Vickrey (1991, 2020) proposed a bathtub model for the evolution of trip flows served by privately operated vehicles inside a road network based on three premises: (i) treatment of the road network as a single bathtub; (ii) the speed-density relation at the network level, also known as the network fundamental diagram of vehicular traffic, and (iii) the time-independent negative exponential distribution of trip distances. However, the distributions of trip distances are generally time-dependent in the real world, and Vickrey’s model leads to unreasonable results for other types of trip distance distributions. Thus there is a need to develop a bathtub model with more general trip distance distribution patterns.

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, 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.

Vickrey（1991，2020）基于三个前提提出了一个浴缸模型，用于分析道路网络内私人车辆服务的出行流量：（i）将道路网络视为一个单独的浴缸；（ii）在网络级别上的速度-密度关系，也称为车辆交通的网络基本图，以及（iii）行驶距离的与时间无关的负指数分布。但是，在现实世界中，行程距离的分布通常与时间有关，Vickrey的模型导致其他类型的行程距离分布的结果不合理。因此，需要开发一种具有更通用的行程距离分布模式的浴缸模型。

在这项研究中，我们提出了一个统一的框架，用于对具有行程距离的一般分布的网络行程流进行建模，包括文献中研究的负指数行程，常数行程和定期排序行程距离。除了像Vickrey模型一样跟踪主动出行的次数外，该模型还跟踪主动出行的剩余距离分布的演变。对于剩余距离不小于任何值的活动行程数，我们从网络基本图和行程守恒律中得出四个等效的微分公式。然后，我们定义并讨论稳态和僵锁状态的特性，并用特征方法导出浴缸模型的积分形式，并分别提出了两种基于微分形式和整数形式求解浴缸模型的数值方法。我们进一步研究了两种特殊的行程距离分布的等效公式和解：与时间无关的负指数或确定性。特别是，我们提出了适用于Vickrey浴缸模型的六个等效条件。