Generalized bathtub model of network trip flows

TL;DR

This paper introduces a comprehensive bathtub model for network trip flows that accommodates various trip distance distributions, extending existing models to include new formulations, properties, and solutions for complex trip flow scenarios.

Contribution

It develops a unified framework for modeling network trip flows with general distributions, deriving multiple equivalent formulations, and extending to multi-commodity trip flows with mobility services.

Findings

Derived four equivalent differential formulations from the fundamental diagram and conservation law.

Presented two numerical methods for solving the bathtub model based on differential and integral forms.

Extended the model to multi-commodity trip flows with mobility service vehicles.

Abstract

In this study, we present a unified framework for modeling network trip flows with general distributions of trip distances, including negative exponential, constant, and regularly sorting trip distances studied in the literature. In addition to tracking the number of active trips as in Vickrey's model, this model also tracks the evolution of the distribution of active trips' remaining distances. We derive four equivalent differential formulations from the network fundamental diagram and the conservation law of trips for the number of active trips with remaining distances not smaller than any value. Then we define and discuss the properties of stationary and gridlock states, derive the integral form of the bathtub model with the characteristic method, obtain equivalent formulations by replacing the time coordinate with the cumulative travel distance, and present two numerical methods to solve the bathtub model based on the differential and integral forms respectively. We further study equivalent formulations and solutions for two special types of distributions of trip distances: time-independent negative exponential or deterministic. In particular, we present six equivalent conditions for Vickrey's bathtub model to be applicable. Finally we demonstrate that the fundamental diagram and the bathtub model can be extended for multi-commodity trip flows with trips served by mobility service vehicles.