Monitoring the Number of Pedestrians in an Area: The Applicability of Counting Systems for Density State Estimation

Crowd monitoring systems are more and more used to support crowd management organizations. Currently, counting systems are often used to provide quantitative insights into the pedestrian traffic state, since they are fairly easy to install and the accuracy is reasonably good under normal conditions. However, there are no sensor systems that are 100% accurate. Detection errors might have severe consequences for the density state estimation at large squares. The consequences of these errors for pedestrian state estimation have not yet been determined. This paper studies the impact of one specific type of detection error on the functionality of counting camera systems for density state estimation, namely, a randomly occurring “false negative” detection error. The impact is determined via two tracks, a theoretical track and a simulation track. The latter track studies the distribution of the cumulative number of pedestrians after 24 hours for three stylized cases by means of Monte Carlo simulations. This paper finds that counting camera systems, which have a detection error that is not correlated with the flow rate, provide a reasonably good estimation of the density within an area. At the same time, if the detection error is correlated with the flow rate, counting camera systems should only be used in the situation where symmetric demand patterns are expected

Trip chain complexity: a comparison among latent classes of daily mobility patterns

This paper studies the relationship between trip chain complexity and daily travel behaviour of travellers. While trip chain complexity is conventionally investigated between travel modes, our scope is the more aggregated level of a person’s activity-travel pattern. Using data from the Netherlands Mobility Panel, a latent class cluster analysis was performed to group people with similar mode choice behaviour in distinct mobility pattern classes. All trip chains were assigned to both a travel mode and the mobility pattern class of the traveller. Subsequently, differences in trip chain complexity distributions were analysed between travel modes and between mobility pattern classes. Results indicate considerable differences between travel modes, particularly between multimodal and unimodal trip chains, but also between the unimodal travel modes car, bicycle, walking and public transport trip chains. No substantial differences in trip chain complexity were found between mobility pattern classes. Independently of the included travel modes, the distributions of trip chain complexity degrees were similar across mobility pattern classes. This means that personal circumstances such as the number of working hours or household members are not systematically translated into specific mobility patterns.Transport and PlanningTransport and Plannin

An overview of the modeling of crowd dynamics

In this chapter we review some of the most important models at microscopic, macroscopic, and mesoscopic scale, which, in our opinion, represent milestones in their respective fields or are of particular interest for this book. We also report some models for rational pedestrians, which make use of techniques from optimal control theory. For the sake of convenience, we present all models in two space dimensions

Multiscale modeling by time-evolving measures

This chapter is devoted to a multiscale approach to the modeling of crowd dynamics, which is the core topic of the book. We begin by presenting, in Sect. 5.1, a general measure-based modeling framework suitable to include the basic features of pedestrian kinematics at any scale. Specifically, we assume that pedestrian motion results from the interplay between the individual will to follow a preferred travel program and the necessity to face the rest of the crowd. We discuss in Sect. 5.2 how to properly model these behavioral aspects. In Sect. 5.3 we show how discrete (microscopic) and continuous (macroscopic) models can be obtained in the proposed framework, before focusing, in Sect. 5.4, on multiscale modeling issues. We also propose a detailed dimensional analysis, which highlights the role of a few significant parameters, and a numerical scheme for the approximate solution of the equations. The scheme is obtained in two steps in Sect. 5.5. First we derive a discrete-in-time model; next we discretize the space variable as well, obtaining an algorithm (cf. Appendix B) which can be implemented on a computer to produce simulations (cf. Chap.  2). Finally, in Sect. 5.6 we extend the previous modeling structures to the case of two interacting crowds

Problems and simulations

In this chapter we give an informal introduction to the multiscale model and present some case studies of interest for applications, along with related numerical simulations. Results presented here are somehow complementary to those usually presented by physicists, engineers, and computer scientists. Indeed, we aim at showing how mathematical modeling can help in developing truthful pedestrian models, and at giving a sample of phenomena which can be simulated without the introduction of artificial or ad hoc effects

Basic theory of measure-based models

This chapter is devoted to the mathematical foundations of the model introduced in Chap.  5. Contents go continuously back and forth between modeling and analysis, however with a more formal approach than that used in the previous chapter. The first three sections, from Sects. 6.1 to 6.3, discuss how the measure-based model can be derived from a particle description of pedestrians, thereby formalizing the link between individualities and collectivity which is at the basis of most of the complexity of crowd behaviors. In addition, in the light of such a derivation they propose a probabilistic reading of the measure-based model, which turns out to be particularly meaningful for applications. The central part of the chapter, encompassing Sects. 6.4–6.7, is concerned with the basic theory of well-posedness and numerical approximation of measure-valued Cauchy problems for first order models based on conservation laws, also in a multiscale perspective. Minimal generic assumptions are stated in order to achieve proofs, to be regarded possibly also as guidelines in the modeling approach. Finally, Sect. 6.8 resumes the discussion about the crowd model presented in Chap.  5 studying under which conditions it is in the scope of the theory set forth in the preceding sections