Abstract
It has been argued that the speed-density diagram of pedestrian movement has an inflection point. This inflection point was found empirically in investigations of closed-loop single-file pedestrian movement.The reduced complexity of single-file movement does not only allow a higher precision for the evaluation of empirical data, but it also significantly simplifies analytical considerations. This is especially true if one assumes homogeneous conditions, i.e. neglects temporal variations (consider time averages, neglect stop-and-go waves), individual differences of pedestrians (all simulated pedestrians have identical parameters) and investigates only steady-state (not the initial phase). As will be shown in this contribution one then can make a transition from the microscopic to a continuous and macroscopic perspective.Building on that it will be shown that certain (common) variants of the Social Force Model (SFM) do not produce an inflection point in the speed-density diagram if – assuming periodic boundary conditions – infinitely many pedestrians contribute to the force computed for one pedestrian. It will furthermore be shown that if – in said 1d movement situation – one only considers nearest neighbors for the computation of the inter-pedestrian forces the Social Force Model in the continuous description results in the so called Kladek formula for the speed-density relation. Since the Kladek formula exhibits the desired inflection point this observation is used as a motivation for an extension of the Social Force Model which allows to transform the continuous description of the SFM continuously to the Kladek formula and which also exhibits the inflection point in the speed density relation. It will be shown then, that this extended SFM yields astonishingly similar speed density relations as the original SFM when only a fixed limited number of (nearest) pedestrians are considered in the computation of the inter-pedestrian force.Finally it will be discussed, if also the description of the speed-density diagram for (motorized, four-wheel) vehicular and/or bicycle traffic could benefit from these measures.
Highlights
I: Empirical Data on Pedestrians’ Speed-Density RelationIn the course of recent years a number of experiments have been conducted in which pedestrians walk single-file in a closed loop [1,2,3,4,5]
Where free speeds are not shown in a diagram (e.g. Fig. 3) and data for very small densities is missing as well the existence of an inflection point may not be immediately obvious, assuming typical free speeds it becomes clear that it must exist, and even more important the curvature at high densities is without doubt positive
In a strict sense the fact that the Kladek formula is meant to describe average momentary speed requires for calibration first, that to measure speed it has to be measured how far pedestrians move in a given, fixed, and very short time span, and second that each data point has to be an average of measurements done within the same time span
Summary
In the course of recent years a number of experiments have been conducted in which pedestrians walk single-file in a closed loop [1,2,3,4,5]. Where free speeds are not shown in a diagram (e.g. Fig. 3) and data for very small densities is missing as well the existence of an inflection point may not be immediately obvious, assuming typical free speeds it becomes clear that it must exist, and even more important the curvature at high densities is without doubt positive. This holds independently of the measurement method, see Fig. 4. Speed density diagram for different loop sizes and professions in India (left) and Germany (right). Sources: left: Figure 3 of [3]; right: Figure 6 of [4]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
