Modeling Density Thresholds and Speed-Density Relationships in Crowd Evacuation

Modeling Density Thresholds and Speed-Density Relationships in Crowd Evacuation

Problem Description
During crowd evacuation, pedestrian density is a key factor affecting movement speed. When density is low, individuals can move freely at speeds close to normal walking speed. However, as density increases, interference between individuals intensifies, causing movement speed to gradually decrease. When density exceeds a certain critical value (the density threshold), complete stagnation or even stampede risks may occur. This problem requires understanding the mathematical relationship between speed and density, mastering methods for determining density thresholds, and learning to apply this relationship in evacuation models to predict crowd dynamics.

Explanation of Solution Process

  1. Introduction of Basic Concepts

    • Density Definition: The number of people per unit area (persons/m²). For example, density may be <1 person/m² in open spaces, and potentially >5 persons/m² in crowded conditions.
    • Speed Observation: When density is low, individual speed can reach 1.2-1.5 m/s. When density increases to 3-4 persons/m², speed may drop below 0.5 m/s.
    • Density Threshold: Refers to the critical density where speed drops sharply or congestion occurs, typically ranging from 3-5 persons/m². The specific value needs to be calibrated through experiments or models.

  2. Analysis of Classic Speed-Density Models

    • Linear Model: The simplest relational assumption: speed \(v = v_0 \cdot (1 - \rho/\rho_{\text{max}})\), where \(v_0\) is the free-flow speed and \(\rho_{\text{max}}\) is the maximum tolerable density (e.g., 5-7 persons/m²).
    • Non-linear Models (More Realistic):
      • Weidmann Model: \(v = v_0 \cdot [1 - \exp(-1.913 \cdot (1/\rho - 1/\rho_{\text{max}}))]\). This model is based on empirical pedestrian flow data and better fits speed changes at medium to high densities.
    • Threshold Introduction: In practical applications, a critical density \(\rho_c\) needs to be defined. When \(\rho > \rho_c\), speed may decay exponentially, e.g., \(v = v_0 \cdot \exp(-\alpha (\rho - \rho_c))\).

  3. Methods for Determining Density Thresholds

    • Experimental Observation: Recording real evacuation scenarios using cameras or sensors to analyze the average crowd speed at different densities and identify the point of sharp speed decline.
    • Simulation Calibration: Adjusting parameters in computer simulations to match model outputs with experimental data. For example, in the social force model, inferring the critical density by adjusting force parameters.
    • Theoretical Calculation: Based on spatial geometric constraints. For instance, assuming a minimum required personal space of 0.3 m² per individual, the theoretical maximum density is approximately 3.3 persons/m². However, the actual threshold may vary due to cultural differences or environmental layout.

  4. Application Steps in Evacuation Models

    • Step 1: Environment Discretization
      Divide the evacuation area into grids and calculate the instantaneous density for each grid in real-time.
    • Step 2: Dynamic Speed Assignment
      Assign movement speeds to individuals based on the speed-density relationship function queried for the current grid density. For example, if the density in an area is 2 persons/m², assign a speed of 1.0 m/s; if density rises to 4 persons/m², reduce speed to 0.3 m/s.
    • Step 3: Congestion Warning and Intervention
      When a grid's density is detected to consistently exceed the threshold (e.g., ρ>4 persons/m²), trigger mitigation strategies such as using guidance signs for分流 (diversion) or temporarily opening new exits.

  5. Case Study

    • Assume a corridor is 2 meters wide and 10 meters long. With 30 people, the average density is 1.5 persons/m² and speed is approximately 1.2 m/s. If the number increases to 60 people (density 3 persons/m²), speed drops to 0.6 m/s. When density exceeds 4 persons/m², speed may fall below 0.2 m/s, requiring immediate intervention to prevent blockage.

Summary
The speed-density relationship is a core mechanism in evacuation simulation. By properly calibrating the density threshold and function parameters, evacuation times can be predicted more accurately, risk areas identified, and a quantitative basis provided for optimizing evacuation strategies.