Equilibrium traffic dynamics in a bathtub model: A special case

Arnott, Richard; Kokoza, Anatolii; Naji, Mehdi

doi:10.1016/j.ecotra.2016.11.001

Cited by 37 publications

(14 citation statements)

References 10 publications

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“…We will be interested in the distribution of trip lengths in an urban network, such as deterministic distributions of trip lengths ( 46 ) and constant trip length ( 47 ).…”

Section: Discussionmentioning

confidence: 99%

Cordon-Based Pricing Schemes for Mixed Urban-Freeway Networks using Macroscopic Fundamental Diagrams

Wang

Gayah

2021

Transportation Research Record: Journal of the Transportation R

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The development of traffic models based on macroscopic fundamental diagrams (MFD) enables many real-time control strategies for urban networks, including cordon-based pricing schemes. However, most existing MFD-based pricing strategies are designed only to optimize the traffic-related performance, without considering the revenue collected by operators. In this study, we investigate cordon-based pricing schemes for mixed networks with urban networks and freeways. In this system, heterogeneous commuters choose their routes based on the user equilibrium principle. There are two types of operational objective for operating urban networks: (1) to optimize the urban network’s performance, that is, to maximize the outflux; and (2) to maximize the revenue for operators. To compare those two objectives, we first apply feedback control to design pricing schemes to optimize the urban network’s performance. Then, we formulate an optimal control problem to obtain the revenue-maximization pricing scheme. With numerical examples, we illustrate the difference between those pricing schemes.

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“…We will be interested in the distribution of trip lengths in an urban network, such as deterministic distributions of trip lengths ( 46 ) and constant trip length ( 47 ).…”

Section: Discussionmentioning

confidence: 99%

Cordon-Based Pricing Schemes for Mixed Urban-Freeway Networks using Macroscopic Fundamental Diagrams

Wang

Gayah

2021

Transportation Research Record: Journal of the Transportation R

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“…The bathtub model [16][17][18][19] analyzes urban hypercongestion at an aggregate level. In the morning rush hour, cars enter the downtown urban center and when density is sufficiently large, traffic flow becomes inefficiently low and the outflow of cars decreases, which makes hypercongestion more persistent.…”

Section: Literature Reviewmentioning

confidence: 99%

Quantifying the phantom jam externality: the case of an Autobahn section in Germany

Goldmann

Sieg

2021

Eur. Transp. Res. Rev.

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If not restricted by tolls, private decisions to drive on a highway result in inefficiently high usage which leads to traffic jams. When traffic demand is high, traffic jams can occur simply because of the interaction of vehicle drivers on the road, a phenomenon called phantom jam. The probability of phantom jams occurring increases with traffic flow. Unpriced externalities lead to inefficiently high road usage. We offer a method for quantifying traffic jam externalities and identifying and isolating the phantom jam externality. We examine the method by applying it to a specific highway section in Germany. The maximal congestion externality for the analyzed highway section is about 38 cents per vehicle and kilometer. Congestion charges that are calculated ignoring phantom jam externalities, can only internalize two-thirds of the true externality.

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“…In (Fosgerau, 2015), a bathtub model was developed for deterministic distributions of trip distances, and trips are "regularly sorted" such that shorter trips enter the network later but exit earlier than longer ones (last-in-first-out); then the departure time user equilibrium was solved with experienced travel times. Yet another bathtub model was developed in (Arnott et al, 2016;Arnott and Buli, 2018), in which all trips are assumed to have the same distance, and the dynamics of the number of active trips and the experienced travel times are described by delay-differential equations, which are solved by an iterative method.…”

Section: Introductionmentioning

confidence: 99%

“…Theoretically, many studies following (Daganzo, 2007) were not aware of the third premise of Vickrey's bathtub model in (Vickrey, 1991) and have attempted to apply Vickrey's bathtub model for constant and other distributions of trip distances; but this has led to physically unreasonable results as information travels too fast (see (Mariotte et al, 2017) and references therein). On the other hand, it has been argued that Vickrey's bathtub model is improper as "the exit rate from downtown traffic (i.e., the arrival rate at work) depends only on the density of downtown traffic, and hence that the first exit occurs as soon as the first entry" in (Arnott et al, 2016;Arnott and Buli, 2018). In addition, there seems to be no bathtub model for trip flows served by mobility service vehicles in the literature; but such a model is essential for understanding congestion dynamics in a shared mobility system.…”

Section: Introductionmentioning

confidence: 99%

Generalized bathtub model of network trip flows

Jin¹

2019

Preprint

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Vickrey, 1991) 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, 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.

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Equilibrium traffic dynamics in a bathtub model: A special case

Cited by 37 publications

References 10 publications

Cordon-Based Pricing Schemes for Mixed Urban-Freeway Networks using Macroscopic Fundamental Diagrams

Cordon-Based Pricing Schemes for Mixed Urban-Freeway Networks using Macroscopic Fundamental Diagrams

Quantifying the phantom jam externality: the case of an Autobahn section in Germany

Generalized bathtub model of network trip flows

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