons: pattern formation and self- organization processes that occur in crowd dynamics, the advancement of simulation techniques and hardware that enable fast and realistic simulations, and finally the growing area of potential applications (planning of pedestrian facilities, crowd management, or evacuation analysis). The field is spanning the borders of various disciplines: physiology, psychology, sociology, civil en- gineering, mathematics, physics, etc. It depends on the point of view which aspects are given the main focus. One approach is to reduce complexity to fundamental principles that make a mathematical (quantitative) formulation possible and at the same time are sufficiently complex to reproduce the major phenomena that can be observed in reality.
The major aim of this dissertation is to define and validate a model for the simulation of evacuation processes and their analysis. To this end the analogy between non-equilibrium many particle systems and crowds is used. However, it will also become clear that this analogy is not sufficient for com- plex scenarios and realistic egress simulations and additional, ‘non-physical’, parameters and principles must be introduced. Even though the investiga- tion is motivated by the applications, the dynamics of crowd movement and model properties are scrutinized. This also includes a thorough review of the data available in the literature, the calibration of the model parameters and the comparison of simulated and empirical flow-density relations.
The core of any evacuation simulation is a set of rules or equations for the movement of people. This is connected to the representation of space, popu- lation, and behavior. These topics will be investigated generally (micro- vs. macroscopic, discrete vs. continuous) and especially with regard to a specific two-dimensional cellular automaton model, where the movement dynamics is based on discrete space and time. This allows an efficient implementation and therefore large scale simulations. The route-choice is done via the orientation along a discrete vector field which can in principal be derived from a discrete potential. It is therefore not explicitly simulated but taken into account in a pre-determined way, i.e., the coupling to the vector field is static (constant coupling parameter). In addition to the model characteristics, extensions like competition, multiple and dynamically varying orientation potentials or coupling parameters, or individual egress routes are discussed.
In order to validate the simulation results and the application to full-scale problems, simulations for realistic scenarios are performed and compared to data from evacuation trials. Design variants, aspects of crowd management, or operational measures to optimize evacuation performance are also men- tioned. However, they are the task of experts (architects, psychologists, safety engineers) who might use simulations as a design and evaluation tool. There- fore, these results are rather case studies supplementing the major topics of the model characteristics and implementation.
Die Bewegung von Menschenmengen ist ein Forschungsgebiet das zunehmend an Aufmerksamkeit gewinnt. Dafür gibt es im wesentlichen drei Gründe: die Ausbildung von Mustern und Selbstorganisationsphänomene, der Fortschritt im Bereich der Simulationstechnik, die schnelle und realitätsnahe Simulationen ermöglicht und schließlich die wachsende Zahl möglicher An- wendungen (die Planung von Fußgängeranlagen, die Steuerung von Per- sonenströmen oder die Evakuierungsanlyse). Das Gebiet überspannt ver- schiedene Disziplinen: Physiologie, Psychologie, Soziologie, Ingenieurwis- senschaften, Mathematik, Physik, etc. Es hängt vom Blickpunkt ab, welche Aspekte dabei im Mittelpunkt stehen. Ein Ansatz ist, die Komplexität dadurch zu reduzieren, dass man sie auf grundlegende Prinzipien zurück führt. Dadurch wird eine mathematische (quantitative) Formulierung möglich und gleichzeitig können die wichtigsten Phänomene, die in der Realität beobachtet werden, reproduziert werden.
Das Hauptziel dieser Dissertation ist die Formulierung und Validierung eines Modells für die Simulation und Analyse von Evakuierungsvorgängen. Zu diesem Zweck wird die Analogie zwischen physikalischen Vielteilchen- Systemen und Menschenmengen benutzt. Es wird jedoch auch deutlich wer- den, dass diese Analogie nicht ausreicht, um komplexe Szenarien zu erfassen und realistische Entfluchtungs-Simulationen durchzuführen. Dazu müssen zusätzliche ‘nicht-physikalische’ Parameter eingeführt werden. Auch wenn die Untersuchungen durch die Anwendung motiviert sind, so stehen doch die Dynamik der Bewegung von Menschenmengen und die charakteristischen Modell-Eigenschaften im Mittelpunkt. Das umfasst auch eineÜbersicht der einschlägigen Literatur zur Kalibrierung der Modell-Parameter und den Ver- gleich von Simulationsergebnissen mit empirischen Fluss-Dichte-Relationen.
Den Kern einer Evakuierungssimulation stellt ein Satz von Regeln oder Gleichungen für die Bewegung der Menschen dar. Verbunden damit ist die Art und Weise, wie Raum, Personen und Verhalten repräsentiert wer- den. Diese Punkte werden sowohl allgemein (mikroskopisch gegenüber makroskopisch, diskret gegenüber kontinuierlich) als auch spezifisch mit Bezug zu einem zweidimensionalen Zellularautomaten-Modell untersucht. Im zweiten Fall sind die Bewegungsgesetze in diskreter Zeit formuliert. Dadurch ergibt sich auch eine Diskretisierung des Raumes. Gleichzeitig werden eine effiziente Implementierung und die Simulation großer Personenzahlen ermöglicht. Die Routenwahl wird auf die Orientierung entlang eines diskreten Vektorfeldes zurückgeführt, das grundsätzlich von einem diskreten Potential abgeleitet werden kann. Die Routenwahl ist daher nicht im Modell enthalten sondern wird von außen vorgegeben, d.h. das Vektorfeld ist statisch. Neben den Modelleigenschaften werden auch mögliche Erweiterungen für Konkur- renzverhalten, mehrere oder sich ändernde Orientierungs-Potentiale oder in- dividuelle Fluchtwege diskutiert.
Um die Simulationsergebnisse zu überprüfen (Validierung) und die An- wendbarkeit zu testen, wird die Simulation auch auf umfangreiche Probleme und realistische Szenarien angewandt. Das erlaubt einen Vergleich mit em- pirischen Daten. Entwurfsvarianten, Aspekte der Steuerung von Menschenmengen oder andere Maßnahmen werden ebenfalls erwähnt, wo sich ein Bezug ergibt. Allerdings sind das Aufgaben für Experten (Architekten, Psychologen, Sicherheits-Ingenieure), die solche Simulationen als Werkzeuge zum Entwurf und zur Auswertung benutzen. Daher sind diese Ergebnisse als Fallstudien zu sehen, die die Anwendbarkeit der Simulation belegen.
This chapter gives an overview of the topics dealt with in this dissertation. It describes the intention, scope, and limitations of simulation models for evacuation analysis and roughly outlines the content of the succeeding chapters.
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The field of pedestrian movement has received growing interest over the last decades. This is due to several reasons:
1. Growing mobility: Even if walking is not the most important form of locomotion when the distance covered is concerned it is necessary for every other form of traveling (e.g., walking to the bus, the car, or to the airport terminal) and it is probably the most time-intensive form of mobility if waiting and queuing are taken into account. Simulations can help to increase the level of comfort and decrease waiting times by assessing alternative layouts or procedures.
2. Large facilities, like theme parks and shopping centers, are usually populated by a large number of persons. In densely packed crowds high ‘pressure’ can occur and pose a threat on peoples health [Smith and Dickie, 1993]. This requires de- tailed planning of the “walkways” and crowd management to avoid such dangerous situations.
3. Events like rock concerts or sport matches often attract a huge number of persons. To manage this situation a profound knowledge of the laws of crowd motion is necessary. Scientific research is one tool to gain this knowledge which can be used to ‘channel’ flows (make them more homogeneous), increase capacity by decreasing orientation problems or holding back persons to avoid peak flows in critical areas.
4. In case of an emergency buildings or passenger vessels have to be evacuated within a short time span under stress conditions. Simulations help to improve the building or vessel layout and to optimize the evacuation performance.
5. The requirements to the safety of passenger vessels are increasing. And there is a tendency to increase the number of persons carried in airplanes (Airbus A 380) or on ships (large passenger ships with more than 5000 persons on board are currently planned).
6. Numerous phenomena can be observed in crowd motion: shock waves, oscillation at bottlenecks, lane-formation, etc. Can they be explained by simple rules and assumptions? To identify those basic principles increases the understanding of crowd dynamics.
7. Finally, crowd motion is an important topic in the investigation of group dynamics. And it is connected to several other fields, like social psychology, traffic engineering and safety science. Deepening this connection might lead to fruitful results and new insights on a level beyond the limitations of the single disciplines.
For all these reasons, a thorough investigation of the laws of crowd motion and the influences on it, e.g., the geometrical layout, the environment, or the procedure, especially in case of an evacuation is required.^{1}
As stated before, the potential applications of a theory for crowd movement are quite numerous. At the same time, there are many fundamental questions concerning the application of such a theory, especially when it comes to the assumptions that have to be made and the implementation into a specific model:
- How can cognitive aspects be modeled? Can psychological and social aspects under certain circumstances be represented by fairly easy assumptions?
- What are the differences between continuous and discrete models?
- How do model characteristics influence macroscopic quantities like the flow of per- sons or the egress time from a room?
- How can the model be validated for the application to evacuation simulations?
There are of course many more questions. Some of them will be addressed in the remaining chapters. Others - like the simulation of decision making by using artificial intelligence - are beyond the scope of this thesis, however.
Considering the requirements stated in table 1.1 computer simulations are the tool of choice for investigating crowd movement and especially assessing evacuation processes. As will be argued in the following chapters, they are able to cover the relevant influences in a uniform and comprehensive way, provide useful information about the dynamics and time evolution, and can be build up from intuitive and comprehensible basic assumptions and rules. Of course, the range covered and influences explicitly taken into account in a model for crowd movement have to be restricted. From a practical as well as a theoretical point of view, a model should be as simple as possible. Of course, the question remains, what as simple as possible means, e.g., whether or not cognition must be explicitly
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Table 1.1. Requirements for a theory of crowd motion and its application to the assessment of evacuation processes. These requirements are very general. If interpreted in a broad sense they describe a desired optimal standard. A specific model might nevertheless focus on certain aspects of such a theory and therefore be based on simplifying assumptions.
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modeled. This topic will be discussed in the following chapters and arguments presented for the hypothesis that under certain circumstances (i.e., mainly when route choice is pre-determined) it is justified, to use a ‘physical’ model for crowd motion.
Taking into account the application to real world structures and processes, another requirement is flexibility, especially concerning the application to various phenomena and settings or environments. The potential areas of application are again numerous: buildings, urban systems like pedestrian crossings, shopping malls, theme parks, airports, railway and subway stations, ships, aircraft, buses, trains, etc. Most of them have already been covered by simulations and references to the literature are given in chapter 4.
In summary, the aims of this thesis are the following:
1. Compilation of the basic principles, the aim, and the scope of computer simulations for evacuation processes (remaining part of this chapter);
2. Developing a consistent theory for crowd movement (chapter 2) and its formulation in a form suitable for an algorithmic representation (chapter 3);
3. Implementing this model into a simulation and applying it to realistic egress and evacuation scenarios - for buildings (chapter 4) as well as for ships (chapter 5);
4. Using empirical data (a literature review is given in section 2.5) and experiments (own experiments are presented in section 2.7) to calibrate the model parameters in chapter 3, assess the scope of its application, and validate the simulation results (chapters 4 and 5).
A crowd is a group of interacting individuals and therefore a social system. Crowd movement can be described on different levels according to the cognitive and social processes involved.
1. Physical/Physiological
2. Psychological
3. Social
Behavior is related to the psychological and social level. However, it can also be taken into account implicitly (cf. fig. 1.1). Therefore, a ‘physical’ model is able to cover
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Fig. 1.1. The representation of behavior can range from its neglecting to artificial intelligence
(AI), which aims at even modeling the decision making process [cf. ISO, 1999]. However, the more complex approaches are not completely different from the basic ones but include additional features. Therefore, the autonomy and interaction increase towards the top but are not completely absent and completely present in one case or the other. These qualities are rather a question of interpretation than of direct representation in a set of rules. the essential aspects of crowd movement in an egress situation when route-choice is predetermined. Traditionally, physics has dealt with matter and the rules governing it, e.g., particles and fields (or forces). In order to master the complexity emerging from rather simple principles, many concepts and methods have been developed. Those tools are not only useful to tackle the problems they have originally been developed for but can fruitfully be extended to other areas.
The motion of a crowd can always be modeled ‘physically’ by a description of the trajectories of all individuals. Therefore it can be viewed as a many particle system governed by appropriate rules or ‘forces’. This does not imply that those forces can be represented in the same way as, e.g., gravitational or electro-magnetic forces. However, they share common features with them that allow to employ some of the concepts used there. What remains necessary, of course, is input from empirical psychology about the decisions of people, e.g., whether or not they follow the provided escape routes.
Another aspect that is often mentioned in connection with evacuation is that of panic. However, Sime [1990] argues that the concept is misguiding and that the behavior of people can usually be explained rationally.^{2} It seems to be irrational due to the lack of information and the pressure people face in a dangerous situation. Harbst and Madsen [1996] argue along the same lines. Panic is mentioned usually in the context of masses.^{3} These mass phenomena (like de-individuation) do not necessarily play an important role in evacuation processes, though. To assess the topic more thoroughly, a distinction has to be made between groups and collectives [Brown, 1986, Forsyth, 1999]
Table 1.2. Characteristics of collectives or crowds [from Forsyth, 1999, abridged].
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Fig. 1.2. Classification of crowds: [Forsyth, 1999]. Crowds are large groups that occupy a single location and share a common focus.
(cf. fig 1.2). The distinction between groups, masses and collectives is the following: Any gathering of two or more persons is a group, a large group is called a mass. And if they occupy a single location and share a common focus, they form a crowd (cf. fig. 1.2). The term collective is used more or less as a synonym for crowd. Many collectives spring up spontaneously, exist only briefly, and then fade away as members go their separate ways. Further investigations concerning the topic of crowd behavior in the context of evacuation can be found in [Canter, 1990, Schreckenberg and Sharma, 2002, Smith and Dickie, 1993].
These general considerations can be further specified when restricting the intented scope of a model. The application of the theories discussed and the model presented here are evacuation processes.
When considering evacuation processes, the movement of a crowd becomes simpler and is therefore easier to model. This is mainly due to the fact, that the destinations and goals of the individuals are determined. Furthermore, the egress routes are known. With respect to models for evacuation processes, two main aims can be identified: simulation and optimization. There is an obvious difference between both: Optimization techniques provide a well-defined quantity, e.g. the value of a function, that can be minimized. Thus, one obtains (within the restrictions of the boundary conditions) an optimal solu- tion. This is not the case for a simulation: Here, the situation is modeled and a prediction about the outcome under certain initial conditions is made. What will be presented here is simulation. It might be used - by using some additional measure - for optimization.
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Fig. 1.3. Influences on an evacuation: The rectangles represent the parameters influencing the population. The movement is described basically by the variables of the individual evacuees which form the population [cf. Gwynne et al., 1999]. The several influences can be treated separately and therefore provide a natural division of the model into sub-models.
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Fig. 1.4. A simplified analysis can basically cover the same range as an evacuation exercise. Aggravating circumstances like hazards can be included via extrapolation of the obtained results. A simulation, however, allows to include those influences via the adaption of the parameters, i.e., the extrapolation is made on the input and not on the output.
However, this will result in an iterative process, where not the optimal solution but one that is close enough to the desired output, is aimed at. The reason is, that there is usually no function for the quantity to optimize. It has therefore to be borne in mind that a simulation by itself is not an optimization tool. It is a description of a system covering the relevant dynamical aspects.
The different influences for the simulation of an evacuation or egress are summarized in figure 1.3. In order to improve, one has to state what should be changed, what to optimize, and how an improvement can be reached. The overall evacuation or egress time is one quantity in this context. Others are the length of queues, the densities occurring, etc. Chapter 4 contains further remarks concerning the assessment of evacuation routes (configuration) and procedures (see fig. 1.3). The environment and the population can usually not be changed and are determined by the situation. Therefore, the major aim of evacuation simulations is to quantify the influence of the layout and the procedure and investigate potential improvements. Changes in the population or the environment are not of immediate concern. Some general aspects of emergency planning can be found in a brochure issued by the Health and Safety Executive [1999] of the UK.
To be able to use simulation results for assessment and improvement of layouts and evacuation procedures they must at least comprise the sequence of the evacuation, the overall evacuation time and especially information about bottlenecks and retardation. Second, the whole evacuation process has to be taken into account spatially, e.g., a complete building or vessel, as well as temporarily, i.e., from the alarm or abandon ship signal until the last person has reached a place of safe refuge.
It should have become clear by now that the movement of crowds is influenced by many factors and its description can easily become very complex because of the many interact- ing individuals. Nevertheless, from the ‘physical’ point of view it can be regarded as an interacting many ‘particle’ system. And there are similarities and analogies to processes far from equilibrium. However, the nature of the interaction between the pedestrians and their driving force are not known in detail. And the system is open with respect to the kinetic energy (i.e., persons might stop abruptly) and the number of pedestrians might not be conserved^{4}.
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Fig. 1.5. Schematic view of a mockup used for crowd flow experiments as described by Müller [1999]. The corridor width and the door width were variable and the experiments were compared to others performed using smaller mockups and metal balls representing the persons. The initial density was 6 persons/m[2] and pulsed outflow as well as the formation of arches in front of the door were observed.
It can therefore be viewed as sort of an ‘exotic fluid’ and there are analogies to the flow of granular material, especially at high densities, when the ‘pressure’ from behind keeps the flow going. And in fact experiments using inclined mockups of rooms and sections of buildings (consisting of several rooms connected by corridors) filled with metal balls have been carried out [Müller, 1999]. In order to compare the results to the movement of real crowds another mockup (cf. fig. 1.5) was used. The major conclusion is that there are no differences for the flow characteristics between balls and persons. This justifies a ‘physical’ approach as a starting point for simple geometries and the outflow problem. For complex geometries, however, building down-sized mockups becomes tedious. Therefore, computer simulations are a more appropriate tool. In the following chapters, general properties of crowd movement will be investigated, different modeling approaches compared to each other, and a specific discrete (cellular automaton) model will be analyzed and applied to the simulation of evacuation processes.
This chapter contains basic remarks on how to model pedestrian movement. It therefore deals with the methodology rather than a specific model in detail. The problem setting, as introduced in the previous chapter, is the investigation, description, and prediction of crowd motion and the aspects of evacuation processes related to it. To this end a theory (a set of assumptions and statements) is developed. Different model classes that comply with the theory will be introduced and briefly described. This is the first step providing the basis for empirical studies, model development, and finally the implementation in a simulation.
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Firstly, the task of formulating a theory and subsequently a model for crowd movement and behavior is approached systematically. A correct interpretation of a theory is called a model, where interpretation is understood as the reduction of choices or degrees of free- dom by specifying one of several alternatives without loosing consistency (cf. fig. 2.1). Examples for such reductions are the representation of space as a discrete grid or the exclusion of direct verbal communication between individuals. This is possible, as long
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Fig. 2.1. The interpretation of a theory provides a model which can be implemented in a simulation. In physics this connection between model and theory is usually taken for granted. However, one could also call a set of consistent models a theory. as those aspects comply with the scope and aim of the underlying theory. By imple- menting a model one obtains a simulation program (in short simulation) which is a valid representation of the model and therefore of the subject matter of the theory. In the case of crowd motion, a theory contains assumptions about the reaction of pedestrians to their environment, orientation and route-choice, physiological constraints, etc. Even if those assumptions have been stated, there are different ways of representing them.
As could already be seen from fig. 1.1 analogies to physical systems can provide a starting point for investigating crowd movement. Moreover, they might provide insight into the basic properties of the models. A crowd of pedestrians might be viewed as a driven many particle system with dissipation. It is a system far from equilibrium, though. There are therefore similarities as well as differences between pedestrian motion and non-equilibrium many-particle systems: the concepts used to treat the physical systems can usually not straightforwardly be applied, since the representation of, e.g., the route choice via external fields becomes very tedious and the dissipation (which is typical for non-equilibrium systems) adds further difficulties. However, from a technical point of view continuous models for crowd motion are - with restrictions - similar to molecular dynamics (MD). MD simulations are based on the numerical solution of the Newtonian equations for many interacting particles. And the social force model, which is briefly described in section 2.3.1 employs exactly this correspondence.
In general, models can be characterized according to scale, resolution, and fidelity [Nagel, 1996]. A high fidelity model is one with many parameters that directly takes into account all the different influences (e.g., parameters like age, height, weight, mobility impairment, etc. in the case of pedestrians), resolution is the level of detail regarding the representation of space, and finally, scale is the size of the problem with respect to time, space, etc. The scale of a pedestrian simulation depends on the application of course. For football stadia or theme parks, a model that ‘scales’ linearly (with respect to computation time and memory requirement) with the number of persons or the size of the layout is desirable. To some extent high resolution low fidelity simulations can do as well as low resolution high fidelity ones, i.e., resolution can make up for fidelity [Nagel, 1996].
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Fig. 2.2. Modeling criteria that can be used for classifying different theories and models [from Gershenfeld, 1999]. The major choices for models and simulations of crowd movement are discrete vs. continuous and stochastic vs. deterministic.
Additional modeling criteria are illustrated in fig. 2.2. A high resolution (microscopic) model is usually also based on first principles and intented to be of general use. It can then be treated only numerically, though. Furthermore, in order to be able to make predictions and assess evacuation processes, quantitative results are required. Finally, since there will remain influences on human behavior that cannot be completely quantified, the outcome of an evacuation as well as crowd movement in general are to some extent uncertain. This is reflected by a stochastic model. Whether a discrete or continuous model is more appropriate cannot be directly answered from a theoretical point of view. The differences between discrete and continuous become important for simulations of real world problems (i.e., complete buildings or ships), especially concerning the scalability (which is not a model characteristic but one of the implementation).
With respect to resolution, models for pedestrian motion can be classified into two major categories: microscopic and macroscopic. Macroscopic models cannot represent a general theory of crowd motion and are restricted to specific applications. Their advantage is that they usually can be treated analytically. An example for a macroscopic model is given by Pauls [1995].
Examples for different microscopic models are given in table 2.1. Microscopic models (for pedestrian motion) can be roughly defined according to the following criteria: They are based on
- a detailed representation of space,
- the representation of individual persons,
- a uniform movement algorithm, and
- the consideration of personal abilities and characteristics.
The third and fourth criterion follow from the first and second, which are the proper characteristics. The connection between the geometry and the population for microscopic models is made via the rules (or equations) of motion (see fig. 2.3). This is different from macroscopic, hydrodynamic, or regression models, where this connection is made via a parameter in the respective flow equation.
According to the criteria just introduced (cf. figure 2.2) the model that will be presented in chapter 3 is microscopic, based on first principles, numerical, stochastic, and quantitative.
The movement of pedestrians can be represented by their trajectories. The number of pedestrians be N. If the coordinates of the N pedestrians are given by vectors r i ∈ ID[2] ′ (ID = IR for spatially continuous and ID = IN for discrete models) the new positions r i are given by r i + v i, where v ∈ ID[2] is the velocity at time t and Δ t denotes the time-step in the discrete case:
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This means that the discrete space is represented by a two-dimensional lattice and the lattice sites can be identified by two numbers. Although this notation is intuitive for a square lattice, the lattice type is not restricted and could also be hexagonal or triangular.
The problem of determining the velocities can be subdivided into three steps: Route choice, orientation, and interaction. Route choice requires the autonomy to set strategic goals. Modeling this decision making process from first principles is outside the scope of the approach presented here. Rather, the routes are assumed to be pre-determined and therefore implicitly contained in the rules or equations of movement (cf. fig. 1.1). Then, the route choice can be represented by a vector field V ∈ ID[2] and v i = V (r i). This leads to the analogy to physical systems: orientation is the coupling to a vector field.
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Fig. 2.3. The connection between geometry and population is made via rules of motion, i.e., they determine the movement with respect to the layout.
With respect to the representation of space in a microscopic model, there are two basic approaches (see fig. 2.2): discrete or continuous. If space is represented in a discrete fashion, usually a regular lattice is used. This notion is quite familiar in statistical mechanics and is often used for modeling systems with restricted degrees of freedom such as solids (lattice gas, Ising-model, percolation models). In reality, the degrees of freedom for pedestrian movement are not restricted in this way, though. Whether to use a discrete or continuous representation of space is closely connected to the implementation (similar to models for road traffic). A strong argument in favor of discrete models is that they are simple and can be used for large scale simulations. Additionally, for pedestrian motion and behavior there is a finite reaction time, which introduces a time scale. If the
Table 2.1. Examples for microscopic models, classified with respect to the dynamics of pedestrian motion. Δ t is the time step, a is the length of a quadratic cell, hence ρ max = 1 /a [2] for a square grid. If the cell length a is not specified, ρ max cannot be compared to empirical data (denoted by n.a. for not applicable in the third column).
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time is chosen to be discrete in the model, too, this naturally (but not necessarily) leads to a discrete representation of space. Firstly, the continuous approach will be outlined, and then the main aspects concerning the discrete or grid-based models are introduced. One special class of the discrete models are the so called Cellular Automata (CA) models. Their properties will be discussed in depth (together with a specific model for pedestrian dynamics in egress simulations) in chapter 3.
Even though the major topic of this work are two-dimensional CA models, reference will be made at various places to continuous models for pedestrian dynamics. Due to this and the importance of continuous models as the alternative approach towards representing space, this section describes the social force model and some of its properties in some detail [Helbing et al., 2002, Helbing and Molnar, 1995]. Further models belonging to this class are those of Hoogendoorn [2000], Hoogendoorn and Bovy [2001], Hoogendoorn et al. [2002] and (for the case of evacuation simulation) Thompson et al. [1996]. The former provides a generalization of the social force model where the way finding is based on an extremal principle, whereas the latter is a full scale implementation covering also complex geometries like floor-plans of large office buildings or passenger vessels.
The social force model is based on continuous space and time:
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where x i denotes the position and v i the velocity of pedestrian i. The pedestrians are represented as disks with radii r i. The sum of forces pedestrian i is subject to is called f i (t), m i is the mass of pedestrian i, and ξ i (t) are individual fluctuations. The equation of motion is then given by:
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Then the task is to determin f i (t) and ξ i (t). The resulting system of partial differential equations can be solved numerically (e.g, by applying methods of Molecular Dynamics
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Simulations^{1} ). The force terms are [Helbing et al., 2002]:
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where the first term describes the adaption to the desired velocity and direction (which is the internal driving force) within a relaxation time, the first sum repulsive social forces, physical forces (‘body force’ and sliding friction), and attractive forces between pedestrians, the second sum the repulsion from the boundary, and finally, the last term attraction to landmarks. If the term containing the desired velocity v [0] i (t) e i [0](t)was absent, then the movement would be accelerated. Therefore, the first term represents a dissipative force. The relaxation time τ can be compared with the reaction time Δ t in a discrete model (eq. 2.2). The social force between individuals as well as the repulsion from walls is assumed to decrease exponentially with the distance and the ‘physical’ force ensures that persons do not penetrate each other (an equivalent term is present in the wall or boundary ter f ib):
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Θ(x) is x for x ≥ 0, zero otherwise (Heaviside-function), i.e., the corresponding terms are only relevant, if the pedestrians touch each other (d ij < r ij). n ij is the normal vector pointing from i to j, t ij the tangential vector perpendicular to it, and Δ v t ij =(v j − v i) · t ij the tangential component of the velocity difference. The distance between pedestrian i and j (i.e., the centers of the disks) is denoted d ij.
The parameters are the interaction strength A i, its range B i, and r ij = r i + r j is the sum of the radii. d ij (t) = ∥ x i (t) − x j (t) ∥ is the distance between the centers of i and j. For so called ‘panic’ situations the fluctuations are set to
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where n i with 0 ≤ n i ≤ 1 is the nervousness of pedestrian i, ξ 0 the normal and ξ max the maximum fluctuation strength. Additionally, the social interaction f soc ij isreducedtoa hard-core potential (i.e., κ ≫ A i , B i) and for many purposes the attraction strength C ij is set to zero. In normal situations, there might be no fluctuations. The numerical values for the parameters in different situations can be found in [Helbing et al., 2000].
In order to be able to assess the influence of the different contributions, in the following typical values are assigned to the parameters: τ i = 0 . 5 s, A i = 2 · 10[3] N, B i = 0 . 08 m, v [0] = 1 . 0 ms − [1] for normal situations, k = 1 . 2 · 10[5] kg s − [2], and κ = 2 . 4 · 10[5] kg m − [1]s − [1]. i Additionally, the masses are m i = 80 kg and the radii of the pedestrians are set to 2 r i ∈ [0 . 5 m , 0 . 7 m] in order to avoid exactly symmetrical configurations. For distances (d ij − r ij) larger than 5 B and v ≈ 1m/s, the first term in eq. 2.5 is about 160 N, whereas the sum of the remaining terms is about 13 . 5 N. This might be used to simplify the implementation by neglecting interaction ranges > 5 B = 0 . 4 m.
Lane formation can be observed in this model even for isotropic interaction forces, oscillation at bottlenecks and clogging can be simulated. One problem that is harder to tackle in this model than in discrete models are complex geometries. Since there are N [2] interaction terms, one either has to evaluate them explicitly or to check, whether some of them can be neglected. For complicated structures (like shopping centers or large passenger ships), this is a major challenge. The same holds for the interactions of the pedestrians with the walls, i.e., assuring that walls are not penetrated.^{2} Similarly, it has to be checked whether the forces are screened by walls, i.e., two pedestrians do not repel (or attract, if A i is negative) each other if there is a wall between them.
Especially for the reasons stated above, discrete models are appealing for simulations of large complex structures. Another factor is simulation speed, since cellular automata are per construction well suited for efficient implementation. The Nagel-Schreckenberg model [Nagel and Schreckenberg, 1992] is a very well understood model for the simulation of road traffic and can therefore provide insights into some aspects of models for crowd movement. Due to its simplicity, it provides a good starting point for relating fundamental properties of the model to its characteristics.^{3} For the sake of completeness, the definition of the Nagel-Schreckenberg (NaSch) model is included here. The rule set (parallel update) for t → t + Δ t is:
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Please note that i denotes cars not cells. Lengths and velocities are measured in units of the cell size a and a/ Δ t, respectively, with Δ t being the time-step. Parallel update means that all cars move synchronously. Therefore, each step (1, 2, and 3) is carried out for all cars first before going to the next step, i.e., the velocities are determined for all cars, before the cars move. This can be implemented (and is equivalent to) a sequential update (of cars) in the direction of movement.^{4} This leads effectively to a car blocking all the cells of its trajectory within the update step Δ t.^{5}
The parameters in this case are the cell size a, the maximum velocity v max, and the deceleration probability (braking noise) p dec. v i denotes the actual velocity of car i and g i the distance to its predecessor in cells (see fig. 2.4). If Δ t is set to 1 s, the maximal velocity v max = 5 corresponds to 37 . 5 m / s = 135 km / h and the acceleration is 7 . 5 m / s[2].
16 Modeling Pedestrian and Crowd Dynamics - Methodology
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Fig. 2.4. Definitions and notations in the Nagel-Schreckenberg model. The numbers in the cells give the current velocity, g i denotes the gap between car i and its predecessor. The cell size in the standard model is 7 . 5 m which leads for Δ t = 1 s to a maximum velocity of 135 km / h and an acceleration of 7 . 5 m / s[2] .
The results obtained for the NaSch model concerning the influence of v max and fundamental flow-density-relations are important for understanding the generalization to two dimensions, where similar decisions concerning the cell size, the type of the update, and v max have to be made. Exact results can be obtained for the case v max = 1, where the NaSch model is equivalent to the asymmetric simple exclusion process [Rajewsky et al., 1998]. In this case, the backward sequential update (against the direction of motion) produces the highest flow^{6}, which is given by
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p is the hopping probability^{7} and J grows with p. If p is set to 1, then J = ρ, i.e., all cars always move. The density for which the flow takes its maximum is shifted to the right when p is increased. This is different for the parallel update:
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In this case, the maximum of J is always at ρ = 1 / 2.^{8} Those results will be useful when comparing the different updates for the 2D model in the next chapter. The limit of high acceleration in the NaSch-model is similar to the case of pedestrians: the acceleration when walking is instant. Rule 1 is then replaced by v t i = v max. In the case of v max = 1 this is of course trivial.
For road traffic, the distance between cars is determined by the time gap, i.e., g i ~ v i. This is necessary, since deceleration from v max to 0 takes time, i.e., cars must obey a ‘safety distance’ proportional to the speed. Therefore, the trajectory of a car is effectively blocked, i.e., cannot be accessed by another car. This is automatically taken into account by the parallel update. This behavior is not the case for pedestrians: Stopping is possible instantly and therefore a pedestrian does not necessarily have to keep a distance to his predecessor. Whether this requires to introduce a different type of update for the 2D case will be discussed in the next section.
Therefore, the update type deserves special consideration. As before, the simpler 1D case provides a good starting point. Figure 3.17 shows a comparison of the parallel update and a so called shuffled update for the NaSch model. In the shuffled update, the sequence of the cars moving is random. However, each car is allowed to move only once
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Fig. 2.5. Particle-hole symmetry holds for v max = 1 but does not for v max > 1. Left: the dynamics remains the same, if holes move instead of particles in the opposite direction but according to the same rules. Right: For v max > 1 this does not hold, since the one but leftmost hole would have to move together with the leftmost one, which is not allowed for a parallel update. The particle-hole symmetry leads to a symmetry in the fundamental diagram as can be seen from eq. 2.15 and is shown in fig. 3.17. in a time-step. Since in this case the particle hole symmetry is broken, the fundamental diagram - other than for the parallel update - is not symmetric with respect to ρ = 1 / 2 for v max = 1. Particle hole symmetry means that switching particles with holes and changing the direction of motion without changing the rules otherwise does not change the time evolution of the system (see fig. 2.5). The importance of the symmetry lies in the fact that it enforces the fundamental diagram (flow vs. density relation) to be symmetric around ρ = 1 / 2. This can be seen from the following equations:
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j ↑ denotes the flow in the direction of motion (of the particles) and j ↓ against the direction of motion (for the parallel update). For ρ particles = 1 / 2+ x, ρ holes = 1 / 2 − x, and therefore
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Please note that particle hole symmetry does never hold for an interaction range larger than 1 cell, i.e., neither for v max > 1 (cf. fig. 2.5) nor for particle sizes A larger than the cell size a as shown in fig. 2.6. This can be summarized:
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This statement is not contradicted by the choice v max = v min = A = n, with n arbitrary, e.g., n = 3, since this can easily be rescaled to v max = A = 1.
The NaSch model is covering in its basic form only single-lane traffic either with periodic or open boundary conditions. This is not sufficient for simulating real traffic scenarios. The model can be improved or extended in various ways:
1. multiple lanes [Nagel et al., 1998],
2. smaller cells [Knospe et al., 2000],
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Fig. 2.6. The particle hole symmetry is broken in the 1D case (i.e., NaSch model) for interaction ranges larger than a (A > a). The arrows above the grid indicate the motion of the particles, the ones below the grid the motion of the holes (see also fig. 2.5). Since the movement of one particle corresponds to the movement of three holes, there is no symmetry between particles and holes.
3. anticipation of preceeding car’s movement, and braking lights [Knospe, 2002],
4. on- and off-ramps [Diedrich et al., 2000],
5. sinks and sources (varying number of cars) [Kaumann et al., 2000], and
6. different sizes for cars and trucks.
Decreasing the cell size makes it possible to distinguish between trucks and cars (adapting the speed limits v max accordingly) and to accelerate more smoothly. The distinction between anticipation and braking lights is basically that the former takes into account the velocity of and the gap to the preceeding car, the latter its deceleration.
The next chapter deals in detail with a specific CA model for pedestrian motion. Here, an overview over some related models is given. Starting from the 1D model, a generalization to 2D models that might be applicable for pedestrian movement is via forming a corridor as a multi-lane structure. This makes it necessary to define lane- changing rules. An overview for road traffic can be found in [Chowdhury et al., 1997, Nagel et al., 1998, Rickert et al., 1996]. However, in this case, there is only one direction of movement. A first step towards generalization is the introduction of two possible walking directions. Blue and Adler [1999] have proposed such a model, which they call bi-directional. This was then extended to a four-directional (in the sense of possible walking directions) one [Blue and Adler, 2000] which simulates pedestrian crossings. The idea is highlighted in fig. 2.7. Since there is hard core exclusion, deadlock situations might occur, in which blocks are formed and not resolved. Therefore, a switching process was introduced: two opponents might change their positions if they occupy cells next to each other and have opposite walking direction.
However, this approach is limited to geometries, where the walking direction does not change. Therefore, it is not possible to simulate situations where movement is not from left to right but, e.g., towards an exit. To do this, the walking direction has to be determined taking into account external information like signage. Just recently Burstedde et al. [2001] have proposed a 2D CA model with different kinds of floor fields: a static (S) and a dynamic (D) one. The static floor field is a scalar field representing the distance to either the exit or the destination cells measured by a Manhattan metric, i.e., the number of steps across edges between this cell and the exit.^{9}
A more extensive description of this model can be found in [Burstedde, 2001]. One of the interesting results of this approach is its ability to reproduce lane formation without
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Fig. 2.7. The preferred walking directions can be expressed via a 3 × 3 matrix. The entries might be interpreted as probabilities, such that i =[1] ,j =[1] i = − 1 ,j = − 1 M i j =[1].Foruni-directionalflow (road traffic) M 0 , 1 = 1 and all particles move in the same direction. For bi-directional flow, there are two different species with uni-directional matrices M[1] and M[2] and for four directional flow, four different ones. explicitly taking into account the interaction between persons. Therefore, lane formation can be found also in (spatially) discrete models. The static field S enables route choice, whereas D is modified by the pedestrians and introduces long range interactions.
The transition probability is obtained by combining those different influences:
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k s and k d are parameters determining the interaction of the pedestrians with the fields∑ j =1 S and D, N is a normalization constant to ensure that [illustration not visible in this excerpt] not denote cells but neighborhood relations (cf. fig. [2].[7]), i.e., i, j ∈ { − 1 , 0 , 1 } and (00) is the current cell. The occupation number n ij is 0, if the cell j is empty, 1 otherwise, and n 00 ≡ 0. Therefore, n ij can be used to represent walls and other obstacles. This means that the transition probabilities are determined by considering only accessible cells. Otherwise it would be necessary to set S j = −∞ for wall cells in order to have the same effect in eq. 2.17 with n ij representing only the occupation by pedestrians. The update is done synchronously (parallel update), i.e., all pedestrians move at the same time. For k d = 0 (no interaction, except the hard core exclusion) the coupling constant k s can be compared to the probability P (v x ≤ 0) which is in a sense the equivalent of p dec in the NaSch-model.^{10}. For four possible destination cells this is given by:
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and the transition probabilities for movement along a corridor can be written in matrix
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Table 2.2. Different influences and their representation
Different influences and their representation in the agent framework. Agents are representing a person in a simulation model and their abilities are represented on different levels.
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This illustrates that the connection made in eq. 2.20 between k s and a deceleration probability p dec like the one used in the NaSch model is rather an estimate for interpreting k s, since k s does not only determine the probability for deceleration but also for stepping to the side. Furthermore there are negative velocities. Nevertheless, eq. 2.20 allows to connect it with the probability P (v max ≤ 0). This is different from the model investigated in the next chapter, where there are two separate probabilities p dec and p sway for stopping and stepping to the side.
Similar to the case of space, there are two basic possibilities for representing the population (cf. fig. 2.2): on the one hand macroscopic models, utilizing aggregated variables and describing the flow of persons as a hydrodynamic system. On the other hand microscopic models, describing the individual movement and behavior (comparable to thermodynamics vs. statistical mechanics). Usually, either space and population are microscopic (detailed geometry and individual persons) or neither.
A general framework for representing a population of individuals are multi-agent- systems. The concept has recently been used to visualize pedestrian activity [Dijkstra et al., 2000] and to investigate the behavior of road users [Wahle, 2002]. An overview over recent developments in the field can be found in [Moss, 2001].
The abilities of an agent can be divided into three different levels:
1. Skill-based - operational level (automatic reaction),
2. Rule-based - tactical level (stereotypic reaction)
3. Knowledge-based - strategic level (cognition, problem-solving, decision making).
Classical Cellular Automata (a definition is given in section 3.1) can be interpreted as multi-agent systems where only the tactical and operational level are present. Examples for tasks and the level they correspond to are shown in table 2.2.
Concerning the population one of the major questions is whether social influences can be quantified and reproduced by a simulation. For one aspect of social behavior, namely
competition vs. cooperation, it will be shown in section 3.7.2 that this is possible. Ex- perimental results [Muir, 1996] show that the motivation level has a significant influence on the egress time from a narrow body aircraft (the empirical results are also described in section 3.7.2).
The question arises, how the motivation level can be modeled. Two approaches, with the second being a generalization of the first, can be utilized to cover the influence of the different behavior:
1. Competition is represented via friction in combination with a more assertive be- havior (increased walking speeds).
2. There is a pay-off for winning the competition and a penalty for loosing it.
The first approach does penalize everyone for competition. However, since the be- havior is more assertive, it might still lead to a more efficient egress (i.e., smaller time). Therefore, there are two opposite contributions: the gain for being more assertive and the loss due to the competition. The latter depends on the exit width, since the number of conflicts decreases with the exit width. The corresponding simulation results are shown in section 3.7.2.
The second approach is not based on increasing the assertiveness of the whole pop- ulation. However, it can not as easily be incorporated in a simple model without in- troducing further model features. On the other hand, in such an extension, individuals could increase their pay-off (i.e., decrease their egress time) by behaving more clever. The ‘currency’ used in this context could be speed, for example. This concept is not further investigated here, since it would lead to a level of complexity beyond the scope of a first principles model.
There are of course further social influences that are important in the case of an emergency. The formation of groups is one prominent example [Kugihara, 2002]. However, these rather complex social processes are (at least for the time being) hard to express in mathematical terms and therefore beyond the scope of this work.
This section gives an overview over the empirical and experimental data available in the literature. A major distinction can be made between empirical observations and experimental investigations. The former contain observations of daily walking patterns or crowd behavior, whereas the latter aim at a controlled laboratory environment, where the influence of one controlled variable on preferably one other variable is investigated. The second criterion for classification is into normal or emergency situations. The different sources for data about crowd motion are illustrated in fig. 2.8.
The connection between reality (observations and experiments) and a microscopic theory is threefold:
1. Calibration (Parameter Settings) Rule-set and functions, e.g., the maximum individual walking speed as a function of the age, etc. (v i max = f (age , ...));
2. Validation of the model Fundamental relations like the one between flow and density (j (ρ));
3. Validation of simulation results Full-scale tests, e.g., evacuation drills.
In general the description of the experimental setting contains population, geometrical layout, hazards, stress factors, information available to the participants, and the sequence of the events. For empirical data the description of those factors is often less detailed. The main sources for empirical data are summarized in table 2.3.
Table 2.3. Summary of the empirical data found in the literature. The results are described in more detail in the text. FWHM is short for Full width at half maximum (θ). If no explicit formula or type of distribution is given θ is used to characterize the width of the distribution (for the probability density it holds f (μ ± θ) = 1 / 2 f (μ)). Additional reviews for the walking speed on stairs can be found in [Frantzich, 1996] and for the flow on stairs and surface level in [Graat et al., 1999].
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Fig. 2.8. Empirical data (including experiments) can be roughly classified according to con- trolled/uncontrolled and emergency/normal situations. Of course, there are other important criteria, like validity, reliability and objectivity. This becomes especially important for the un- controlled situations, where those criteria are harder to ensure due to the lack of an operational definition of the situation.
The empirical observations on pedestrian movement patterns in non emergency situations can be classified into four major categories:
- Oscillations at bottlenecks, resp. pulsed outflow at doors [Helbing, 2001, Müller, 1999],
- Lane formation [Yamori, 2001],
- Round-about traffic [AlGadhi et al., 2002] and the formation of stable walking patterns,
- Jam waves for high densities around 8 P/m[2] [Pauls, 1995] and density fluctuations.
It is difficult to perform experiments on crowd motion, which is mainly due to the fact that reliability requires a statistical analysis and therefore a high number of repetitions. Either one has to restrict oneself to small numbers of participants, which then is rather an experiment on single pedestrians and physiology. Even in this case, the effort can be immense [Bles et al., 2002]. Or one has to accept the fact that the situation is not controlled (in the sense of varying parameters or control variables and observing the controlled variables) and the evaluation is restricted to observation.
Frequency Distribution
Walking speed, like any other physiological quantity, can best be described as a statistical distribution. Whether an analytical expression is used to fit the data is mainly a ques- tion of practicality, as long as there is no theoretical foundation in favor of it. Weidmann [1992] has evaluated about 150 references. His report contains information about the de- pendency of walking speed on physiology, height of the persons, space requirement, level of service, etc. The results are usually obtained by averaging over all the different values given in the literature. Fundamental diagrams j (ρ) (ρ is the density and j the specific flow) are given for movement on surface level and stairs. By summing up the different distributions found in the literature, a walking speed of 1 . 34m / s ± 0 . 26m / s (mean value ± standard deviation) for flat terrain results. This gives the frequencies of the walking speed in an average population. They are assumed to be normally distributed.
Henderson [1971] derives distribution types for pedestrians from Maxwell Boltzmann theory. For the speeds in the case of no directed movement (v = |v|) the probability density is given by:
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where v ′ is the velocity for which f has its maximum.^{11} The parameter v ′ was obtained for children on a playground to be 0 . 67 m/s (walk mode) and 1 . 9 m/s (run mode). In this case, there is no directed movement, i.e., 〈 v x 〉 = 〈 v y 〉 = 0.
In the case of directed movement the distribution for the velocities (x-component) is
Gaussian: [ ]
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with μ = 〈 v x 〉 being the velocity of the flow. The parameters obtained for students walking on the campus are μ = 1 . 44 m/s and σ = 0 . 228 m/s. For a zebra crossing, the values are μ = 1 . 53 m/s and σ = 0 . 201 m/s. Other authors have suggested skewed Gaussian distributions [Werenskiold, 1998]. However, the analytical form is not specified and therefore ambiguities remain. In addition to the asymmetry, the probability density would have to be known to compare this suggestion with the previous formulae.
A further study we performed to check the assumptions concerning the distribution of walking speeds is presented in section 2.6 below.
Flow Density Relation
A second important fundamental relation is the one between the density and the flow. The shape of the fundamental diagram given by Weidmann [1992] for uni-directional pedestrian movement on walkways is shown in fig. 2.9. The analytical expression for the specific flow obtained via fitting to empirical data is given by:
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where the fit-parameter γ is calculated to be 1 . 913. Finally from eqs. 2.24 and 2.23 a flow-speed relation can be obtained
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Additional flow density relations can be found in [Pauls, 1995, Transportation Re- search Board, 1994]. Predtetschenski and Milinski [1971] carried out extensive experi- ments on flow of persons for different geometries. The so called ‘macroscopic’ models
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Fig. 2.9. Flow density relation for pedestrian movement. The analytical expression is shown at the top of the figure. The curve is a fit to empirical data [Weidmann, 1992].
employ those fundamental properties - mainly the relation of density to flow and walking speed - to calculate egress times.
Empirical observations on the influence of the width (e.g., of a corridor) on the specific flow could not be found in the literature. The specific flow j spec = j/W = ρ· v is obtained from the overall flow by dividing it by the width W (j = ρ · v · W). This assumes that the specific flow does not depend on the width.
Experiments carried out in Sweden [Frantzich, 1996] addressed the walking speed on stairs up and down. Especially the case of spiral staircases was examined, which had not been done to a larger extent before. Whereas in the earlier studies of Predtetschenski and Milinski [1971] and Fruin [1971] the main results were flow-density relations obtained from observations, in this case flow-distance relations were measured under controlled laboratory conditions. The participants were students of age 20-30. The distance to the person ahead (gap) can be transformed into a local density via
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where d gap includes the ‘size’ of a pedestrian in walking direction, i.e., for a distance of around 0 . 25 m (the minimal distance that occurred during the experiments) there was body contact. This gap of 0 . 25 m corresponds (if the minimal stair width is assumed to be 60 cm) to a maximum density of ρ maxlocal =[6] . [7]m − [2].
The walking speed did not show a dependence on the inter-person distance for dis- tances in the range between 0.5 and 2 . 5 m for a narrow stair and walking downstairs (0 . 72 ± 0 . 29 m/s). Basically the same result was obtained for wider stairs (0 . 69 ± 0 . 15 m/s)
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Fig. 2.10. Flow density relation for pedestrian movement on stairs. The dependence on the slope is most prominent for low densities, where steeper stairs perform worst. The experiments have been carried out in a Dutch football stadium. Additionally, the influence of the motivational level on the flow has been investigated [Graat et al., 1999]. and up stair direction (0 . 51 ± 0 . 10 m/s for a narrow and 0 . 56 ± 0 . 14 m/s for a wider stair). The minimum interpersonal distance was measured to be around 0 . 25 m for both upstairs and downstairs. This range of distances can be transformed into densities between 0.4 and 2 m − [2] via eq. 2.26. Therefore, the results are in a sense contradictory to the as- sumption of a flow density relation for stairs similar to the one in fig. 2.9 and the speed density relation (horizontal component of v) for movement on stairs [Weidmann, 1992]:
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with c = 0 . 61 and γ = − 3 . 7 for up and c = 0 . 69 and γ = − 3 . 8 for downstairs.
However, the movement on stairs could be fundamentally different from plain areas. Especially sort of ‘synchronization’ effects can be observed in railway and subway stations when dense crowds move up or down to a platform with considerable walking speed. The separation of the area into steps might support such an effect.
Furthermore, the report of Frantzich [1996] contains useful remarks about the video analyzing technique (using the software package Persias) as well as the experimental setup.
Movement through doors has mainly been investigated with respect to the capacity of the door, i.e., based on the concept of a specific flow. It is for example common, to assume a maximum specific flow in egress calculations of 1 . 33 P/ms [Health and Safety Executive, 1996]. The underlying assumption is a smooth functional relation between the width and the flow, i.e., there are no special widths where there is a jump in the capacity of the door. However, this assumption is limited to a certain range of widths as can be seen from the fact that a bottleneck smaller than the body size can no longer be passed.^{12}
The flow of persons with respect to the stair angle of the tribune in a football stadium was investigated by [Graat et al., 1999]. Figure 2.10 shows that the specific flow increases more strongly with the density for steep stairs (38 ◦) than for those with a normal angle (30 ◦).
Even though most of these results will not be implemented in the simulations (cf. chap- ter 4) it is important to know the different influences and to be able to estimate the error that results when they are neglected. A straightforward approach to include these spe- cial aspects could be via multiplying the walking speeds with an appropriate reduction factor. This will be done for the maximum individual walking speed v i max on stairs (cf. section 4.1.1). This allows to use one parameter for the walking speed. Otherwise, a separate parameter for walking speed on stairs would have to be introduced.
Evacuation exercises have been carried out for different vessels and buildings:
- Aircraft [Jungermann and Göhlert, 2000, Muir, 1996, Owen et al., 1998],
- Land based passenger vessels: trains [Galea and Galparsoro, 1994],
- Passenger Ships [Harbst and Madsen, 1996, Marine Safety Agency, 1997, Wood, 1997],
- Residential, Office, and Public Buildings [Proulx, 1995, Weckman et al., 1999].
The results vary greatly depending on the occupants and the type of the building. One major influence is whether the occupants are familiar with the surrounding or not, i.e., office and residential buildings on the one and public buildings on the other hand. It has been reported for nursing homes and residential buildings that it took up to 30 minutes for some occupants to respond to the alarm and they basically had to be forced by fire fighters to leave the building [Proulx, 1995]. Such a case is not within the scope of a simulation. One could, however, adapt the reaction time distribution for the simulation accordingly, if the necessary data is provided (cf. eq. 4.1).
A long response time is connected to the decision making process. Figure 2.11 shows different possible strategies in an emergency situation. Since the scope of this work is not the decision making but the movement dynamics, it is put into the model as an assumption that the persons are able and decide to egress.
Another topic in this context is the one of panic. Canter [1990] basically discards it and argues that behavior that seems to be strange from the outside is understandable by the restricted amount of information available to those who are actually in an emergency situation. Also Proulx [1995] did not find any hints for flight panic in her investigations. It was rather observed that people become lethargic when facing immense danger. Therefore, the major consequences for representing these special scenarios are higher fluctuations for the parameter values (like walking speed or reaction time) seem to be higher under those unusual and unfamiliar circumstances.
An overview over accidents for buildings (including football stadiums and the like) can be found in [Helbing et al., 2002]. Remarks on safety management for football stadiums and large events in general are contained in [Health and Safety Executive, 1996, 1999].
Route Choice in an Evacuation Exercise
Abe [1986] has investigated the route choice behavior in a mimicked emergency situation. The fire alarm was triggered and artificial smoke occurred. People were asked what the 28 Modeling Pedestrian and Crowd Dynamics - Methodology decreasing mobility
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Fig. 2.11. Egress vs refuge in building evacuation: The more complex a building is and the less the mobility the more difficult is the egress from a building. This leads to a distinction between four different strategies: egress, slow egress, move to refuge, and defend in place [Abrahams, 1994].
major motivation for choosing the exit in a large supermarket was. The answers were given with the following frequencies:
1. Following exit signs, announcements via the public address system, or from the staff (53.3%).
2. Choosing the nearest exit (24.7%).
3. Escaping from fire and smoke, taking the direction away from it (12%).
4. Following other persons (6.7%).
5. Using the familiar door (1.7%).
6. A window near to the door, it’s bright there (1.0%).
7. The door wasn’t crowded (0.7%).
8. Others.
It is remarkable that the familiarity with the exit did not play a very important role concerning the exit choice. A similar observation has been made in an evacuation exercise we carried out in a movie theater that will be presented together with a comparison to simulation results in section 4.3.2.
In 2000 the world exhibition (Expo) took place in Hannover. At this event we observed pedestrian movement at different scenes. The most useful one was a pedestrian bridge,
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Fig. 2.12. Shown is the walking speed distribution for the pedestrian bridge at the World Exhibition (Expo) 2000 in Hannover. The length that was walked by the pedestrians was 15.5m. A Gaussian distribution is fitted to the results with μ and σ obtained either from the sample or using adapted values where the medians of the fitted and empirical curves are closer to each other. The data is shown in the table below the horizontal axis. Groups are represented by one data point (cf. table 2.4). where the frequency distribution of the walking speeds and its relation to the group size was measured. The measurement of the walking speed on the pedestrian bridge comprised 700 persons. There was no slope on the bridge. The measurement area was a square of 7 × 7 m[2]. The width of the bridge was 14 m altogether, i.e., twice the length of this square. The length of the bridge was much larger than its width. The speed of the pedestrians was obtained by v i = l/ Δ t i, where Δ t i is the time for walking from one edge of the square to the other in the longitudinal direction (across the bridge). Due to the low densities there was no obstruction by other pedestrians, i.e., the walking speeds were that of free flow.
A normal distribution has been fit to the data using the mean and standard deviation of the empirical distribution as well as adapted values. The data and the fitted curves are shown in fig. 2.12. The mean value obtained was μ = 〈 v x 〉 = 1 . 30 m/s and the standard deviation σ = 0 . 21 m/s. The third moment of the distribution E (x [3] − x [3] ) is 0 . 41 (m/s)[3]. This shows that the distribution is not symmetric but slightly skewed towards the origin. The parameters for the second fit-curve shown in fig. 2.12 are μ =
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A second aspect of this investigation is the dependence of the walking speed on the group size. Several persons were identified as a group if the distance between at least two of them was not larger than about 1 m, they walked at the same speed, and in the same ‘formation’, i.e., they actually formed a social group.
Table 2.4 shows the decrease of the walking speed with increasing group size. It is interesting to note that groups larger than 6 persons were not observed. Of course the
30 Modeling Pedestrian and Crowd Dynamics - Methodology
Table 2.4. Walking speed vs. group size for the pedestrian bridge. The speed is obtained by dividing the distance of 7 m by the travel time, i.e., 〈 v x 〉, if the ‘direction’ of the bridge is denoted x.
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statistics for the larger groups are less reliable since they rarely occurred.
Nevertheless, this information could be useful when integrating the influence of group size into the model. This could - as a first approximation - be done by reducing the walking speed according to the group size. Of course, this would also require knowledge about the division of the population into groups and the distribution of group sizes.
This section introduces a specific microscopic model for pedestrian and crowd motion. This is embedded in the context of microscopic models, i.e., its properties and features are investigated and compared to those of other similar models that have been described in the previous section.
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[...]
^{1} The relevance of the topic for society and technology is reflected by its coverage in newspapers as well as magazines [GEO, 2001, NZZ Folio, 2002, Spektrum der Wissenschaft, 2001, Spiegel, 2001, SZ, 2001].
^{2} This is in a sense similar to deterministic chaos, where the indeterminism is not inherent but due to the lack of information (incomplete knowledge of the initial conditions).
^{3} Le Bon [1895] claimed that masses behave completely different from individuals, e.g., they are less rational and are transformed into another state of mind. Freud [1924] followed his argu-mentation. However, the concept of de-individuation via a mass soul was vehemently criticized later [Hofstätter, 1990].
^{4} This is not understood in the sense of persons being created or annihilated, of course, but leaving the system.
^{1} Information about MD techniques (containing algorithms) can be found in, e.g., [Rapaport, 1995].
^{2} A typical plan of a large building may contain more than 10,000 line elements. If there are 10,000 persons in a simulation then this means 10[8] checks every time-step, if no further simplifications are utilized.
^{3} Nagel [1996] provides a framework for road traffic in the language of particle hopping models.
^{4} For periodic boundary conditions it has to be checked whether a car crosses the boundary. In this case, it must be ensured that it does not drive onto a cell that has been left by another car in the same update step.
^{5} This equivalence will become useful when generalizing the model to 2D.
^{6} In this case, updating cells or cars is equivalent.
^{7} corresponding to 1 − p dec in the NaSch-model
^{8} The case of different p for each particle has been investigated by Evans [1997].
^{9} An extended Manhattan metric (where steps across edges and corners are possible) is illustrated in fig. 3.12.
^{10} p dec denotes the probability for deceleration (stopping in the v max = 1-case). Pedestrians can stop immediately, therefore, p dec is - other than for cars - the probability for stopping and not for reducing the speed by 1. However, for v max = 1 there is of course no difference between both.
^{11} [illustration not visible in this excerpt]
^{12} Pauls [1995] suggests a value of 1 . 0 P/s for a door of width 910 mm and moderate flow conditions.
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