Path Selection and Congestion Control in Crowd Evacuation

Path Selection and Congestion Control in Crowd Evacuation

Problem Description
During emergencies such as fires or earthquakes, large crowds need to evacuate quickly from buildings or open areas to safe locations. Due to limited exit capacities and uneven crowd densities, certain paths may experience severe congestion, which can reduce overall evacuation efficiency and even lead to stampede incidents. This problem requires designing a method to dynamically guide people in choosing different paths, aiming to minimize total evacuation time and avoid congestion.

Detailed Knowledge Points

  1. Core Conflict

    • Individual Rationality: Each person tends to choose the shortest path (e.g., the exit with the shortest straight-line distance). However, if everyone rushes to the same path simultaneously, congestion can cause actual travel time to surge.
    • System Optimality: It is necessary to disperse the flow of people so that the "travel times" on various paths become relatively balanced, thereby maximizing overall evacuation efficiency.

  2. Key Parameters

    • Path Capacity: The maximum number of people that can pass through a certain path per unit time (e.g., determined by exit width).
    • Dynamic Density: Real-time monitoring of crowd density at the entrances of each path. Higher density leads to slower passage speed (usually following a nonlinear relationship).
    • Expected Travel Time: The travel time for a path calculated based on current density, used for guidance decisions.

Solution Steps
Step 1: Establish a Travel Time Model
Assume the travel time \(T\) for a path is related to the current number of people \(N\) and the path capacity \(C\). A commonly used formula is:

\[T = t_0 \left(1 + \alpha \left(\frac{N}{C}\right)^\beta\right) \]

  • \(t_0\) is the travel time under idle conditions (e.g., normal walking time).
  • \(\alpha, \beta\) are congestion coefficients (usually \(\beta > 1\), indicating the accelerating effect of congestion on time).
  • Example: If an exit has \(C=100\) people/minute and \(N=150\) people, \(T\) may become three times the idle time.

Step 2: Real-time Monitoring and Information Updates

  • Deploy sensors at key nodes in the evacuation area (e.g., corridor intersections) to count the current number of people \(N\) on each path in real time.
  • Update the expected travel time \(T\) for each path every 10-15 seconds and notify people via electronic signage or mobile phone push notifications.

Step 3: Path Allocation Strategy
Adopt the System Optimal Allocation principle instead of the shortest path principle:

  1. Calculate Diversion Ratios:

    • Assume there are \(k\) alternative paths with current expected travel times \(T_1, T_2, ..., T_k\).
    • The goal is to allocate the number of people so that the travel times of all paths tend to be equal (i.e., \(T_1 \approx T_2 \approx ... \approx T_k\)).
    • Simplified algorithm: Allocate people to each path in proportion to \(\frac{1}{T_i}\) (paths with shorter travel times receive more people).

  2. Dynamic Adjustment Example:

    • Initial state: Path A (\(T_A=2\) minutes), Path B (\(T_B=5\) minutes).
    • Allocation ratio: Path A is allocated approximately \(\frac{1/2}{1/2+1/5} \approx 71\%\) of the people, Path B receives 29%.
    • If Path A's travel time \(T_A\) increases due to excessive allocation, its allocation ratio is automatically reduced in the next adjustment.

Step 4: Enhanced Measures to Avoid Local Congestion

  • Segmented Control: Divide long passages into multiple sections, independently monitor the density of each section. If the density in any section exceeds a safety threshold, temporarily close its entrance and guide people to detour.
  • Reverse Incentives: Provide prompts to individuals who choose non-shortest paths but can alleviate congestion (e.g., "This path is expected to save 5 minutes"), leveraging behavioral economics to encourage cooperation.

Summary
Path selection in crowd evacuation is essentially a resource competition problem in a multi-agent system. By dynamically monitoring, quantifying travel times, and balancing the flow of people, it is possible to overcome the dilemma of "individual rationality leading to collective irrationality." Practical applications require integrating sensor networks, real-time algorithms, and behavioral guidance to achieve safe and efficient evacuation.