Thesis - Chapter 1

(updated  2nd September 2007)

Keywords: Pedestrian Evacuation, simulating people, simulating crowds, simulating crowd dynamics

"Crowd Dynamics" - PhD Thesis - University of Warwick - August 2000

Introduction

Legion is the name used to describe the collection of programmes developed to analyse the dynamics of crowds. The heart of this suite there is an algorithm which models the dynamics of the crowd by using a least effort algorithm. It treats every person (entity) in the crowd as an individual, calculating their positions by scanning their local environment and choosing an appropriate, humanlike, action.

1.1.1 The least effort algorithm.

Suppose we have N entities, with an entity i at position (x_i, y_i) inside a region R of the plane R², representing the accessible parts of a building. Each entity's path P_i through the building is constrained: first, by the entity's speed distribution, and secondly by the requirement that entity i visit certain places or subregions of the building in some order. Call the set of all these constraints on the entity i's path C_i.

There are also non-collision constraints K_ij which asserts that entities i and j cannot occupy the same position at the same time. There is a cost function u(P_i) for example, length, total time, effort. For a set P of paths P_i satisfying constraints C_i and K_ij, there is a total cost U(P) = u(P₁) + ... + u(P_n). The problem is to minimize U(P) subject to those constraints, thereby finding the set of paths (flow pattern of the crowd) that requires the least effort (in total). This optimization problem can be solved by a type of simulated annealing, iteratively starting from a set of paths P, randomly varying it, seeing if the cost goes down, and if so choosing the cheaper set of paths; then repeat. The algorithm stops when it fails to improve the solution.

It can be shown that the algorithm solves the problems of calculating the dynamics of a large population (>100,000) in the simulation in polynomial time.

Further detail of the algorithm is subject to a commercial non disclosure agreement. However, we discuss its inputs, outputs and the validation process in this thesis. We also examine the emergent phenomena, unique to crowds, which can be examined using the simulation suite and the least effort algorithm.

1.1.2 Introduction to the Legion tools

The Legion Crowd Dynamics tool kit consist of a prototype development suite, a C library, a model builder and simulator (the latter two were the first attempts to develop commercial products developed for client projects) and a replayer, which allows the clients to view, review and analyse their models.

With the Legion simulation it is possible to alter various parameters and study the effects of, for example, increasing the crowd density. The use of a simulation provides us with two important perspectives. Firstly, the simulation provides us with insights to the nature of crowd dynamics, often it is the insight to the problem that leads us to the solution. Secondly, the simulation can be used to prove or disprove a variety of relationships observed in the crowd, for example, it is presently assumed that doubling the width of an egress route will double the flow of people on that route, but this turns out to be wrong.

Applications include modelling multi-venue events, such as the Olympics Games. Figure 1 shows the screen display from the Legion simulation replayer and Figure 2 shows the plan of the same area of the Sydney Olympic Park.

The boulevard and rail station are a common domain for all venues. Loading and unloading crowds are simulated using the Legion system (Figure 1).

Figure 3 shows the validation tool displaying the entities (yellow dots) moving along two corridors (grey = walls). During the simulation run the entities move around the screen, avoiding each other and the local geometry.

In chapters 2 and 3 we discuss the various problems relating to crowd movement and crowd safety. We discuss the problems of applying the present guidelines and review the various methodologies used to assess crowd safety. We also discuss the application of a variety of techniques presently used to model crowds and review the existing literature.

Chapter 4 discusses the principles of computer simulation, the problems of developing a suitable model of crowd dynamics and the parameters required for accurate crowd modelling. We relate the psychological factors of human decisions to a mathematical framework, using the four rules which define the least effort algorithm: Objective, Motility, Constraint and Assimilation.

In chapter 5 we discuss the various inputs to the Legion simulation suite, why they were chosen and the relevance they have in determining the parameters for assessing crowd risk and crowd safety. The entity behaviour, and its significance to simulating crowd dynamics are also discussed in chapters 5 and 6.

Chapter 6 discusses the validation of the algorithms and compares the results to field observations and historical studies. We examine the qualitative and quantitative data, the results of the Legion crowd simulation analysis and the emergent phenomena which are unique to crowds.

Chapters 7 and 8 discuss case studies using the tools we have developed.

Chapter 9 draws the previous chapters to a conclusion and indicates the scope of the simulation, how and where it should be applied and the potential future developments of the Legion pedestrian simulation methodology.

It is important to understand the framework for the development of these tools and we need to outline the various guidelines which are used to define crowd safety in places of public assembly.

1.3 The Guides

There are a number of documents which have been produced to advise on crowd safety issues. The relevant documents are the Green, Purple and Primrose Guides. The guides are advisory documents for use by competent persons. They are the distillation of many years of research and experience of the safe management and design of places of public assembly. The guides have no statutory force but many of their recommendations are given force of law at individual sites by their inclusion in safety certificates issued under the Safety of Sports Grounds Act 1975 or the Fire Safety and Safety of Places of Sport Act 1987. The advice given in the Guides is without prejudice to the application of the appropriate building regulations, the Health and Safety at Work Act 1974, and any other relevant legislation.

1.4 Crowds and occupant capacity

The maximum size of the audience for a particular event is generally determined by the licensing authority (taking advice from the fire authority) and is known technically as the "occupant capacity". This will include all ticket holders, pass holders and guests. The method for establishing this capacity is calculated by finding the total area available to the public (in square metres) and multiplying by 2, where 2 = two people per square metre. For example: An outdoor site measuring 100 metres x 50 metres with all areas available to the public could accommodate a maximum of 10,000 people (100 x 50 = 5,000 sq. metres x 2 = 10,000 people). We will discuss the naivety of this type of calculation in later chapters.

Crowds have certain interactions which are part fluid, part granular and part psychological reaction. The nomenclature we will use as follows.