Continuous-time link-based kinematic wave model: formulation, solution existence, and well-posedness

References

• Aubin, J.-P., A. M. Bayen, and P Saint-Pierre. 2008. “Dirichlet Problems for Some Hamilton–Jacobi Equations with Inequality Constraints.” SIAM Journal on Control and Optimization 47 (5): 2348–2380. doi: 10.1137/060659569
   Web of Science ®Google Scholar
• Balijepalli, N. C., D. Ngoduy, and D. P Watling. 2014. “The Two-Regime Transmission Model for Network Loading in Dynamic Traffic Assignment Problems.” Transportmetrica A 10 (7): 563–584. doi: 10.1080/18128602.2012.751680
   Web of Science ®Google Scholar
• Ban, X., J. S. Pang, H. Liu, and R Ma. 2012. “Modeling and Solving Continuous-Time Instantaneous Dynamic User Equilibria: A Differential Complementarity System Approach.” Transportation Research Part B 46 (3): 389–408. doi: 10.1016/j.trb.2011.11.002
   Google Scholar
• Bressan, A.. 2000. Hyperbolic Systems of Conservation Laws. The One Dimensional Cauchy Problem. Oxford: University Press.
   Google Scholar
• Bretti, G., R. Natalini, and B Piccoli. 2006. “Numerical Approximations of a Traffic Flow Model on Networks.” Networks and Heterogeneous Media 1, 57–84. doi: 10.3934/nhm.2006.1.57
   Web of Science ®Google Scholar
• Chitour, Y., and B Piccoli. 2005. “Traffic Circles and Timing of Traffic Lights for Cars Flow.” Discrete and Continuous Dynamical Systems, Series B 3, 599–630.
   Google Scholar
• Claudel, C. G., and A. M. Bayen 2010. “Lax-Hopf Based Incorporation of Internal Boundary Conditions Into Hamilton–Jacobi Equation. Part I: Theory.” IEEE Transactions on Automatic Control 55 (5): 1142–1157. doi: 10.1109/TAC.2010.2041976
   Web of Science ®Google Scholar
• Claudel, C. G., and A. M. Bayen 2010. “Lax-Hopf Based Incorporation of Internal Boundary Conditions Into Hamilton–Jacobi Equation. Part II: Computational methods.” IEEE Transactions on Automatic Control 55 (5): 1158–1174. doi: 10.1109/TAC.2010.2045439
   Web of Science ®Google Scholar
• Coclite, G. M., M. Garavello, and B. Piccoli 2005. “Traffic Flow on a Road Network.” SIAM Journal on Mathematical Analysis 36 (6): 1862–1886. doi: 10.1137/S0036141004402683
   Web of Science ®Google Scholar
• Dafermos, C. M. (2000). Hyperbolic conservation laws in continuum physics, Grundlehren der Mathematischen Wissenschaften, Vol. 325. Berlin: Springer.
   Google Scholar
• Dafermos, C. M. 1972. “Polygonal Approximations of Solutions of the Initial Value Problem for a Conservation Law.” Journal of Mathematical Analysis and Applications 38 (1): 33–41. doi: 10.1016/0022-247X(72)90114-X
   Web of Science ®Google Scholar
• Daganzo, C. F. 1994. “The Cell Transmission Model: A Simple Dynamic Representation of Highway Traffic.” Transportation Research Part B 28 (4): 269–287. doi: 10.1016/0191-2615(94)90002-7
   Google Scholar
• Daganzo, C. F. 1995. “The Cell Transmission Model, Part II: Network Traffic.” Transportation Research Part B 29 (2): 79–93. doi: 10.1016/0191-2615(94)00022-R
   Google Scholar
• Daganzo, C. F. 2005. “A Variational Formulation of Kinematic Waves: Basic Theory and Complex Boundary Conditions.” Transportation Research Part B 39, 187–196. doi: 10.1016/j.trb.2004.04.003
   Google Scholar
• Daganzo, C. F. 2006. “On the Variational Theory of Traffic Flow: Well-Posedness, Duality and Application.” Network and Heterogeneous Media 1 (4): 601–619. doi: 10.3934/nhm.2006.1.601
   Web of Science ®Google Scholar
• D'Apice, C., and B. Piccoli 2008. “Vertex Flow Models for Vehicular Traffic on Networks.” Mathematical Models and Methods in Applied Schences 18, 1299–1315. doi: 10.1142/S0218202508003042
   Web of Science ®Google Scholar
• Evans, L. C. 2010. Partial Differential Equations (2nd ed.). Providence, RI: American Mathematical Society.
   Google Scholar
• Friesz, T. L., K. Han, P. A. Neto, A. Meimand, and T. Yao 2013. “Dynamic User Equilibrium Based on a Hydrodynamic Model.” Transportation Research Part B 47 (1): 102–126. doi: 10.1016/j.trb.2012.10.001
   Google Scholar
• Friesz, T. L., T. Kim, C. Kwon, and M. A. Rigdon 2011. “Approximate Network Loading and Dual-Time-Scale Dynamic User Equilibrium.” Transportation Research Part B 45, 176–207. doi: 10.1016/j.trb.2010.05.003
   Google Scholar
• Friesz, T. L., and A Meimand. 2013. “A Differential Variational Inequality Formulation of Dynamic Network User Equilibrium with Elastic Demand.” Transportmetrica A: Transport Science 10 (7): 661–668. doi: 10.1080/18128602.2012.751684
   Web of Science ®Google Scholar
• Garavello, M., and B. Piccoli 2006. Traffic Flow on Networks. Conservation Laws Models. AIMS Series on Applied Mathematics, Springfield, MO: American Institute of Mathematical Sciences.
   Google Scholar
• Garavello, M., and B. Piccoli 2009. “Conservation Laws on Complex Networks.” Annales De L Institut Henri Poincare-analyse Non Lineaire 26 (5): 1925–1951. doi: 10.1016/j.anihpc.2009.04.001
   Web of Science ®Google Scholar
• Godunov, S. K. 1959. “A Finite Difference Method for the Numerical Computation of Discontinuous Solutions of the Equations of Fluid Dynamics.” Matematicheskii Sbornik 47, 271–290.
   Google Scholar
• Han, K., T. L. Friesz, and T. Yao 2013a. “A Partial Differential Equation Formulation of Vickrey's Bottleneck model, Part I: Methodology and Theoretical Analysis.” Transportation Research Part B 49, 55–74. doi: 10.1016/j.trb.2012.10.003
   Google Scholar
• Han, K., T. L. Friesz, and T. Yao 2013b. “A Partial Differential Equation Formulation of Vickrey's Bottleneck Model, Part II: Numerical Analysis and Computation.” Transportation Research Part B, 49, 75–93. doi: 10.1016/j.trb.2012.10.004
   Google Scholar
• Han, K., T. L. Friesz, and T. Yao 2013c. “Existence of Simultaneous Route and Departure Choice Dynamic User Equilibrium.” Transportation Research Part B 53, 17–30. doi: 10.1016/j.trb.2013.01.009
   Google Scholar
• Han, K., T. L. Friesz, and T. Yao 2014a. Vehicle spillback on dynamic traffic networks and what it means for dynamic traffic assignment models. Paper presented at the 5th international symposium on dynamic traffic assignment. June 18–20, Salerno, Italy.
   Google Scholar
• Han, K., V. Gayah, B. Piccoli, T. L. Friesz, and T. Yao 2014b. “On the Continuum Approximation of the On-and-Off Signal Control on Dynamic Traffic Networks.” Transportation Research Part B, 61, 73–97. doi: 10.1016/j.trb.2014.01.001
   Google Scholar
• Han, K., B. Piccoli, and T. L. Friesz 2015. “Continuity of the Path Delay Operator for LWR-based Network Loading with Spillback.” Submitted. Preprint available online at http://arxiv.org/abs/1501.04241.
   Google Scholar
• Han, K., W. Y. Szeto, and T. L. Friesz 2015. “Formulation, Existence, and Computation of Boundedly Rational Dynamic User Equilibrium with Fixed or Endogenous User Tolerance.” Transportation Research Part B, 79, 16–49. doi: 10.1016/j.trb.2015.05.002
   Google Scholar
• Herty, M., and A. Klar 2003. “Modeling, Simulation, and Optimization of Traffic Flow Networks.” SIAM Journal on Scientific Computing 25, 1066–1087. doi: 10.1137/S106482750241459X
   Web of Science ®Google Scholar
• Holden, H., and N. H. Risebro 1995. “A Mathematical Model of Traffic Flow on a Network of Unidirectional Roads.” SIAM Journal on Mathematical Analysis 26 (4): 999–1017. doi: 10.1137/S0036141093243289
   Web of Science ®Google Scholar
• Holden, H., and N. H. Risebro 2011. Front Tracking for Hyperbolic Conservation Laws. Applied Mathematical Sciences. Vol. 152. Berlin: Springer. http://link.springer.com/book/10.1007%2F978-3-642-23911-3http://link.springer.com/book/10.1007%2F978-3-642-23911-3
   Google Scholar
• Jin, W.-L. 2010. “Continuous Kinematic Wave Models of Merging Traffic Flow.” Transportation Research Part B 44, 1084–1103. doi: 10.1016/j.trb.2010.02.011
   Google Scholar
• Kachani, S., and G. Perakis 2002. “Fluid Dynamics Models and Their Application in Transportation and Pricing.” Technical Report, Sloan School of Management, MIT.
   Google Scholar
• Lebacque, J., and M. Khoshyaran 1999. “Modeling Vehicular Traffic Flow on Networks Using Macroscopic Models.” In Finite Volumes for Complex Applications II, edited by Roland Vilsmeier, Fayssal Benkhaldoun, and Dieter Hanel, 551–558. Paris: Hermes Science Publication; 1999.
   Google Scholar
• Lebacque, J., and M. Khoshyaran 2002. “First Order Macroscopic Traffic Flow Models for Networks in the Context of Dynamic Assignment.” In Transportation Planning State of the Art. M. Patriksson and K. A. P. M. Labbe, eds., Kluwer Academic Publushers, Norwell, MA.
   Google Scholar
• LeVeque, R. J. 1992. Numerical Methods for Conservation Laws. Basel: Birkhäuser.
   Google Scholar
• Lighthill, M., and G. Whitham 1955. “On Kinematic Waves. II. A Theory of Traffic Flow on Long Crowded Roads.” Proceedings of the Royal Society of London: Series A 229, 317–345.
   Google Scholar
• Liu, H., K. Han, V. Gayah, T. L. Friesz, and T. Yao 2015. “Data-driven Linear Decision Rule Approach for Distributionally Robust Optimization of On-line Signal Control.” Transportation Research Part C. doi:10.1016/j.trc.2015.05.016.
   Web of Science ®Google Scholar
• Lo, H. K., and W. Y. Szeto 2002. “A Cell-Based Dynamic Traffic Assignment Model: Formulation and Properties.” Mathematical and Computer Modelling 35 (7–8): 849–865. doi: 10.1016/S0895-7177(02)00055-9
   Web of Science ®Google Scholar
• Mesa-Arango, R., and S. V. Ukkusuri 2014. “Modeling the Car-Truck Interaction in a System-Optimal Dynamic Traffic Assignment Model.” Journal of Intelligent Transportation Systems 18 (4): 327–338. doi: 10.1080/15472450.2013.806846
   Web of Science ®Google Scholar
• Ma, R., X. Ban, J. S. Pang, and H. Liu 2014a. “Convergence of Time Discretization Schemes for Continuous-time Dynamic Network Loading Models. Networks and Spatial Economics. doi: 10.1007/s11067-014-9239-5.
   Web of Science ®Google Scholar
• Ma, R., X. Ban, and J.-S. Pang 2014b. “Continuous-Time Dynamic System Optimum for Single-Destination Traffic Networks with Queue Spillbacks.” Transportation Research Part B 68, 98–122. doi: 10.1016/j.trb.2014.06.003
   Google Scholar
• Moskowitz, K. 1965. “Discussion of ‘Freeway Level of Service as Influenced by Volume and Capacity Characteristics’ by D.R. Drew and C.J. Keese.” Highway Research Record, 99, 43–44.
   Google Scholar
• Newell, G. F. 1993a. “A Simplified Theory of Kinematic Waves in Highway Traffic. Part I: General Theory.” Transportation Research Part B 27 (4): 281–287. doi: 10.1016/0191-2615(93)90038-C
   Google Scholar
• Newell, G. F. 1993b. “A Simplified Theory of Kinematic Waves in Highway Traffic. Part II: Queuing at Freeway Bottlenecks.” Transportation Research Part B 27 (4): 289–303. doi: 10.1016/0191-2615(93)90039-D
   Google Scholar
• Newell, G. F. 1993c. “A Simplified Theory of Kinematic Waves in Highway Traffic. Part III: Multi-Destination Flows.” Transportation Research Part B 27 (4): 305–313. doi: 10.1016/0191-2615(93)90040-H
   Google Scholar
• Ni, D., and J. D. Leonard 2005. “A Simplified Kinematic Wave Model at a Merge Bottleneck.” Applied Mathematical Modeling 29 (11): 1054–1072. doi: 10.1016/j.apm.2005.02.008
   Web of Science ®Google Scholar
• Osorio, C., G. Flotterod, and M. Bierlaire 2011. “Dynamic Network Loading: A Stochastic Differentiable Model that Derives Link State Distributions.” Transportation Research Part B 45 (9): 1410–1423. doi: 10.1016/j.trb.2011.05.014
   Google Scholar
• Perakis, G., and G. Roels 2006. “An Analytical Model for Traffic Delays and the Dynamic User Equilibrium Problem.” Operations Research 54 (6): 1151–1771. doi: 10.1287/opre.1060.0307
   Web of Science ®Google Scholar
• Richards, P. I. 1956. “Shockwaves on the Highway.” Operations Research 4, 42–51. doi: 10.1287/opre.4.1.42
   Web of Science ®Google Scholar
• Sumalee, A., R. X. Zhong, T. L. Pan, and W. Y. Szeto 2011. “Stochastic Cell Transmission Model (SCTM): A Stochastic Dynamic Traffic Model for Traffic State Surveillance and Assignment.” Transportation Research Part B 45 (3): 507–533. doi: 10.1016/j.trb.2010.09.006
   Google Scholar
• Szeto, W. Y. 2003. “Dynamic Traffic Assignment: Formulations, Properties, and Extensions.” Ph.D. thesis, Hong Kong University of Science and Technology, Hong Kong.
   Google Scholar
• Szeto, W. Y. 2008. “The Enhanced Lagged Cell Transmission Model for Dynamic Traffic Assignment.” Transportation Research Record 2085, 76–85. doi: 10.3141/2085-09
   Web of Science ®Google Scholar
• Szeto, W. Y., Y. Jiang, and A. Sumalee 2011. “A Cell-Based Model for Multi-Class Doubly Stochastic Dynamic Traffic Assignment.” Computer-Aided Civil and Infrastructure Engineering 26 (8): 595–611. doi: 10.1111/j.1467-8667.2011.00717.x
   Web of Science ®Google Scholar
• Szeto, W. Y., and H. K. Lo 2006. “Dynamic Traffic Assignment: Properties and Extensions.” Transportmetrica 2 (1): 31–52. doi: 10.1080/18128600608685654
   Web of Science ®Google Scholar
• Yperman, I., S. Logghe, and L. Immers 2005. “The Link Transmission Model: An Efficient Implementation of the Kinematic Wave Theory in Traffic Networks, Advanced OR and AI Methods in Transportation.” Proceedings of the 10th EWGT meeting and 16th Mini-EURO conference, Poznan, Poland, 122–127, Publishing House of Poznan University of Technology.
   Google Scholar
• Zhang, L., Y. Yin, and S. Chen 2013. “Robust Signal Timing Optimization with Environmental Concerns.” Transportation Research Part C 29, 55–71. doi: 10.1016/j.trc.2013.01.003
   Web of Science ®Google Scholar
• Ziliaskopoulos, A. K. 2000. “A Linear Programming Model for the Single Destination System Optimal Dynamic Traffic Assignment Problem.” Transportation Science 34 (1): 37–49. doi: 10.1287/trsc.34.1.37.12281
   Web of Science ®Google Scholar
• Zhu, F., H. K. Lo, and H.-Z. Lin 2013. “Delay and Emissions Modelling for Signalised Intersections.” Transportmetrica B 1 (2): 111–135.
   Web of Science ®Google Scholar