Abstract
This paper describes a continuous-flow, continuous-time model for which a dynamic Wardrop equilibrium provably exists. This formulation is general, but is particularly designed to include the link and node models of Yperman’s Link Transmission Model as a special case. Rather than using path flows to describe route choice, travelers are aggregated by destination and node-specific routing parameters are used to reduce the number of control variables needed. Furthermore, this formulation allows efficient solution methods from static traffic assignment, such as Linear User Cost Equilibrium (LUCE), to be applied in a fairly straightforward manner. Demonstrations on a small network verify the effectiveness of this dynamic LUCE algorithm in our model, showing favorable performance comapred to the method of successive averages.
Similar content being viewed by others
Notes
1 Note that this may change the character of the underlying solutions somewhat, as discussed in Wie et al. (2002)
References
Astarita V (1996) A continuous link time model for dynamic network loading based on travel time function. In: Lesort J-B (ed)Transportation and traffic theory. Pergamon, pp 79–102
Ban XJ, Liu HX, Ferris MC, Ran B (2008) A link-node complementarity model and solution algorithm for dynamic user equilibria with exact flow propagations. Transp Res B 42: 823–842
Ban XJ, Pang J-S, Liu HX, Ma R (2012) Modeling and solving continuous-time instantaneous dynamic user equilibria: a differential complementarity systems approach. Transp Res B 46(3): 389–408
Bar-Gera H (2005) Continuous and discrete trajectory models for dynamic traffic assignment. Netw Spat Econ 5:41–70
Bar-Gera H, Boyce D (2006) Solving a non-convex combined travel forecasting model by the method of successive averages with constant step sizes. Transp Res B 40: 351–367
Bliemer MCJ, Bovy PHL (2003) Quasi-variational inequality formulation of the multiclass dynamic traffic assignment problem. Transp Res B 37(6): 501–519
Blumberg M, Bar-Gera H (2009) Consistent node arrival order in dynamic network loading models. Transp Res B 43(3): 285–300
Carey M (1992) Nonconvexity of the dynamic assignment problem. Transp Res B 26(2): 127–133
Chiu Y-C, Bottom J, Mahut M, Paz A, Balakrishna R, Waller T, Hicks J (2010) A primer for dynamic traffic assignment. Prepared by the Transportation Network Modeling Committee of the Transportation Research Board (ADB30)
Daganzo CF (1994) The cell transmission model: a dynamic representation of highway traffic consistent with the hydrodynamic theory. Transp Res B 28(4): 269–287
Daganzo CF (1995) The cell transmission model, part II: network traffic. Transp Res B 29(2): 79–93
Fan K (1952) Fixed-point and minimax theorems in locally convex topological linear spaces. Proc Natl Acad Sci USA 38(2): 121–126
Friesz T, Bernstein D, Smith TE, Tobin RL, Wie BW (1993) A variational inequality formulation of the dynamic network user equilibrium problem. Oper Res 41(1): 179–191
Friesz T, Luque J, Tobin R, Wie B (1989) Dynamic network traffic assignment considered as a continuous time optimal control problem. Oper Res 37: 893–901
Friesz TL, Han K, Neto PA, Meimand A, Yao T (2013) Dynamic user equilibrium based on a hydrodynamic model. Transp Res B 47: 102–126
Friesz TL, Kim T, Kwon C, Rigdon MA (2011) Approximate network loading and dual-time-scale dynamic user equilibrium.Transp Res B 45(1): 176–207
Gentile G (2009) Linear User Cost Equilibrium: a new algorithm for traffic assignment. Working paper
Gentile G (2010) The general link transmission model for dynamic network loading and a comparison with the DUE algorithm. In: Immers LGH, Tampere CMJ, Viti F (eds)New developments in transport planning advances in dynamic traffic assignment. Edward Elgar Publishing
Glicksberg IL (1952) A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium. Proc Am Math Soc 3(1): 170–174
Iryo T (2013) Investigating factors for existence of multiple equilibria in dynamic traffic network. Accepted for publication in Networks and Spatial Economics. http://link.springer.com/article/10.1007/s11067-013-9206-6
Janson BN, Robles J (1995) Quasi-continuous dynamic traffic assignment model. Transp Res Rec 1493: 199–206
Kakutani S (1941) A generalization of Brouwer’s fixed point theorem. Duke Math J 8(3): 457–459
Lighthill MJ, Whitham GB (1955) On kinematic waves II. A theory of traffic flow on long crowded roads. Proc R Soc A 229(1178): 314–345
Long J, Huang H-J, Gao Z, Szeto WY (2011) An intersection-movement-based dynamic user optimal route choice problem. Oper Res 61(5): 1134–1147
Merchant D, Nemhauser G (1978a) A model and an algorithm for the dynamic traffic assignment problems. Transp Sci 12: 183–199
Merchant D, Nemhauser G (1978b) Optimality conditions for a dynamic traffic assignment model. Transp Sci 12: 200–207
Mounce R (2006) Convergence in a continuous dynamic queueing model for traffic networks. Transp Res B 40(9): 779–791
Newell GF (1993a) A simplified theory of kinematic waves in highway traffic, part I: general theory. Transp Res B 27(4): 281–287
Newell GF (1993b) A simplified theory of kinematic waves in highway traffic, part II: queueing at freeway bottlenecks. Transp Res B 27(4): 289–303
Newell GF (1993c) A simplified theory of kinematic waves in highway traffic, part III: multi-destination flows. Transp Res B 27(4): 305–313
Nezamuddin (2011) Improving the efficiency of dynamic traffic assignment through computational methods based on combinatorial algorithm. Ph.D. thesis, The University of Texas at Austin
Nie Y (2012) A note on Bar-Gera’s algorithm for the origin-based traffic assignment problem. Transp Sci 46(1): 27–38
Nie YM (2010) Equilibrium analysis of macroscopic traffic oscillations. Transp Res B 44: 62–72
Peeta S, Ziliaskopoulos AK (2001) Foundations of dynamic traffic assignment: the past, the present, and the future. Netw Spat Econ 1:233–265
Qian Z(S), Zhang HM (2013) A hybrid route choice model for dynamic traffic assignment. Netw Spat Econ 13(2):183–203
Ramadurai G, Ukkusuri SV (2011) B-dynamic: an efficient algorithm for dynamic user equilibrium assignment in activity-travel networks. Comput Aided Civ Infrastruct Eng 26(4): 254–269
Ran B, Boyce DE (1996) A link-based variational inequality formulation of ideal dynamic user-optimal route choice problem. Transp Res C 4(1): 1–12
Ran B, Boyce DE, LeBlanc LJ (1993) A new class of instantaneous dynamic user-optimal traffic assignment models. Oper Res 41(1): 192–202
Ran B, Hall RW, Boyce DE (1996) A link-based variational inequality model for dynamic departure time/route choice. Transp Res B 30: 31–46
Tampère CMJ, Corthout R, Cattrysse D, Immers LH (2011) A generic class of first order node models for dynamic macroscopic simulation of traffic flows. Transp Res B 45: 289–309
Waller ST, Fajardo D, Duell M, Dixit V (2013) Linear programming formulation for strategic dynamic traffic assignment. Netw Spat Econ 13(4):427–443
Wie BW (1991) Dynamic analysis of user optimized network flows with elastic travel demand. Presented at the 70th Annual Meeting of the Transportation Research Board, Washington, DC
Wie B-W, Tobin RL, Carey M (2002) The existence, uniqueness and computation of an arc-based dynamic network user equilibrium formulation. Transp Res B 36(10): 897–918
Yperman I (2007) The link transmission model for dynamic newtork loading. Ph.D. thesis, Katholieke Universiteit Leuven, Belgium
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Nezamuddin, N., Boyles, S.D. A Continuous DUE Algorithm Using the Link Transmission Model. Netw Spat Econ 15, 465–483 (2015). https://doi.org/10.1007/s11067-014-9234-x
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11067-014-9234-x