Applied Mathematics
Models for Vehicular Traffic on Networks
By Mauro Garavello, Ke Han and Benedetto Piccoli
ISBN-10: 1-60133-019-7
ISBN-13: 978-1-60133-019-2
This book presents a theory of modeling, analyzing, computing, and optimizing traffic flows on networks. It provides comprehensive and self-contained descriptions and analyses on the widely used conservation law models for dynamic trafic flows on networks.
The book covers theoretical aspects of these models using novel and rigorous mathematical techniques, while seeking a balance between theory and practice by including a considerable number of examples and applications.
Book Order Information
Contents Author Bio.
The main topics covered in this books are:
- First- and high-order conservation law models
- Network extensions of fluid-based models
- Dynamic network traffic assignment
- Control and optimization of traffic flow at urban intersections
- Extension of network models to incorporate heterogeneous road conditions
- Numerical schemes and computational examples of traffic network models
- Applications to traffic sensing,traffic control, and road network design.
This book encompasses a wide range of topics in mathematics and engineering and is meant to be a reference and guide to scholars and practitioners. The book also offers materials that support theme-based design of 1-2 semesters of curriculum.
To View /Download Contents and Chapter1 (Introduction)
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1
1.1 Review of some trafic flow models . . . . . . . . . . . . . . . . . . .2
1.1.1 Non-physical queue models . . . . . . . . . . . . . . . . . . . . . .2
1.1.2 Physical-queue models . . . . . . . . . . . . . . . . . . . . . . . .4
1.2 Examples of trafic modeling . . . . . . . . . . . . . . . . . . . . . .6
1.2.1 Shock wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6
1.2.2 Spillback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6
1.2.3 Gridlock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9
1.2.4 Trafic signal control . . . . . . . . . . . . . . . . . . . . . . . .10
1.2.5 Car trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . .14
1.2.6 Second-order extension of the LWR model . . . . . . . . . . . . . . .15
1.3 State of the art and challenges . . . . . . . . . . . . . . . . . . . .17
1.4 The materials ahead . . . . . . . . . . . . . . . . . . . . . . . . . .19
2 Macroscopic and Mesoscopic Trafic Models . . . . . . . . . . . . . . . . .21
2.1 The LWR model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22
2.1.1 Derivation of the equation . . . . . . . . . . . . . . . . . . . . . .22
2.1.2 Fundamental diagrams . . . . . . . . . . . . . . . . . . . . . . . . .23
2.1.3 Riemann problems. . . . . . . . . . . . . . . . . . . . . . . . . . .25
2.1.4 LWR model with viscosity . . . . . . . . . . . . . . . . . . . . . . .26
2.1.5 Hamilton-Jacobi formulation . . . . . . . . . . . . . . . . . . . . .28
2.1.6 LWR model in other coordinate systems. . . . . . . . . . . . . . . . .30
2.2 The Payne-Whitham model . . . . . . . . . . . . . . . . . . . . . . . .33
2.3 Drawbacks of second order models . . . . . . . . . . . . . . . . . . . .35
2.4 The Aw-Rascle-Zhang model . . . . . . . . . . . . . . . . . . . . . . .36
2.4.1 Characteristic fields . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4.2 Invariant domains . . . . . . . . . . . . . . . . . . . . . . . . . .37
2.5 Third order models . . . . . . . . . . . . . . . . . . . . . . . . . . .39
2.6 Hyperbolic phase transition model . . . . . . . . . . . . . . . . . . .40
2.6.1 The Riemann problem . . . . . . . . . . . . . . . . . . . . . . . . .42
2.7 The Greenberg-Klar-Rascle multilane model . . . . . . . . . . . . . . .44
XIV Contents
2.8 A multipopulation model . . . . . . . . . . . . . . . . . . . . . . . .45
2.8.1 The case n = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . .47
2.9 A multiclass model with creeping . . . . . . . . . . . . . . . . . . . .48
2.10 The Aw-Rascle-Zhang model with phase transition . . . . . . . . . . . .49
2.11 The Siebel-Mauser balanced vehicular trafic model . . . . . . . . . . 49
2.12 Generalized Aw-Rascle-Zhang models . . . . . . . . . . . . . . . . . .50
2.13 No-gap phase transition models . . . . . . . . . . . . . . . . . . . .51
2.13.1 LWR and follow-the-leader model . . . . . . . . . . . . . . . . . . .53
2.14 Qualitative features of models for urban trafic . . . . . . . . . . . 54
2.14.1 First order models . . . . . . . . . . . . . . . . . . . . . . . . .55
2.14.2 Second order models . . . . . . . . . . . . . . . . . . . . . . . . .57
2.14.3 The Helbing third order model . . . . . . . . . . . . . . . . . . . .63
2.14.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .66
2.15 Mesoscopic models . . . . . . . . . . . . . . . . . . . . . . . . . . .67
2.15.1 The Prigogine model . . . . . . . . . . . . . . . . . . . . . . . . .68
2.15.2 The Paveri-Fontana model . . . . . . . . . . . . . . . . . . . . . .69
2.15.3 Enskog-like models . . . . . . . . . . . . . . . . . . . . . . . . .69
2.15.4 Discrete velocity models . . . . . . . . . . . . . . . . . . . . . .70
2.16 Bibliographical note . . . . . . . . . . . . . . . . . . . . . . . . .71
3 Models for Trafic on Networks . . . . . . . . . . . . . . . . . . . . . . 73
3.1 Discrete models . . . . . . . . . . . . . . . . . . . . . . . . . . . .74
3.1.1 The cell transmission model . . . . . . . . . . . . . . . . . . . . .74
3.1.2 The link transmission model . . . . . . . . . . . . . . . . . . . . .78
3.2 Single conservation law models . . . . . . . . . . . . . . . . . . . . .80
3.2.1 Solution for 1 1 junction . . . . . . . . . . . . . . . . . . . . .83
3.2.2 Solution for 2 1 (merge) junction . . . . . . . . . . . . . . . . .83
3.2.3 Solution for 1 2 (diverge) junction . . . . . . . . . . . . . . . .85
3.2.4 Solution for 2 2 junction . . . . . . . . . . . . . . . . . . . . .86
3.2.5 A solver redirecting trafic . . . . . . . . . . . . . . . . . . . . . 87
3.2.6 A solver keeping priorities and not redirecting trafic . . . . . . . .89
3.2.7 A solver for T-junctions . . . . . . . . . . . . . . . . . . . . . . .90
3.3 Comparison of discrete and conservation laws models . . . . . . . . . .91
3.3.1 Comparison of the CTM with the conservation law model . . . . . . . .92
3.3.2 Comparison of the LTM with the variational formulation . . . . . . . .93
3.4 The Aw-Rascle-Zhang model on networks . . . . . . . . . . . . . . . . .100
3.4.1 A model conserving the number of cars . . . . . . . . . . . . . . . .100
3.4.2 A model conserving the number of cars and the generalized momentum . .102
3.5 Phase transition models on networks . . . . . . . . . . . . . . . . . . 103
3.6 Coupling of models . . . . . . . . . . . . . . . . . . . . . . . . . . .105
3.6.1 Follow-the-leader model . . . . . . . . . . . . . . . . . . . . . . . 106
3.6.2 Follow-the-leader and LWR . . . . . . . . . . . . . . . . . . . . . . 106
3.6.3 Follow-the-Leader and ARZ . . . . . . . . . . . . . . . . . . . . . . 107
Contents XV
3.6.4 Follow-the-leader and phase transition . . . . . . . . . . . . . . . 109
3.6.5 LWR and phase transition models . . . . . . . . . . . . . . . . . . . 109
3.6.6 CTM and LWR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.7 Bibliographical note . . . . . . . . . . . . . . . . . . . . . . . . . .111
4 Mathematical Theory for Trafic on Networks . . . . . . . . . . . . . . . .113
4.1 Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .114
4.2 The LWR model at a junction . . . . . . . . . . . . . . . . . . . . . . 114
4.2.1 The Riemann problem . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.2.2 Examples of Riemann solvers . . . . . . . . . . . . . . . . . . . . . 119
4.2.3 The Cauchy problem. . . . . . . . . . . . . . . . . . . . . . . . . . 134
4.2.4 Existence of a wave-front tracking solution . . . . . . . . . . . . 139
4.2.5 Dependence of solutions on initial data . . . . . . . . . . . . . . . 143
4.2.6 Some technical lemmas . . . . . . . . . . . . . . . . . . . . . . . . 147
4.3 Time-varying trafic distribution coeficients . . . . . . . . . . . . . .148
4.3.1 Family of Riemann solvers . . . . . . . . . . . . . . . . . . . . . . 148
4.3.2 Examples of families of Riemann solvers . . . . . . . . . . . . . . . 153
4.4 Entropy conditions at junctions . . . . . . . . . . . . . . . . . . . . 159
4.4.1 Riemann solvers satisfying an entropy condition . . . . . . . . . . . 162
4.4.2 Riemann solvers at a general junction . . . . . . . . . . . . . . . . 164
4.5 The system case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
4.6 Bibliographical note . . . . . . . . . . . . . . . . . . . . . . . . . .174
5 Source Destination and Bu er Models . . . . . . . . . . . . . . . . . . . 175
5.1 Basic source destination model . . . . . . . . . . . . . . . . . . . . .176
5.1.1 Trafic distribution at junctions . . . . . . . . . . . . . . . . . . .177
5.1.2 Evolution equations for trafic-type functions . . . . . . . . . . . . 178
5.1.3 The Riemann problem . . . . . . . . . . . . . . . . . . . . . . . . . 179
5.1.4 Wave-front tracking algorithm . . . . . . . . . . . . . . . . . . . . 185
5.2 Models with bufiers . . . . . . . . . . . . . . . . . . . . . . . . . . 186
5.2.1 Single bu er models . . . . . . . . . . . . . . . . . . . . . . . . . 186
5.2.2 Multi-bu er models . . . . . . . . . . . . . . . . . . . . . . . . . .188
5.2.3 The Bressan-Nguyen model with bu ers . . . . . . . . . . . . . . . . .197
5.3 Bibliographical note . . . . . . . . . . . . . . . . . . . . . . . . . .198
6 Dynamic Trafic Assignment . . . . . . . . . . . . . . . . . . . . . . . . 199
6.1 Notion of dynamic user equilibrium . . . . . . . . . . . . . . . . . . .200
6.2 Mathematical representations of dynamic user equilibrium . . . . . . . .200
6.2.1 Notation and essential background . . . . . . . . . . . . . . . . . . 201
6.2.2 Dynamic user equilibrium . . . . . . . . . . . . . . . . . . . . . . .203
6.3 Dynamic network loading . . . . . . . . . . . . . . . . . . . . . . . . 205
6.3.1 Formulation as a system of partial difierential algebraic equations . 205
6.3.2 Formulation as a system of difierential algebraic equations . . . . .212
XVI Contents
6.4 Continuity of the path delay operator . . . . . . . . . . . . . . . . . 218
6.4.1 The two junction models . . . . . . . . . . . . . . . . . . . . . . . 219
6.4.2 Well-posedness of the diverge junction model . . . . . . . . . . . . .221
6.4.3 Well-posedness of the merge junction model . . . . . . . . . . . . . .226
6.4.4 Continuity of the delay operator . . . . . . . . . . . . . . . . . . .226
6.5 Existence of dynamic user equilibrium . . . . . . . . . . . . . . . . . 235
6.5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
6.5.2 The existence result . . . . . . . . . . . . . . . . . . . . . . . . .236
6.6 Computation of dynamic user equilibrium . . . . . . . . . . . . . . . . 242
6.6.1 The di erential variational inequality formulation . . . . . . . . . .242
6.6.2 Solution algorithm based on the xed-point iterations . . . . . . . . 243
6.7 Bibliographical note . . . . . . . . . . . . . . . . . . . . . . . . . .246
7 Control of Trafic at Junctions . . . . . . . . . . . . . . . . . . . . . .247
7.1 Modeling signalized intersections . . . . . . . . . . . . . . . . . . . 248
7.2 The on-and-o and the continuum signal models . . . . . . . . . . . . . 250
7.2.1 The Hamilton-Jacobi representation of signal models . . . . . . . .. .251
7.2.2 When spillback is absent . . . . . . . . . . . . . . . . . . . . . . .252
7.2.3 When spillback is present and sustained . . . . . . . . . . . . . . . 255
7.2.4 When spillback is present and transient . . . . . . . . . . . . . . . 264
7.3 Optimization of trafic signal timing . . . . . . . . . . . . . . . . . .266
7.3.1 Mixed integer linear program approach . . . . . . . . . . . . . . . . 267
7.3.2 Metaheuristic approach . . . . . . . . . . . . . . . . . . . . . . . .270
7.4 Flow models for lights vs circles . . . . . . . . . . . . . . . . . . . 275
7.4.1 Flux control with trafic lights . . . . . . . . . . . . . . . . . . . 275
7.4.2 Single-lane trafic circle with low trafic . . . . . . . . . . . . . . 279
7.4.3 Single-lane trafic circle with heavy trafic . . . . . . . . . . . . . 281
7.4.4 Multi-lane trafic circle with no interaction . . . . . . . . . . . . .284
7.4.5 Trafic light vs trafic circle . . . . . . . . . . . . . . . . . . . . 285
7.4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
7.5 Bibliographical note . . . . . . . . . . . . . . . . . . . . . . . . . .288
8 Car Paths, Bottlenecks and Flux Limiters . . . . . . . . . . . . . . . . .289
8.1 Car paths for conservation laws models . . . . . . . . . . . . . . . . .290
8.1.1 Wave front tracking for car paths . . . . . . . . . . . . . . . . . . 290
8.2 Moving bottlenecks . . . . . . . . . . . . . . . . . . . . . . . . . . .296
8.2.1 Solutions by fractional step . . . . . . . . . . . . . . . . . . . . 298
8.3 Fixed bottlenecks and ux limiters for the LWR model . . . . . . . . . . 302
8.3.1 Flux limiters as 1-1 junctions with ux constraint . . . . . . . . . . 302
8.4 Flux limiters for the Aw-Rascle-Zhang model . . . . . . . . . . . . . . 304
8.4.1 The rst constrained Riemann solver . . . . . . . . . . . . . . . . . 306
8.4.2 The second constrained Riemann solver . . . . . . . . . . . . . . . . 306
8.5 Moving bottlenecks via ux limiters . . . . . . . . . . . . . . . . . . .307
8.6 Bibliographical note . . . . . . . . . . . . . . . . . . . . . . . . . .308
Contents XVII
9 Numerical Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . .311
9.1 Numerical approximation of conservation law models . . . . . . . . . . .312
9.1.1 The Godunov scheme . . . . . . . . . . . . . . . . . . . . . . . . . .312
9.1.2 Kinetic method for a boundary value problem . . . . . . . . . . . .. .313
9.1.3 Boundary conditions and conditions at junctions . . . . . . . . .. . .316
9.2 Simulation results for trafic ow on networks . . . . . . . . . . . . . .318
9.2.1 Bottleneck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .320
9.2.2 Trafic circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
9.3 Numerical computation of car paths . . . . . . . . . . . . . . . . . . .328
9.3.1 Simluation results for car paths . . . . . . . . . . . . . . . . . . .334
9.3.2 Application to highway accidents . . . . . . . . . . . . . . . . . . .339
9.4 Simulations for moving bottlenecks . . . . . . . . . . . . . . . . . . .340
9.5 Dynamic user equilibria on networks . . . . . . . . . . . . . . . . . . 342
9.6 Signal optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 344
9.6.1 Mixed integer linear program . . . . . . . . . . . . . . . . . . . . .344
9.6.2 Metaheuristic methods . . . . . . . . . . . . . . . . . . . . . . . . 347
9.7 Bibliographical note . . . . . . . . . . . . . . . . . . . . . . . . . .351
10 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
10.1 Trafic monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . .354
10.1.1 Mobile Millennium . . . . . . . . . . . . . . . . . . . . . . . . . .354
10.1.2 Octotelematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
10.2 Signal control with equilibrium conditions . . . . . . . . . . . . . . 376
10.2.1 Dynamic network loading with continuum signal control . . . . . . . 376
10.2.2 The MPEC formulation . . . . . . . . . . . . . . . . . . . . . . . . 378
10.2.3 Numerical example I . . . . . . . . . . . . . . . . . . . . . . . . .383
10.2.4 Numerical example II . . . . . . . . . . . . . . . . . . . . . . . . 385
10.2.5 Numerical example III . . . . . . . . . . . . . . . . . . . . . . . .387
10.3 On-line signal control . . . . . . . . . . . . . . . . . . . . . . . . 389
10.3.1 The linear decision rule . . . . . . . . . . . . . . . . . . . . . . 389
10.3.2 Data-driven calibration of the uncertainty set . . . . . . . . . . . 392
10.3.3 Numerical test . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
10.4 Trafic data fusion using generalized Lax-Hopf formula . . . . . . . . .396
10.4.1 Hamilton-Jacobi equation in Lagrangian coordinates . . . . . .. . .. 396
10.4.2 Afine value conditions . . . . . . . . . . . . . . . . . . . . . . . 397
10.4.3 Explicit Lax-Hopf formula . . . . . . . . . . . . . . . . . . . . . .397
10.4.4 Application to highway trafic estimation . . . . . . . . . . . . . . 399
10.5 Bibliographical note . . . . . . . . . . . . . . . . . . . . . . . . . 402
A Theory of Conservation Laws. . . . . . . . . . . . . . . . . . . . . . . .405
A.1 Basic de nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
A.2 Weak solutions and the method of characteristics . . . . . . . . . . . .406
A.3 Entropy admissible solutions . . . . . . . . . . . . . . . . . . . . . .415
A.3.1 Kruzkov entropy admissibility condition . . . . . . . . . . . . . . . 419
A.4 The Riemann problem . . . . . . . . . . . . . . . . . . . . . . . . . . 420
XVIII Contents
A.4.1 The scalar case . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
A.5 The Cauchy problem . . . . . . . . . . . . . . . . . . . . . . . . . . .428
A.5.1 Wave-front tracking for the scalar case . . . . . . . . . . . . . . . 428
A.5.2 The system case . . . . . . . . . . . . . . . . . . . . . . . . . . . 433
A.6 Boundary conditions for scalar conservation laws . . . . . . . . . . . .434
A.6.1 The left boundary condition for the Riemann problem . . . .. . .. . ..436
A.6.2 The right boundary condition for the Riemann problem . . .. . .. . .. 438
A.7 Bibliographical note . . . . . . . . . . . . . . . . . . . . . . . . . .440
B Generalized Solutions to Hamilton-Jacobi Equations . . . . . . . . . .. . 441
B.1 Basic definition . . . . . . . . . . . . . . . . . . . . . . . . . . . .441
B.2 Classical Hopf-Lax formula . . . . . . . . . . . . . . . . . . . . . . .443
B.3 Generalized Lax-Hopf formula . . . . . . . . . . . . . . . . . . . . . .449
B.4 Bibliographical note . . . . . . . . . . . . . . . . . . . . . . . . . .455
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .457
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
Author Bio.
Mauro Garavello received the Ph.D. degree in Functional Analysis and Applications from SISSA-ISAS in 2004. He held a postdoctoral position at the University of Milano Bicocca from 2005 to 2006. He was a researcher at the University of Piemonte Orientale from 2007 to 2011 and then at the University of Milano Bicocca from 2011 to 2015. Currently he is Associate Professor of Mathematical Analysis at the University of Milano Bicocca. He has published more than 40 research articles in referee journals, books and proceedings. His main research interests are in the field of conservation laws and control problems.
Ke Han received his B.S. degree in Applied Mathematics from the University of Science and Technology of China in 2008. He graduated from the Pennsylvania State University in 2013 with a Ph.D. degree in Mathematics. He then joined the Department of Civil and Environmental Engineering at Imperial College London in 2013 as an Assistant Professor. His research interests span a variety of aspects of transportation science and engineering, including traffic flow theory, network modeling, traffic control and management, intelligent transportation system, sustainable transportation, and network resilience. He has published over 60 papers in international journals and conferences.
Benedetto Piccoli received his B.S. from University of Padova in 1991 and P.h.D from SISSA-ISAS in 1994. He is currently Distinguished Professor and Joseph and Loretta Lopez Chair at Rutgers University - Camden and serves as the Associate Provost for Research. He published five books, more than 200 papers and proceedings and is the founding editor and Editor in Chief of Networks and Heterogeneous Media. He received the Fubini prize in 2009, was nominated AMS Fellow in 2012 and was invited speaker at ICIAM 2011. His research interests span various areas of applied mathematics including: networks flows, vehicular trafffic, crowd dynamics, control theory, partial differential equations, math finance and systems biology.
[Back to Top]