A meso-to-macro cross-resolution performance approach for connecting polynomial arrival queue model to volume-delay function with inflow demand-to-capacity ratio

Although the macroscopic volume-delay function (VDF) has been widely used in static traffic assignment for transportation planning, the planning community has long recognized its deficiencies as a static function in capturing traffic flow dynamics and queue evolution process. In the existing literature, many queueing-based and simulation-based dynamic traffic assignment (DTA) models involving various traffic flow parameters have been proposed to capture traffic system dynamics on different spatial scales; however, how to calibrate these DTA models could still be a challenging task in its own right, especially for real-world congested networks with complex traffic dynamics. By extending the fluid-based polynomial arrival queue (PAQ) model with quadratic inflow rates proposed by Newell (1982) and cubic inflow rates by Cheng et al. (2022), this paper attempts to propose a cross-resolution Queueing-based Volume-Delay Function (QVDF) to explicitly establish a coherent connection between (a) the macroscopic average travel delay performance function in a long-term planning horizon and (b) the mesoscopic dynamic queuing model during a single oversaturated period. By introducing two types of elasticity functional forms, this paper develops a relationship from the macroscopic inflow demand-to-capacity (D/C) ratio to the congestion duration of a bottleneck, from the congestion duration to the magnitude of speed reduction. The QVDF can be directly utilized to provide closed-form expressions for both average travel delay performance and the time-dependent speed profiles. The proposed cross-resolution QVDF provides a numerically reliable and theoretically rigorous performance function to characterize oversaturated bottlenecks at both macroscopic and mesoscopic scales.