  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.
