A multi-stage spatial queueing model with logistic arrivals and departures consistent with the microscopic fundamental diagram and hysteresis

Abstract

This paper introduces a spatial queueing model for a single bottleneck during morning peak hours. Utilizing the logistic function and after appropriate calibration, it articulates the arrival and departure flows in continuous, differentiable terms. By validating the model across different peak periods and locations, the demand model’s robustness is superior to other commonly used functions. This model also incorporates constant or varying capacity scenarios. It effectively captures key aspects of morning peak traffic, including the emergence of hysteresis loops in fundamental diagrams (FDs) of density and flow. The model’s multi-stage approach recognizes three distinct phases in traffic flow: freeflow, transition, and queued segments, ensuring spatial consistency in flow and density across these stages. It accounts for the growth of the queued segment and vehicle spillback under various bottleneck intensities, with the resulting FDs for speed and density also displaying hysteresis loops. The calibration of model parameters utilizes time-series data of traffic flow and density space–time maps derived from real-world data. The validation results accurately reflect real traffic scenarios, emulating the counterclockwise hysteresis loops observed in density and its heterogeneity, and provide both planar and three-dimensional FDs at different points along the traffic link, each mirroring real-life traffic patterns. Additionally, a comparison with the cell transmission model (CTM) reveals that the proposed model exhibits superior generalization and robustness.