Analysis of Exit Queueing Behavior and Fairness in Crowd Evacuation

Analysis of Exit Queueing Behavior and Fairness in Crowd Evacuation

Problem Description
During emergency evacuations, when a large number of individuals rush towards limited exits, queuing naturally occurs. The analysis of exit queueing behavior focuses on individual decision-making within queues (such as queue selection and whether to switch queues), the impact of queuing rules (e.g., first-in-first-out) on overall evacuation efficiency, and the fairness of different individuals' access to exits (e.g., whether individuals at varying distances from an exit have a fair chance of escape). This topic requires integrating queuing theory, spatial game theory, and fairness criteria to study how to optimize the queuing system to reduce total evacuation time and enhance fairness among individuals.

Problem-Solving Process

  1. Problem Definition and Key Parameters

    • Queuing Scenario: Independent queues form in front of multiple exits. Individuals choose a queue based on their current position, queue length, and movement speed.
    • Key Metrics:
      • Efficiency: Total evacuation time, number of people passing per unit time (throughput).
      • Fairness: Variance in individual waiting times, time difference between the first and last evacuated individuals, priority for vulnerable groups (e.g., those with slower mobility).
    • Points of Conflict: Individuals tend to choose the shortest queue, but information asymmetry or herd behavior may lead to excessively long queues in some lines, reducing efficiency.

  2. Establishing a Basic Queuing Model

    • Single-Exit Queue: Use the M/M/1 queuing model (arrival rate λ, service rate μ), where inter-arrival and service times follow exponential distributions. The average waiting time formula is \(W = \frac{1}{μ - λ}\) (requiring λ < μ).
    • Multi-Exit Queuing: Extend to a multi-queue M/M/c model (c exits). If queues are not interconnected and individuals cannot switch after choosing, initial uneven distribution can easily lead to efficiency loss.
    • Example: Assuming two exits with the same service rate (μ=10 people/minute), if 60 people are initially evenly distributed between two queues (30 per queue), the total evacuation time is approximately 3 minutes; if 40 people gather in one queue and 20 in the other, the total time increases to 4 minutes (the long queue takes 4 minutes, the short queue 2 minutes, and the system time is determined by the longer queue).

  3. Modeling Individual Queuing Decision Behavior

    • Shortest Queue Strategy: Individuals choose based only on the currently visible queue length, ignoring subsequent arrivals and differences in movement speed.
    • Issue: If exit service rates differ (e.g., left exit μ=12 people/minute, right exit μ=8 people/minute), individuals may still cluster at the right queue because it "appears shorter," actually increasing waiting time.
    • Improved Strategies:
      • Global Information Guidance: Use displays to show estimated waiting times for each queue (requiring calculation incorporating service rates).
      • Dynamic Queue-Switching Rules: Allow individuals to re-choose based on new information while in a queue, but the cost of交叉冲突 caused by switching must be weighed.

  4. Quantitative Methods for Fairness

    • Temporal Fairness Metrics:
      • Gini Coefficient: Calculate the Lorenz curve for the waiting times of all individuals. A value closer to 0 indicates greater fairness (complete equality when all individuals have the same waiting time).
      • Maximum Time Difference: The time difference between the last and first evacuated individuals. A smaller value indicates greater fairness.
    • Consideration of Spatial Fairness: Individuals far from exits may lag behind due to queuing rules. It is necessary to introduce "priority weights" or "batch scheduling" (e.g., alternately releasing people from different areas).

  5. Optimization Strategies and Algorithms

    • Centralized Scheduling: Use a central system to assign individuals to exits, such as the Hungarian algorithm to minimize total waiting time. However, this requires complete information and control, which is difficult to achieve in reality.
    • Distributed Guidance:
      • Virtual Queueing: Require all individuals to form a single queue in a central hall, then assign them to the fastest available exit (similar to a bank ticketing system). This sacrifices some efficiency for improved fairness.
      • Dynamic Signal Control: Set traffic lights at exits to periodically switch the right-of-way for queue discharge, preventing a single queue from monopolizing exit resources.
    • Fairness-Efficiency Trade-off: Complete fairness may reduce efficiency (e.g., forcing priority for distant individuals). A Pareto-optimal solution needs to be defined (e.g., a 1% loss in fairness for a 10% gain in efficiency).

  6. Simulation Verification and Case Studies

    • Simulation Tools: Use AnyLogic or NetLogo to build multi-agent models, input different decision rules, and compare output metrics.
    • Typical Results:
      • Under a pure shortest-queue strategy, the Gini coefficient may reach 0.3 (highly unfair), while virtual queueing can reduce it to below 0.1, but total time increases by 15%.
      • In a stadium evacuation case study, using "zonal rotation release" (dividing the venue into扇形 sectors and evacuating them in sequence) can reduce the maximum time difference by 40%.

Summary
The exit queuing problem requires balancing efficiency and fairness. By quantifying the impact of decision-making behaviors and introducing dynamic information guidance or scheduling rules, congestion and unfairness caused by盲目排队 can be mitigated. Practical applications need to adjust strategy parameters based on architectural layout and crowd characteristics (e.g., proportion of elderly individuals).