A Multi-stage Spatial Queueing Model with Logistic Arrivals and Departures Consistent with the Microscopic Fundamental Diagram and Hysteresis

This paper introduces a spatial queueing model for a single bottleneck during morning peak hours. Using a logistic function for arrival and departure flows (after calibration), we express these flows in continuous, differentiable terms. The model, validated across different peak periods and locations, shows superior robustness compared to common functions and accommodates constant or varying capacity. It captures key aspects of morning peaks, including the emergence of hysteresis loops in flow-density fundamental diagrams. The multi-stage approach identifies three traffic phases (free-flow, transition, queued) and maintains spatial consistency in flow and density. It accounts for queue growth and spillback under various bottleneck intensities, producing speed-density diagrams with hysteresis loops. Calibration with real traffic flow/density time-series data shows the model accurately reproduces observed counterclockwise hysteresis loops and can generate both planar and 3D fundamental diagrams along a traffic link, matching real patterns. Compared to the cell transmission model, the proposed model shows superior generalization and robustness.