标题： 单调三维曲面和广义浴缸模型的等价公式 显示英文标题

标题： Monotone three-dimensional surface and equivalent formulations of the generalized bathtub model

摘要： 在单车道交通的Lighthill-Whitham-Richards（LWR）模型中，车辆轨迹遵循先进先出（FIFO）原则，并且可以由累积车辆数量的单调三维曲面表示。相比之下，广义浴缸模型通过相对空间描述交通网络中的拥堵动态，通常违反FIFO原则，使得其表示更具挑战性。基于特征距离排序的概念，我们观察到广义浴缸模型中的行程可以根据它们的特征距离（剩余行程距离加上网络行驶距离）进行排序。我们定义了一个新的累计行程数，在给定剩余距离的情况下，在某一时间点之前的所有行程，表明它形成了一个尽管有FIFO违规但仍为单调的三维曲面。利用反函数定理，我们推导了不同坐标和相关变量的等效公式，包括Vickrey浴缸模型和基本浴缸模型的特殊情况。我们展示了基于这些公式的数值方法，并讨论了离散需求模式下的基于行程的方法。本研究增强了对广义浴缸模型特性的理解，促进了其在网络交通流建模、拥堵定价和交通规划中的应用。 显示英文摘要

摘要： In the Lighthill-Whitham-Richards (LWR) model for single-lane traffic, vehicle trajectories follow the first-in-first-out (FIFO) principle and can be represented by a monotone three-dimensional surface of cumulative vehicle count. In contrast, the generalized bathtub model, which describes congestion dynamics in transportation networks using relative space, typically violates the FIFO principle, making its representation more challenging. Building on the characteristic distance ordering concept, we observe that trips in the generalized bathtub model can be ordered by their characteristic distances (remaining trip distance plus network travel distance). We define a new cumulative number of trips ahead of a trip with a given remaining distance at a time instant, showing it forms a monotone three-dimensional surface despite FIFO violations. Using the inverse function theorem, we derive equivalent formulations with different coordinates and dependent variables, including special cases for Vickrey's bathtub model and the basic bathtub model. We demonstrate numerical methods based on these formulations and discuss trip-based approaches for discrete demand patterns. This study enhances understanding of the generalized bathtub model's properties, facilitating its application in network traffic flow modeling, congestion pricing, and transportation planning.