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

TL;DR

This work extends the monotone three-dimensional surface concept to the generalized bathtub model by introducing a cumulative trips-ahead function $N(t,x)$ ordered by the characteristic distance $\theta=x+z(t)$, enabling a monotone surface despite FIFO violations. It derives 20 equivalent formulations across time, space, network travel distance, and cumulative order coordinates using the inverse function theorem, including the $N$- and $K$-models and several $(t,\theta)$, $(z,x)$, and $(x,\theta)$ representations. A new, stable numerical method in $(z,x)$ coordinates is proposed to avoid artificial gridlock, and the framework is extended to discrete demand with trip-based formulations and event-driven approaches. The results enhance analytic and computational tractability of congestion dynamics, with potential applications in congestion pricing and planning for multi-modal transportation networks, while preserving a rigorous link to the classical bathtub and Vickrey bathtub special cases.

Abstract

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.

Figures (8)

• Figure 1: Illustration of trips in different spaces: (a) Absolute space within a grid network; (b) Relative space based on remaining distances to individual trip destinations
• Figure 2: Illustration of aggregate variables in the generalized bathtub model
• Figure 3: Equivalent formulations of the generalized bathtub model
• Figure 4: Two discretization methods
• Figure 5: Monotone three-dimensional surfaces of $F(t,x)$ and $N(t,x)$

Theorems & Definitions (2)

• Theorem 2.1
• Lemma 2.2