Fifty years of the bottleneck model: A bibliometric review and future research directions

Abstract The bottleneck model introduced by Vickrey in 1969 has been recognized as a benchmark representation of the peak-period traffic congestion due to its ability to capture the essence of congestion dynamics in a simple and tractable way. This paper aims to provide a 50th anniversary review of the bottleneck model research since its inception. A bibliometric analysis approach is adopted for identifying the distribution of all journal publications, influential papers, top contributing authors, and leading topics in the past half century. The literature is classified according to recurring themes into travel behavior analysis, demand-side strategies, supply-side strategies, and joint strategies of demand and supply sides. For each theme, typical extended models developed to date are surveyed. Some potential directions for further studies are discussed.

[1]  Gordon F. Newell The Morning Commute for Nonidentical Travelers , 1987, Transp. Sci..

[2]  Feng Xiao,et al.  The Morning Commute Problem with Coarse Toll and Nonidentical Commuters , 2011 .

[3]  D. Bernstein,et al.  UNDERSTANDING THE COMPETING SHORT-RUN OBJECTIVES OF PEAK PERIOD ROAD PRICING , 1993 .

[4]  Chairman.,et al.  The Value of Reliability , 1986, IEEE Power Engineering Review.

[5]  Erik T. Verhoef,et al.  Airlines' Strategic Interactions and Airport Pricing in a Dynamic Bottleneck Model of Congestion , 2014 .

[6]  Feng Xiao,et al.  Managing morning commute traffic with parking , 2012 .

[7]  Hai-Jun Huang,et al.  On the morning commute problem with carpooling behavior under parking space constraint , 2016 .

[8]  Joseph I. Daniel,et al.  The untolled problems with airport slot constraints , 2014 .

[9]  André de Palma,et al.  Public transport reliability and commuter strategy , 2014 .

[10]  Eric J. Gonzales Coordinated pricing for cars and transit in cities with hypercongestion , 2015 .

[11]  Jonathan D. Hall Pareto Improvements from Lexus Lanes: The effects of pricing a portion of the lanes on congested highways , 2015 .

[12]  Maria Börjesson,et al.  Estimating exponential scheduling preferences , 2015 .

[13]  R. Lindsey,et al.  Comparison of Morning and Evening Commutes in the Vickrey Bottleneck Model , 2002 .

[14]  C. Lindsey,et al.  Traffic Congestion And Congestion Pricing , 2000 .

[15]  Huijun Sun,et al.  Modeling the Morning Commute Problem in a Bottleneck Model Based on Personal Perception , 2019, Journal of Advanced Transportation.

[16]  Wei Liu,et al.  Managing morning commute with parking space constraints in the case of a bi-modal many-to-one network , 2016 .

[17]  Nicolas Coulombel,et al.  The marginal social cost of travel time variability , 2014 .

[18]  Tao Yao,et al.  A partial differential equation formulation of Vickrey’s bottleneck model, part II: Numerical analysis and computation , 2013 .

[19]  Makoto Yonekawa,et al.  Flextime, Traffic Congestion and Urban Productivity , 2006 .

[20]  Hai Yang,et al.  Day-to-day evolution of departure time choice in stochastic capacity bottleneck models with bounded rationality and various information perceptions , 2019, Transportation Research Part E: Logistics and Transportation Review.

[21]  Hai-Jun Huang,et al.  Dynamic ridesharing with variable-ratio charging-compensation scheme for morning commute , 2019, Transportation Research Part B: Methodological.

[22]  Yafeng Yin,et al.  Tradable Credit Scheme to Control Bottleneck Queue Length , 2016 .

[23]  Hai-Jun Huang,et al.  Fares and tolls in a competitive system with transit and highway: the case with two groups of commuters , 2000 .

[24]  Moshe Ben-Akiva,et al.  Dynamic network models and driver information systems , 1991 .

[25]  Paul Koster,et al.  Long‐Run Versus Short‐Run Perspectives on Consumer Scheduling: Evidence from a Revealed‐Preference Experiment Among Peak‐Hour Road Commuters , 2015 .

[26]  Xiaoning Zhang,et al.  Integrated scheduling of daily work activities and morning–evening commutes with bottleneck congestion , 2005 .

[27]  André de Palma,et al.  Congestion in a city with a central bottleneck , 2012 .

[28]  Hai-Jun Huang,et al.  Pareto-improving policies for an idealized two-zone city served by two congestible modes , 2018, Transportation Research Part B: Methodological.

[29]  Yu Nie,et al.  A New Tradable Credit Scheme for the Morning Commute Problem , 2015 .

[30]  André de Palma,et al.  Properties of Dynamic Traffic Equilibrium Involving Bottlenecks, Including a Paradox and Metering , 1993, Transp. Sci..

[31]  J. Daniel,et al.  Congestion Pricing of Canadian Airports , 2011 .

[32]  Michael J. Smith,et al.  The Existence of a Time-Dependent Equilibrium Distribution of Arrivals at a Single Bottleneck , 1984, Transp. Sci..

[33]  M. Ben-Akiva,et al.  STOCHASTIC EQUILIBRIUM MODEL OF PEAK PERIOD TRAFFIC CONGESTION , 1983 .

[34]  Xiangmin Zhang,et al.  Efficiency Comparison of Various Parking Charge Schemes Considering Daily Travel Cost in a Linear City , 2011, European Journal of Transport and Infrastructure Research.

[35]  A. Palma,et al.  Economics of a bottleneck , 1986 .

[36]  Jiancheng Long,et al.  Optimal official work start times in activity-based bottleneck models with staggered work hours , 2019 .

[37]  Amnon Rapoport,et al.  Departure Times in Y-Shaped Traffic Networks with Multiple Bottlenecks , 2009 .

[38]  Mogens Fosgerau,et al.  Commuting and land use in a city with bottlenecks: Theory and evidence , 2019, Regional Science and Urban Economics.

[39]  Jeffrey L. Adler,et al.  A direct redistribution model of congestion pricing , 2001 .

[40]  A. Palma,et al.  SCHEDULE DELAY AND DEPARTURE TIME DECISIONS WITH HETEROGENEOUS COMMUTERS , 1988 .

[41]  Paul Koster,et al.  Regulating Dynamic Congestion Externalities With Tradable Credit Schemes: Does a Unique Equilibrium Exist? , 2019, Transportation Research Part B: Methodological.

[42]  Richard Arnott,et al.  Equilibrium traffic dynamics in a bathtub model: A special case , 2016 .

[43]  M. Fosgerau Congestion in the bathtub , 2015 .

[44]  Richard Arnott,et al.  Morning Commute in a Single-Entry Traffic Corridor with No Late Arrivals , 2012 .

[45]  Hai Yang,et al.  Modeling and managing morning commute with both household and individual travels , 2017 .

[46]  Vincent A.C. van den Berg Step-tolling with price-sensitive demand: Why more steps in the toll make the consumer better off , 2012 .

[47]  Yuichiro Yoshida,et al.  Commuter arrivals and optimal service in mass transit: Does queuing behavior at transit stops matter? , 2008 .

[48]  Amnon Rapoport,et al.  Vickrey's model of traffic congestion discretized , 2008 .

[49]  Kuan-Yu Chen,et al.  Effects of the optimal port queuing pricing on arrival decisions for container ships , 2007 .

[50]  A. Palma,et al.  Modelling and evaluation of road pricing in Paris , 2006 .

[51]  Mogens Fosgerau,et al.  Travel time variability and rational inattention , 2017 .

[52]  Kenneth A. Small,et al.  THE SCHEDULING OF CONSUMER ACTIVITIES: WORK TRIPS , 1982 .

[53]  André de Palma,et al.  Route choice with heterogeneous drivers and group-specific congestion costs , 1992 .

[54]  Hai Yang,et al.  Interactive travel choices and traffic forecast in a doubly dynamical system with user inertia and information provision , 2017 .

[55]  Ziyou Gao,et al.  Morning commute problem with queue-length-dependent bottleneck capacity , 2019, Transportation Research Part B: Methodological.

[56]  Ziyou Gao,et al.  A single-step-toll equilibrium for the bottleneck model with dropped capacity , 2016 .

[57]  G. F. Newell Traffic Flow for the Morning Commute , 1988, Transp. Sci..

[58]  Yang Liu,et al.  Morning commute problem considering route choice, user heterogeneity and alternative system optima , 2011 .

[59]  Ian Savage,et al.  Tolling roads to improve reliability , 2019, Journal of Urban Economics.

[60]  Peter P. Belobaba,et al.  Survey Paper - Airline Yield Management An Overview of Seat Inventory Control , 1987, Transp. Sci..

[61]  André de Palma,et al.  Private operators and time-of-day tolling on a congested road network , 2008 .

[62]  Hai Yang,et al.  Managing rail transit peak-hour congestion with a fare-reward scheme , 2018 .

[63]  R. Braid Uniform versus peak-load pricing of a bottleneck with elastic demand , 1989 .

[64]  Mogens Fosgerau,et al.  Road pricing with complications , 2013 .

[65]  M. Fosgerau,et al.  Valuing travel time variability: Characteristics of the travel time distribution on an urban road , 2012 .

[66]  Carlos F. Daganzo,et al.  A Pareto Improving Strategy for the Time-Dependent Morning Commute Problem , 1999, Transp. Sci..

[67]  Robert B. Noland,et al.  Commuter Responses to Travel Time Uncertainty under Congested Conditions: Expected Costs and the Provision of Information , 1997 .

[68]  Kenneth Button,et al.  Road pricing, traffic congestion and the environment : issues of efficiency and social feasibility , 1998 .

[69]  Ralph M. Braid Partial peak-load pricing of a transportation bottleneck with homogeneous and heterogeneous values of time , 2018 .

[70]  Marc Willinger,et al.  Road Traffic Congestion and Public Information: An Experimental Investigation , 2006 .

[71]  Eleni Christofa,et al.  Empirical assessment of bottleneck congestion with a constant and peak toll: San Francisco–Oakland Bay Bridge , 2014, EURO J. Transp. Logist..

[72]  Robin Lindsey,et al.  Trip-timing decisions with traffic incidents , 2013 .

[73]  Xiaoning Zhang,et al.  Analysis of User Equilibrium Traffic Patterns on Bottlenecks with Time-Varying Capacities and their Applications , 2010 .

[74]  André de Palma,et al.  Dynamic Model of Peak Period Traffic Congestion with Elastic Arrival Rates , 1986, Transp. Sci..

[75]  Feng Xiao,et al.  Managing bottleneck congestion with tradable credits , 2013 .

[76]  A. Palma,et al.  The economics of crowding in rail transit , 2017 .

[77]  Kuan-Yu Chen,et al.  Economics on the optimal port queuing pricing to bulk ships , 2009 .

[78]  Satish V. Ukkusuri,et al.  Linear Complementarity Formulation for Single Bottleneck Model with Heterogeneous Commuters , 2010 .

[79]  Hai-Jun Huang,et al.  Morning commute in a single-entry traffic corridor with early and late arrivals , 2017 .

[80]  Joseph I. Daniel,et al.  Congestion Pricing and Capacity of Large Hub Airports: A Bottleneck Model with Stochastic Queues , 1995 .

[81]  Chen-Hsiu Laih,et al.  Effects of the optimal n-step toll scheme on bulk carriers queuing for multiple berths at a busy port , 2013 .

[82]  Wei Shen,et al.  Pareto-Improving Ramp Metering Strategies for Reducing Congestion in the Morning Commute , 2009 .

[83]  Carlos F. Daganzo System optimum and pricing for the day-long commute with distributed demand, autos and transit , 2013 .

[84]  Romeo Danielis,et al.  Bottleneck road congestion pricing with a competing railroad service , 2002 .

[85]  Hong Kam Lo,et al.  Punctuality-based route and departure time choice , 2014 .

[86]  Feng Xiao,et al.  The morning commute under flat toll and tactical waiting , 2012 .

[87]  A. Palma,et al.  The valuation of travel time reliability: does congestion matter? , 2017 .

[88]  Erik T. Verhoef,et al.  Give or take? Rewards vs. Charges for a Congested Bottleneck , 2012 .

[89]  Carlos F. Daganzo,et al.  The Uniqueness of a Time-dependent Equilibrium Distribution of Arrivals at a Single Bottleneck , 1985, Transp. Sci..

[90]  Sofia F. Franco Downtown Parking Supply, Work-Trip Mode Choice and Urban Spatial Structure , 2015 .

[91]  Mogens Fosgerau,et al.  How a fast lane may replace a congestion toll , 2011 .

[92]  Peng Liu,et al.  Optimal Information Provision at Bottleneck Equilibrium with Risk-Averse Travelers , 2018, Transportation Research Record: Journal of the Transportation Research Board.

[93]  Zhen Qian,et al.  The morning commute problem with heterogeneous travellers: the case of continuously distributed parameters , 2013 .

[94]  Xiaolei Guo,et al.  Constrained optimization for bottleneck coarse tolling , 2019 .

[95]  Vincent A. C. van den Berg,et al.  Winning or losing from dynamic bottleneck congestion pricing?: The distributional effects of road pricing with heterogeneity in values of time and schedule delay , 2011 .

[96]  Wei Liu,et al.  Effectiveness of variable speed limits considering commuters’ long-term response , 2015 .

[97]  Mogens Fosgerau,et al.  The value of travel time variance , 2010 .

[98]  Huijun Sun,et al.  Overcoming the Downs-Thomson Paradox by Transit Subsidy Policies , 2017 .

[99]  Mike Mingcheng Wei,et al.  Congestion derivatives for a traffic bottleneck , 2010 .

[100]  Kab Sung Kim,et al.  Traffic Congestion at a Two-Tandem Bottleneck , 1999 .

[101]  Hai-Jun Huang,et al.  Stochastic bottleneck capacity, merging traffic and morning commute , 2014 .

[102]  Julie Bulteau,et al.  Revisiting the bottleneck congestion model by considering environmental costs and a modal policy , 2016 .

[103]  Hai-Jun Huang Pricing and logit-based mode choice models of a transit and highway system with elastic demand , 2002, Eur. J. Oper. Res..

[104]  Masao Kuwahara Equilibrium Queueing Patterns at a Two-Tandem Bottleneck during the Morning Peak , 1990, Transp. Sci..

[105]  Jeppe Rich,et al.  Testing the slope model of scheduling preferences on stated preference data , 2017 .

[106]  Yurii Nesterov,et al.  METROPOLIS: Modular System for Dynamic Traffic Simulation , 1997 .

[107]  Joseph I. Daniel,et al.  Pricing the major US hub airports , 2009 .

[108]  Ludo Waltman,et al.  Visualizing Bibliometric Networks , 2014 .

[109]  Richard Arnott,et al.  The corridor problem: Preliminary results on the no-toll equilibrium , 2011 .

[110]  Robin Lindsey Existence, Uniqueness, and Trip Cost Function Properties of User Equilibrium in the Bottleneck Model with Multiple User Classes , 2004, Transp. Sci..

[111]  Robert B. Noland,et al.  Simulating Travel Reliability , 1997 .

[112]  Joseph I. Daniel,et al.  (When) Do hub airlines internalize their self-imposed congestion delays? , 2008 .

[113]  Changmin Jiang,et al.  Airport congestion pricing and terminal investment: Effects of terminal congestion, passenger types, and concessions , 2015 .

[114]  Tao Yao,et al.  A partial differential equation formulation of Vickrey’s bottleneck model, part I: Methodology and theoretical analysis , 2013 .

[115]  E. Jenelius The value of travel time variability with trip chains, flexible scheduling and correlated travel times , 2012 .

[116]  H. Michael Zhang,et al.  Numerical solution procedures for the morning commute problem , 2009, Math. Comput. Model..

[117]  Kentaro Wada,et al.  The corridor problem with discrete multiple bottlenecks , 2015 .

[118]  Nikolaos Geroliminis,et al.  Modeling the morning commute for urban networks with cruising-for-parking: An MFD approach , 2016 .

[119]  Xiaoning Zhang,et al.  Integrated daily commuting patterns and optimal road tolls and parking fees in a linear city , 2008 .

[120]  David A. Hensher,et al.  Handbook of Transport Systems and Traffic Control , 2001 .

[121]  N. Geroliminis,et al.  On the use of reservation-based autonomous vehicles for demand management , 2017 .

[122]  Se-il Mun PEAK-LOAD PRICING OF A BOTTLENECK WITH TRAFFIC JAM , 1999 .

[123]  Erik T. Verhoef,et al.  Private Road Supply in Networks with Heterogeneous Users , 2017, Transportation Research Part A: Policy and Practice.

[124]  Erik T. Verhoef,et al.  Bottleneck Congestion: Differentiating the Coarse Charge , 2010 .

[125]  Hai Yang,et al.  Efficiency of a highway use reservation system for morning commute , 2015 .

[126]  André de Palma,et al.  The dynamics of urban traffic congestion and the price of parking , 2013 .

[127]  Yang Liu,et al.  Departure time and route choices in bottleneck equilibrium under risk and ambiguity , 2017, Transportation Research Part B: Methodological.

[128]  André de Palma,et al.  On the Existence and Uniqueness of Equilibrium in the Bottleneck Model with Atomic Users , 2014, Transp. Sci..

[129]  Hong Kam Lo,et al.  Punctuality-based departure time scheduling under stochastic bottleneck capacity: formulation and equilibrium , 2013 .

[130]  Chen-Hsiu Laih,et al.  QUEUEING AT A BOTTLENECK WITH SINGLE- AND MULTI-STEP TOLLS , 1994 .

[131]  Eric J. Gonzales,et al.  Analytical equilibrium of bicriterion choices with heterogeneous user preferences: application to the morning commute problem , 2017 .

[132]  David Z.W. Wang,et al.  Traffic managements for household travels in congested morning commute , 2016 .

[133]  André de Palma,et al.  Usage-dependent peak-load pricing , 1986 .

[134]  Hai-Jun Huang,et al.  Congestion Behavior and Tolls in a Bottleneck Model with Stochastic Capacity , 2015, Transp. Sci..

[135]  Yu Nie,et al.  A Semi-Analytical Approach for Solving the Bottleneck Model with General User Heterogeneity , 2014 .

[136]  Hui Zhao,et al.  Impact of Capacity Drop on Commuting Systems under Uncertainty , 2018, Journal of Advanced Transportation.

[137]  André de Palma,et al.  Information and Time-of-Usage Decisions in the Bottleneck Model with Stochastic Capacity and Demand , 1999 .

[138]  Kuan-Yu Chen,et al.  Economics on the optimal n-step toll scheme for a queuing port , 2008 .

[139]  Xiaoning Zhang,et al.  Improving travel efficiency by parking permits distribution and trading , 2011 .

[140]  Ralph M. Braid,et al.  Peak-Load Pricing of a Transportation Route with an Unpriced Substitute , 1996 .

[141]  Hai Yang,et al.  Tradable credit schemes for managing bottleneck congestion and modal split with heterogeneous users , 2013 .

[142]  Marvin Kraus,et al.  A new look at the two-mode problem , 2003 .

[143]  Hai-Jun Huang,et al.  Equilibrium and Modal Split in a Competitive Highway/Transit System Under Different Road-use Pricing Strategies , 2014 .

[144]  T. Rutherford,et al.  Regional labor markets, commuting, and the economic impact of road pricing , 2018, Regional Science and Urban Economics.

[145]  Haijun Huang,et al.  Mathematical and Economic Theory of Road Pricing , 2005 .

[146]  Carlos F. Daganzo,et al.  Morning Commute with Competing Modes and Distributed Demand: User Equilibrium, System Optimum, and Pricing , 2012 .

[147]  William H. K. Lam,et al.  Step tolling in an activity-based bottleneck model , 2017 .

[148]  Erik T. Verhoef,et al.  Robot Cars and Dynamic Bottleneck Congestion: The Effects on Capacity, Value of Time and Preference Heterogeneity , 2016 .

[149]  Mogens Fosgerau,et al.  Additive measures of travel time variability , 2011 .

[150]  Y. Cohen,et al.  COMMUTER WELFARE UNDER PEAK-PERIOD CONGESTION TOLLS : WHO GAINS AND WHO LOSES? , 1987 .

[151]  Kenneth A. Small,et al.  The bottleneck model: An assessment and interpretation , 2015 .

[152]  Mogens Fosgerau,et al.  Endogenous Scheduling Preferences and Congestion , 2017 .

[153]  Maria Börjesson,et al.  Valuations of travel time variability in scheduling versus mean-variance models , 2012 .

[154]  Ziyou Gao,et al.  Modal Split and Commuting Pattern on a Bottleneck-Constrained Highway , 2007 .

[155]  Hai-Jun Huang,et al.  Tradable permit schemes for managing morning commute with carpool under parking space constraint , 2019, Transportation.

[156]  Wei Liu,et al.  An equilibrium analysis of commuter parking in the era of autonomous vehicles , 2018, Transportation Research Part C: Emerging Technologies.

[157]  Nikolas Geroliminis,et al.  The morning commute in urban areas with heterogeneous trip lengths , 2018, Transportation Research Part B: Methodological.

[158]  F. Mirabel,et al.  Bottleneck congestion pricing and modal split: Redistribution of toll revenue , 2011 .

[159]  Joseph I. Daniel,et al.  Comparison of Three Empirical Models of Airport Congestion Pricing , 2000 .

[160]  Yu Xiao,et al.  On the cost of misperceived travel time variability , 2015 .

[161]  André de Palma,et al.  Private roads, competition, and incentives to adopt time-based congestion tolling , 2002 .

[162]  Erik T. Verhoef,et al.  Regulating dynamic congestion externalities with tradable credit schemes: Does a unique equilibrium exist? , 2019, Transportation Research Part B: Methodological.

[163]  Chris Hendrickson,et al.  Schedule Delay and Departure Time Decisions in a Deterministic Model , 1981 .

[164]  Erik T. Verhoef,et al.  Dynamic bottleneck congestion and residential land use in the monocentric city , 2014 .

[165]  Richard Arnott,et al.  CONGESTION TOLLING AND URBAN SPATIAL STRUCTURE , 1998 .

[166]  Ziyou Gao,et al.  Investigation of the traffic congestion during public holiday and the impact of the toll-exemption policy , 2017 .

[167]  Jinwon Kim,et al.  Estimating the Social Cost of Congestion Using the Bottleneck Model , 2019, Economics of Transportation.

[168]  Nanne J. Van Der Zijpp,et al.  Multiclass Continuous-Time Equilibrium Model for Departure Time Choice on Single-Bottleneck Network , 2002 .

[169]  Hai Yang,et al.  On the morning commute problem with bottleneck congestion and parking space constraints , 2013 .

[170]  André de Palma,et al.  The Welfare Effects Of Congestion Tolls With Heterogeneous Commuters , 1993 .

[171]  David M Levinson,et al.  Traveler delay costs and value of time with trip chains, flexible activity scheduling and information , 2011 .

[172]  Eric Pels,et al.  The User Costs of Air Travel Delay Variability , 2013, Transp. Sci..

[173]  Hai Yang,et al.  A novel permit scheme for managing parking competition and bottleneck congestion , 2014 .

[174]  R. Lindsey,et al.  Private toll roads: Competition under various ownership regimes , 2000 .

[175]  Takatoshi Tabuchi,et al.  Bottleneck Congestion and Modal Split , 1993 .

[176]  Moshe Ben-Akiva,et al.  Dynamic model of peak period congestion , 1984 .

[177]  Zhen Sean Qian,et al.  Modeling Multi-Modal Morning Commute in a One-to-One Corridor Network , 2011 .

[178]  L. Engelson,et al.  The cost of travel time variability: three measures with properties , 2016 .

[179]  André de Palma,et al.  Does providing information to drivers reduce traffic congestion , 1991 .

[180]  Vincent A. C. van den Berg,et al.  Congestion Tolling in the Bottleneck Model with Heterogeneous Values of Time , 2011 .

[181]  Shing Chung Josh Wong,et al.  Reliability Evaluation for Stochastic and Time-dependent Networks with Multiple Parking Facilities , 2008 .

[182]  Hai Yang,et al.  Analysis of the time-varying pricing of a bottleneck with elastic demand using optimal control theory , 1997 .

[183]  Saurabh Amin,et al.  Bottleneck model with heterogeneous information , 2018, Transportation Research Part B: Methodological.

[184]  A. Palma,et al.  A STRUCTURAL MODEL OF PEAK-PERIOD CONGESTION: A TRAFFIC BOTTLENECK WITH ELASTIC DEMAND. IN: RECENT DEVELOPMENTS IN TRANSPORT ECONOMICS , 1993 .

[185]  Hai-Jun Huang,et al.  User equilibrium of a single-entry traffic corridor with continuous scheduling preference , 2018 .

[186]  Masao Kuwahara,et al.  Bottleneck congestion and residential location of heterogeneous commuters , 2017 .

[187]  Ronghui Liu,et al.  Pareto-improving social optimal pricing schemes based on bottleneck permits for managing congestion at a merging section , 2017 .

[188]  M. Saberi,et al.  Congestion pricing practices and public acceptance: A review of evidence , 2018 .

[189]  Carlos F. Daganzo,et al.  Spillovers, Merging Traffic and the Morning Commute , 2007 .

[190]  William P. O'Dea,et al.  Optimal Metering in the Bottleneck Congestion Model , 1999 .

[191]  Kenneth A. Small Trip scheduling in urban transportation analysis , 1992 .

[192]  Mike Mingcheng Wei,et al.  Congestion derivatives for a traffic bottleneck with heterogeneous commuters , 2012 .

[193]  Robert B. Noland,et al.  Travel-time uncertainty, departure time choice, and the cost of morning commutes , 1995 .

[194]  Yuki Takayama,et al.  Bottleneck congestion and distribution of work start times: The economics of staggered work hours revisited , 2014 .

[195]  Richard Arnott,et al.  A Bathtub Model of Downtown Traffic Congestion , 2013 .

[196]  Erik T. Verhoef,et al.  Step tolling with bottleneck queuing congestion , 2012 .

[197]  Mogens Fosgerau On the relation between the mean and variance of delay in dynamic queues with random capacity and demand , 2008 .

[198]  Erik T. Verhoef,et al.  Carpooling with heterogeneous users in the bottleneck model , 2019, Transportation Research Part B: Methodological.

[199]  André de Palma,et al.  A Temporal and Spatial Equilibrium Analysis of Commuter Parking , 1991 .

[200]  H. M. Zhang,et al.  The morning commute problem with ridesharing and dynamic parking charges , 2017 .

[201]  Yafeng Yin,et al.  Optimal multi-step toll design under general user heterogeneity , 2015 .

[202]  K. Small,et al.  Hypercongestion in downtown metropolis , 2013 .

[203]  S. Travis Waller,et al.  Modelling and managing the integrated morning-evening commuting and parking patterns under the fully autonomous vehicle environment , 2019, Transportation Research Part B: Methodological.

[204]  Yuichiro Yoshida,et al.  The Commuter's Time-of-Use Decision and Optimal Pricing and Service in Urban Mass Transit , 2002 .

[205]  W. Vickrey PRICING, METERING, AND EFFICIENTLY USING URBAN TRANSPORTATION FACILITIES , 1973 .

[206]  Hai-Jun Huang,et al.  The morning commute problem with endogenous shared autonomous vehicle penetration and parking space constraint , 2019, Transportation Research Part B: Methodological.

[207]  Hai Yang,et al.  Day-To-Day Departure Time Choice under Bounded Rationality in the Bottleneck Model , 2017 .

[208]  Robin Lindsey,et al.  DEPARTURE TIME AND ROUTE CHOICE FOR THE MORNING COMMUTE , 1990 .

[209]  Hai Yang,et al.  Managing network mobility with tradable credits , 2011 .

[210]  Fabrice Marchal,et al.  Real Cases Applications of the Fully Dynamic METROPOLIS Tool-Box: An Advocacy for Large-Scale Mesoscopic Transportation Systems , 2002 .

[211]  Vincent A. C. van den Berg Coarse Tolling with Heterogeneous Preferences , 2014 .

[212]  André de Palma,et al.  Trip-timing decisions and congestion with household scheduling preferences , 2015 .

[213]  Satish V. Ukkusuri,et al.  On the existence of pricing strategies in the discrete time heterogeneous single bottleneck model , 2011 .

[214]  R. Arnott,et al.  Financing Capacity in the Bottleneck Model , 1995 .

[215]  Hai Yang,et al.  Optimal utilization of a transport system with auto/transit parallel modes , 1999 .

[216]  Hai-Jun Huang,et al.  An ordinary differential equation formulation of the bottleneck model with user heterogeneity , 2015 .

[217]  Kenneth A. Small,et al.  Valuation of Travel Time , 2012 .

[218]  Erik T. Verhoef,et al.  Equilibrium at a Bottleneck When Long-Run and Short-Run Scheduling Preferences Diverge , 2013 .

[219]  Yu-Chen Tsai,et al.  Optimal non-queuing pricing for the Suez Canal , 2015 .

[220]  Erik T. Verhoef,et al.  Value of Time by Time of Day: A Stated-Preference Study , 2007 .

[221]  Fabien Leurent,et al.  User Equilibrium in a Bottleneck under Multipeak Distribution of Preferred Arrival Time , 2009 .

[222]  Hai Yang,et al.  Equilibrium properties of the morning peak-period commuting in a many-to-one mass transit system , 2007 .

[223]  Carlos F. Daganzo,et al.  The Evening Commute with Cars and Transit: Duality Results and User Equilibrium for the Combined Morning and Evening Peaks , 2013 .

[224]  Chen-Hsiu Laih,et al.  Effects of the optimal step toll scheme on equilibrium commuter behaviour , 2004 .

[225]  Yafeng Yin,et al.  Expirable parking reservations for managing morning commute with parking space constraints , 2014 .

[226]  Mogens Fosgerau,et al.  Vickrey meets Alonso: Commute scheduling and congestion in a monocentric city , 2018 .

[227]  Hai-Jun Huang,et al.  Modeling the modal split and trip scheduling with commuters' uncertainty expectation , 2015, Eur. J. Oper. Res..

[228]  E. Verhoef,et al.  Congestion pricing in a road and rail network with heterogeneous values of time and schedule delay , 2014 .

[229]  André de Palma,et al.  Equilibrium in a dynamic model of congestion with large and small users , 2019, Transportation Research Part B: Methodological.

[230]  André de Palma,et al.  Recent Developments in the Bottleneck Model , 1995 .

[231]  Ludo Waltman,et al.  A smart local moving algorithm for large-scale modularity-based community detection , 2013, The European Physical Journal B.

[232]  Yang Liu,et al.  Solving the Step-Tolled Bottleneck Model with General User Heterogeneity , 2015 .

[233]  André de Palma,et al.  Congestion pricing on a road network: A study using the dynamic equilibrium simulator METROPOLIS , 2005 .

[234]  W. Vickrey Congestion Theory and Transport Investment , 1969 .

[235]  Yafeng Yin,et al.  Managing rush hour travel choices with tradable credit scheme , 2013 .

[236]  Mogens Fosgerau,et al.  Commuting for meetings , 2014 .

[237]  Kazuhiro Yoshikawa,et al.  Optimal cordon pricing in a non-monocentric city , 2005 .

[238]  Sze Chun Wong,et al.  Bottleneck model revisited: An activity-based perspective , 2014 .

[239]  Yuanyuan Li,et al.  Pricing scheme design of ridesharing program in morning commute problem , 2017 .

[240]  Xeuhao Chu,et al.  Endogenous Trip Scheduling: The Henderson Approach Reformulated and Compared with the Vickrey Approach , 1993 .

[241]  Feng Xiao,et al.  The economics of parking provision for the morning commute , 2011 .

[242]  Bo Wang,et al.  A Pareto-improving and revenue-neutral scheme to manage mass transit congestion with heterogeneous commuters , 2020, Transportation Research Part C: Emerging Technologies.

[243]  Hai Yang,et al.  A new look at the morning commute with household shared-ride: how does school location play a role? , 2017 .

[244]  Shing Chung Josh Wong,et al.  Modeling time-dependent travel choice problems in road networks with multiple user classes and multiple parking facilities , 2006 .

[245]  Lingling Xu,et al.  Equilibrium trip scheduling in single bottleneck traffic flows considering multi-class travellers and uncertainty – a complementarity formulation , 2016 .