Abstract
We investigate the properties of travel times when the latter are derived from traffic-flow models. In particular we consider exit-flow models, which have been used to model time-varying flows on road networks, in dynamic traffic assignment (DTA). But we here define the class more widely to include, for example, models based on finite difference approximations to the LWR (Lighthill, Whitham and Richards) model of traffic flow, and ‘large step’ versions of these. For the derived travel times we investigate the properties of existence, uniqueness, continuity, first-in-first-out (FIFO), causality and time-flow consistency (or intertemporal consistency). We assume a single traffic type and assume that time may be treated as continuous or as discrete, and for each case we obtain conditions under which the above properties are satisfied, and interrelations among the properties. For example, we find that FIFO is easily satisfied, but not strict causality, and find that if we redefine travel time to ensure strict causality then we lose time-flow consistency, and that neither of these conditions is strictly necessary or sufficient for FIFO. All of the models can be viewed as an approximation to a model that is continuous in time and space (the LWR model), and it seems that any loss of desirable properties is the price we pay for using such approximations. We also extend the exit-flow models and results to allow ‘inhomogeneity’ over time (link capacity or other parameters changing over time), and show that FIFO is still ensured if the exit-flow function is defined appropriately.
Similar content being viewed by others
References
Adamo, V., V. Astarita, M. Florian, M. Mahut, and J.H. Wu. (1998). "An Analytical and Applicative Framework for Spillback Congestion Modelling in the Continuous Time Link Based Dynamic Network Loading Models." Tristan 98, Puerto Rico.
Adamo, V., V. Astarita, M. Florian, M. Mahut, and J.H. Wu. (1999a). "Analytical Modelling of Intersections in Traffic Flow Models with Queue Spill-Back." In IFORS' 99, 15th Triennial Conference, Hosted by the Operations Research Society of China (ORSC), Beijing, P.R. China, August 16-20, 1999.
Adamo, V., V. Astarita, M. Florian, M. Mahut, and J. H. Wu. (1999b). "Modelling the Spill-back of Congestion in Link Based Dynamic Network Loading Models: A Simulation Model with Application." In 14th International Symposium on Theory of Traffic Flow, Jerusalem, July 1999. Published by Elsevier, pp. 555-573.
Astarita, V. (1995). "Flow Propagation Description in Dynamic Network Loading Models." In Y.J. Stephanedes and F. Filippi (Eds.), Proceedings of IV International Conference on Application of Advanced Technologies in Transportation Engineering (AATT), pp. 599-603, ASCE.
Astarita, V. (1996). "A Continuous Time Link Model for Dynamic Network Loading Based on Travel Time Function." In J.-B. Lesort (Ed.), Proceedings of the 13th International Symposium on Theory of Traffic Flow, Elsevier, pp. 79-102.
Beckmann, M., C.B. McGuire, and C.B. Winsten. (1956). Studies in the Economics of Transportation. NewHaven, CT: Yale University Press.
Boyce, D., D.-H. Lee, and B. Ran. (2001). "Analytical Models of the Dynamic Traffic Assignment Problem." Networks and Spatial Economics 1, 377-390.
Carey, M. (1986). "A Constraint Qualification for a Dynamic Traffic Assignment Model." Transportation Science 20(1), 55-58.
Carey, M. (1987). "Optimal Time-Varying Flows on Congested Networks." Operations Research 35(1), 56-69.
Carey, M. (1990). "Extending and Solving a Multi-Period Congested Network Flow Model." Computers and Operations Research 17(5), 495-507.
Carey, M. (1999). "A Framework for User Equilibrium Dynamic Traffic Assignment." Research Report. Faculty of Business and Management, University of Ulster, BT37 0QB. Being revised for publication.
Carey, M. (2001). "Dynamic Traffic Assignment with more Flexible Modelling within Links." Networks and Spatial Economics 1(4), 349-375.
Carey, M. (2004). "Link Travel Times I: Desirable Properties." Networks and Spatial Economics 4(3), 257-268.
Carey, M. and Y. Ge. (2004). "Comparing Whole-Link Travel Time Models Used in DTA." Transportation Research 37B(10), 905-926.
Carey, M. and M. McCartney. (2000). "A Class of Traffic Flow Models Used in Dynamic Assignment." Computers & Operations Research 31(10), 1583-1602.
Carey, M. and M. McCartney. (2002). "Behaviour of a Whole-Link Travel Time Model Used in Dynamic Traffic Assignment." Transportation Research 36(1), 83-95.
Carey, M. and A. Srinivasan. (1982). Modelling Network Flows with Time-Varying Demands.Working Paper. School of Urban and Public Affairs, Carnegie-Mellon University, Pittsburgh, PA. Report to the U.S. Department of Transportation, Urban Mass Transportation Authority, 73 pages.
Carey, M. and A. Srinivasan. (1993). "Externalities, Average and Marginal Costs, and Tolls on Congested Networks with Time-Varying Flows." Operations Research 41(1), 217-231.
Daganzo, C.F. (1995). "A Finite Difference Approximation of the Kinematic Wave Model of Traffic Flow." Transp Res 29B(4), 261-276.
Friesz T.L., J. Luque, R.L. Tobin, and B-Y. Wie. (1989). "Dynamic Network Traffic Assignment Considered as a Continuous Time Optimal Control Problem." Operations Research 37(6), 893-901.
Friesz, T.L., D. Bernstein, T.E. Smith, R.L. Tobin, and B.W. Wie. (1993). "A Variational Inequality Formulation of the Dynamic Network user Equilibrium Problem." Operations Research 41, 179-191.
Friesz, T.L., D. Bernstein, Z. Suo, and R.L. Tobin. (2001). "Dynamic Network user Equilibrium with State-Dependent Time Lags." Networks and Spatial Economics 1, 319-347.
Jayakrishnan, R., W.K. Tsai, and A. Chen. (1995). "A Dynamic Traffic Assignment Model with Traffic-Flow Relationships." Transportation Research 3C, 51-72.
Lam, W.H.K. and H.-J. Huang. (1995). Dynamic user Optimal Traffic Assignment Model for many to one Travel Demand." Transportation Research 29B(4), 243-259.
Lighthill, M.J. and G.B. Whitham. (1955). "On Kinematic Waves. I: Flow Movement in Long Rivers II: A Theory of Traffic Flow on Long Crowded Roads." Proceedings of the Royal Society A 229, 281-345.
Lo, H.K. (1999). "A Dynamic Traffic Assignment Model that Encapsulates the Cell Transmission Model." In A. Ceder (Ed.), Traffic and Transportation Theory, pp. 327-350.
Lo, H.K. and W.Y. Szeto. (2002). "A Cell-based Variational Inequality Formulation of the Dynamic User Optimal Assignment Problem." Transportation Research 36B, 421-443.
Merchant, D.K. and G.L. Nemhauser. (1978a). "A Model and an Algorithm for the Dynamic Traffic Assignment Problem." Transportation Science 12(3), 183-199.
Merchant, D.K. and G.L. Nemhauser. (1978b). "Optimality Conditions for a Dynamic Traffic Assignment Model." Transportation Science 12(3), 200-207.
Nie, X. and H.M. Zhang. (2002). A Comparative Study of Some Macroscopic Link Models Used in Dynamic Traffic Assignment.Forthcoming in Networks and Spatial Economics.
Ran, B., D.E. Boyce, and L.J. LeBlanc. (1993). "A New Class of Instantaneous Dynamic User-Optimal Traffic Assignment Models." Operations Research 41, 192-202.
Ran, B. and D. Boyce. (1996). Modelling Dynamic Transportation Networks. Heidelberg: Springer-Verlag.
Ran, B., D.-H. Lee, and M.S.-I. Shin. (2002). "New Algorithm for a Multiclass Dynamic Traffic Assignment Model." Journal of Transportation Engineering 128, 323-335.
Richards, P.I. (1956). "Shock Waves on the Highway." Operations Research 4, 42-51.
Wie, B.W. and R.L. Tobin, (1998). "Dynamic Congestion Pricing for General Traffic Networks." Transportation Research B 32(5), 313-327.
Wie, B.W., R.L. Tobin, and T.L. Friesz. (1994). "The Augmented Lagrangian Method for Solving Dynamic Network Traffic Assignment Models in Discrete Time." Transpn. Sci. 28, 204-220.
Wie, B.W., R.L. Tobin, D. Bernstein, and T.L. Friesz. (1995). "A Comparison of System Optimum and User Equilibrium Traffic Assignments with Schedule Delay." Transpn. Res., 3C, 389-411.
Wu J.H., Y. Chen, and M. Florian. (1995). "The Continuous Dynamic Network Loading Problem: A Mathematical Formulation and Solution Method." In Presented at the 3rd EURO WORKING GROUP Meeting on Urban Traffic and Transportation, Barcelona 27-29 September.
Wu, J.H., Y. Chen, and M. Florian. (1998). "The Continuous Dynamic Network Loading Problem: AMathematical Formulation and Solution Method." Transportation Research, 32B, 173-187.
Xu, Y.W., J.H. Wu, M. Florian, P. Marcotte, and D.L. Zhu. (1999). "Advances in the Continuous Dynamic Network Loading Problem." Transportation Science 33(4), 341-353.
Yang, H. and H.-J. Huang. (1997). "Analysis of the Time-Varying Pricing of a Bottleneck with Elastic Demand using Optimal Control Theory." Transportation Research B 31(6), 425-440.
Zhu, D. and P. Marcotte. (2000). "On the Existence of Solutions to the Dynamic User Equilibrium Problem." Transportation Science 34(4), 402-414.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Carey, M. Link Travel Times II: Properties Derived from Traffic-Flow Models. Networks and Spatial Economics 4, 379–402 (2004). https://doi.org/10.1023/B:NETS.0000047114.31259.3d
Issue Date:
DOI: https://doi.org/10.1023/B:NETS.0000047114.31259.3d