# Traffic Flow Theory: An introduction with exercises

## Synopsis

Traffic processes cause several problems in the world. Traffic delay, pollution are some of it. They can be solved with the right road design or traffic management (control) measure. Before implementing these designs of measures, though, their effect could be tested. To this end, knowledge of traffic flow theory is needed.

## References

Ahn, S. and Cassidy,M. (2007). Freeway traffic oscillations and vehicle lane-change maneuvers.

In Allsop, R. E., Bell, M. G. H., and Heydecker, B. G., editors, Proceedigs of the

International Symposium of Transportation and Traffic Theory, pages 691–710. Elsevier,

Amsterdam.

Aw, A. and Rascle (2000). Resurrection of the “second order” models of traffic flow. SIAM

Journal on AppliedMathematics, 60:916–938.

Bando, M., Hasebe, K., Nakayama, A., Shibata, A., and Sugiyama, Y. (1995). Dynamical

model of traffic congestion and numerical simulation. Physical Review E, 51(2):1035.

Copyright (C) 2007 The American Physical Society Please report any problems to

prola@aps.org PRE.

Bliemer, M. (2007). Dynamic queuing and spillback in analytical multiclass dynamic

network loading model. Transportation Research Record: Journal of the Transportation

Research Board, (2029):14–21.

Brackstone, M. and McDonald, M. (1999). Car-Following: A Historical Review. Transportation

Research F, 2:181 – 186.

Carlson, R. C., Papamichail, I., Papageorgiou, M., and Messmer, A. (2010). Optimal motorway

traffic flow control involving variable speed limits and ramp metering. Transportation

Science, 44(2):238–253.

Cassidy, M. and Bertini, R. (1999). Some traffic features at freeway bottlenecks. Transportation

Research Part B:Methodological, 33(1):25 – 42.

Cassidy, M., Jang, K., and Daganzo, C. (2011). Macroscopic fundamental diagram for

freeway networks: Theory and observation. In Proceedings of the 90th AnnualMeeting

of the Transportation Research Board.

Cassidy, M. J. and Rudjanakanoknad, J. (2005). Increasing the capacity of an isolated

merge by metering its on-ramp. Transportation Research Part B: Methodological,

(10):896 – 913. ISSN 0191-2615.

Chiabaut, N., Leclercq, L., and Buisson, C. (2010). From heterogeneous drivers to

macroscopic patterns in congestion. Transportation Research Part B:Methodological,

(2):299 – 308. ISSN 0191-2615.

Chung, K., Rudjanakanoknad, J., and Cassidy,M. J. (2007). Relation between traffic density

and capacity drop at three freeway bottlenecks. Transportation Research Part B:

Methodological, 41(1):82–95.

Corthout, R., Flötteröd, G., Viti, F., and Tampère, C. M. (2012). Non-unique flows in

macroscopic first-order intersection models. Transportation Research Part B:Methodological,

(3):343–359.

Courant, R., Friedrichs, K., and Lewy, H. (1928). Über die partiellen differenzengleichungen

der mathematischen physik. Mathematische annalen, 100(1):32–74.

Daganzo, C. (2007). Urban gridlock: Macroscopic modeling and mitigation approaches.

Transportation Research Part B:Methodological, 41(1):49–62.

Daganzo, C. F. (1994). The Cell TransmissionModel: a Dynamic Representation of Highway

Traffic Consistent With the Hydrodynamic Theory. Transportation research part

B, 28B(4):269–287.

Daganzo, C. F. (1995a). The cell transmission model, part ii: network traffic. Transportation

Research Part B:Methodological, 29(2):79–93.

Daganzo, C. F. (1995b). Requiem for second-order fluid approximations of traffic flow.

Tranportation Research Part B:Methodological, 29:277–286.

Daganzo, C. F. (1997). Fundamentals of Transportation and Traffic Operations. Pergamon.

Daganzo, C. F. (2002a). A behavioral theory of multi-lane traffic flow. part i: Long homogeneous

freeway sections. Transportation Research Part B: Methodological, 36(2):131

– 158. ISSN 0191-2615.

Daganzo, C. F. (2002b). A behavioral theory ofmulti-lane traffic flow. part ii: Merges and

the onset of congestion. Transportation Research Part B: Methodological, 36(2):159 –

ISSN 0191-2615.

Daganzo, C. F. (2005). A variational formulation of kinematic waves: basic theory

and complex boundary conditions. Transportation Research Part B: Methodological,

(2):187–196.

Drake, J. S., Schöfer, J. L., and May, A. (1967). A statistical analysis of speed density hypotheses.

In Edie, L. C., Herman, R., and Rothery, R., editors, Proceedings of the Third

International Symposium on the Theory of Traffic Flow. Elsevier North-Holland, New

York.

Driels,M. R. and Shin, Y. S. (2004). Determining the number of iterations for monte carlo

simulations of weapon effectiveness.

Dynasmart (2003). DYNAMSMART-X – User’s Guide. Technical report, University of

Maryland.

Flötteröd, G. and Rohde, J. (2011). Operational macroscopic modeling of complex urban

road intersections. Transportation Research Part B:Methodological, 45(6):903–922.

Fortuijn, L. G. H. and Hoogendoorn, S. P. (2015). Turbo roundabouts: Comparing capacity

estimation on gap and flow level. In Proceedings of the 94th Annual Meeting of the

Transportation Research Board.

Geroliminis, N. and Daganzo, C. F. (2008). Existence of urban-scale macroscopic fundamental

diagrams: Some experimental findings. Transportation Research Part B:

Methodological, 42(9):759–770.

Gipps, G. P. (1986). A model for the structure of lane-changing descisions. Tranportation

Research Part B:Methodological, 20:403–414.

Godunov, S. K. (1959). A difference scheme for numerical computation of of discontinuous

solutions of equations of fluid dynamics. Math. Sb., 47:271 – 290.

Greenshields, B. D. (1934). A Study of Traffic Capacity. Proceedings Highway Research

Board, 14:448–477.

Hajiahmadi, M., Knoop, V. L., De Schutter, B., and Hellendoorn, H. (2013). Optimal dynamic

route guidance: A model predictive approach using the macroscopic fundamental

diagram. In Intelligent Transportation Systems-(ITSC), 2013 16th International

IEEE Conference on, pages 1022–1028. IEEE.

Hall, F. L. and Agyemang-Duah, K. (1991). Freeway Capacity Drop and the Definition

of Capacity. Transportation Research Record: Journal of the Transportation Research

Board No.1320, pages 91–98.

Heikoop, H., editor (2011). Capaciteitswaarden Infrastructuur Autosnelwegen. Dienst

Verkeer en Scheepvaart.

Helbing, D. (2003). A section-based queueing-theoretical traffic model for congestion

and travel time analysis in networks. Journal of Physics A: Mathematical and General,

(46):L593.

Helbing, D., Hennecke, A., and Treiber, M. (1999). Phase diagram of traffic states in the

presence of inhomogeneities. Physical Review Letters, 82(21):4360.

Helly, W. . (1959). Simulation of bottlenecks in single lane traffic flow. In Proceedings

of the Symposium on Theory of Traffic Flow, pages 207– 238. GeneralMotors Research

Laboratories, Elsevier, New York.

Hoogendoorn, S., Hoogendoorn, R. G., and Daamen, W. (2011). Wiedemann revisited.

Transportation Research Record: Journal of the Transportation Research Board,

(1):152–162.

Hoogendoorn, S. P. (2005). Unified approach to estimating free speed distributions.

Transportation Research Part B:Methodological, 39(8):709–727.

Ji, Y. and Geroliminis, N. (2012). On the spatial partitioning of urban transportation networks.

Transportation Research Part B:Methodological, 46(10):1639–1656.

Kerner, B. S. (2004). The Physics Of Traffic: Empirical Freeway Pattern Features, Engineering

Applications, And Theory. Springer, Berlin.

Keyvan-Ekbatani, M., Daamen, W., and Knoop, V. (2016). Categorization of the lanechange

decision process on freeways. Transportation Research part C.

Knoop, V., Tamminga, G., , and Leclercq, L. (2016). Network transmission model: Application

to a real world city. In proceedings of the 95th AnnualMeeting of the Transportation

Reseach Board.

Knoop, V. L., Duret, A., Buisson, C., and Van Arem, B. (2010). Lane distribution of traffic

near merging zones – influence of variable speed limits. In Proceedings of IEEE Intelligent

Transportation Systems.

Knoop, V. L. and Hoogendoorn, S. P. (2015). An area-aggregated dynamic traffic simulation

model. European Journal of Transport and Infrastructure Research (EJTIR), 15 (2),

Knoop, V. L., Hoogendoorn, S. P., and Van Zuylen, H. J. (2007). Empirical Differences between

Time Mean Speed and Space Mean Speed. In Proceedings of Traffic and Granular

Flow 07. Springer, Paris, France.

Knoop, V. L., van Lint, H., and Hoogendoorn, S. P. (2015). Traffic dynamics: Its impact

on the macroscopic fundamental diagram. Physica A: Statistical Mechanics and its

Applications, 438:236–250.

Knoop, V. L., Van Zuylen, H. J., and Hoogendoorn, S. P. (2009). Proceeding of the International

Symposium of Transportation and Traffic Theory, chapter Microscopic Traffic

Behavior near Accidents. Springer, New York.

Koshi, M., Iwasaki, M., and Ohkura, I. (1981). Some findings and an overview on vehicular

flow characteristics. In Proceedings of the 8th International Symposiumon Transportation

and Traffic Theory, pages 403–426. Univ. of Toronto Press, Toronto.

Kotsialos, A., Papageorgiou, M., Diakaki, C., Pavlis, Y., and Middelham, F. (2002). Traffic

flowmodeling of large-scale motorway networks using themacroscopicmodeling tool

metanet. Intelligent Transportation Systems, IEEE Transactions on, 3(4):282–292.

Laval, J. and Castrillón, F. (2015). Stochastic approximations for the macroscopic fundamental

diagram of urban networks. In Transportation Research Procedia, volume 7,

page 615–630.

Laval, J. A. (2011). Hysteresis in traffic flowrevisited: An improved measurementmethod.

Transportation Research Part B:Methodological, 45(2):385 – 391.

Laval, J. A. and Daganzo, C. F. (2006). Lane-changing in traffic streams. Transportation

Research Part B:Methodological, 40(3):251–264.

Laval, J. A. and Leclercq, L. (2013). The hamilton–jacobi partial differential equation and

the three representations of traffic flow. Transportation Research Part B:Methodological,

:17–30.

Lebacque, J.-P. (2005). First-order macroscopic traffic flow models: Intersection modeling,

network modeling. In Transportation and Traffic Theory. Flow, Dynamics and

Human Interaction. 16th International Symposium on Transportation and Traffic Theory.

Leclercq, L., Chiabaut, N., Laval, J. A., and Buisson, C. (2007). Relaxation Phenomenon

After Lane Changing. Transportation Research Record: Journal of the Transportation

Research Board, No. 1999, pages 79–85.

Leclercq, L. and Geroliminis, N. (2013). Estimating mfds in simple networks with route

choice. Transportation Research Part B:Methodological, 57:468–484.

Leclercq, L., Parzani, C., Knoop, V. L., Amourette, J., and Hoogendoorn, S. P. (2015).

Macroscopic traffic dynamics with heterogeneous route patterns. Transportation Research

Part C: Emerging Technologies, 59:292–307.

Lighthill, M. J. and Whitham, G. B. (1955). On Kinematic Waves. II. A Theory of Traffic

Flow on Long Crowded Roads,. Proceedings of the Royal Society of London. Series A,

Mathematical and Physical Sciences, 229(1178):317 – 345.

Nagalur Subraveti, H. H. S., Knoop, V. L., and van Arem, B. (2019). First order multi-lane

traffic flow model–an incentive based macroscopic model to represent lane change

dynamics. Transportmetrica B: Transport Dynamics, 7(1):1758–1779.

Newell, G. (2002). A simplified car-following theory: a lower order model. Transportation

Research Part B:Methodological, 36(3):195–205.

Newell, G. F. (1993). A simplified theory of kinematic waves in highway traffic, part ii:

Queueing at freeway bottlenecks. Transportation Research Part B: Methodological,

(4):289 – 303.

Oh, S. and Yeo, H. (2012). Estimation of capacity drop in highway merging sections.

Transportation Research Record: Journal of the Transportation Research Board,

(1):111–121.

Ossen, S. J. L. (2008). Longitudinal Driving Behavior: Theory and Empirics. Trail thesis

series, Delft University of Technology.

Payne, H. J. (1971). Models of freeway traffic and control, mathematicalmodels of public

systems. Simulation Council Proceedings Series, 28(1):51–61.

Pueboobpaphan, R. and van Arem, B. (2010). Understanding the relation between

driver/vehicle characteristics and platoon/traffic flow stability for the design and assessment

of cooperative cruise control. In Proceedings of the 89th Annual Meeting of

the Transportation Research Board.

Richards, P. I. (1956). Shock waves on the highway. Operations Research 4, 4:42 – 51.

Schakel, W. J., Van Arem, B., and Knoop, V. (2012). Lmrs: An integrated lane change

model with relaxation and synchronization. In Proceedings of the 91st AnnualMeeting

of the Transportation Research Board.

Schreiter, T., Van Lint, H., Yuan, Y., and P., H. S. (2010). Propagation wave speed estimation

of freeway traffic with image processing tools. In Proceedings of the 89th Annual

Meeting.

Smits, E.-S., Bliemer, M. C., Pel, A. J., and van Arem, B. (2015). A family of macroscopic

nodemodels. Transportation Research Part B:Methodological, 74:20–39.

Smulders, S. (1989). Control of freeway traffic flow. CWI Tract, 80.

Srivastava, A. andGeroliminis,N. (2013). Empirical observations of capacity drop in freeway

merges with ramp control and integration in a first-order model. Transportation

Research Part C: Emerging Technologies, 30:161–177.

Tampère, C. M., Corthout, R., Cattrysse, D., and Immers, L. H. (2011). A generic class of

first order node models for dynamic macroscopic simulation of traffic flows. Transportation

Research Part B:Methodological, 45(1):289–309.

Toledo, T., Koutsopoulos, H. N., and Ben-Akiva, M. (2007). Integrated driving behavior

modeling. Transportation Research Part C: Emerging Technologies, 15(2):96 – 112. ISSN

-090X.

Toledo, T., Koutsopoulos, H. N., and Ben-Akiva, M. (2009). Estimation of an integrated

driving behavior model. Transportation Research Part C: Emerging Technologies,

(4):365 – 380. ISSN 0968-090X.

Transportation Research Board, (2000). Highway CapacityManual. Technical report.

Treiber, M., Hennecke, A., and Helbing, D. (2000). Congested traffic states in empirical

observations and microscopic simulations. Physical Review E, 62(2):1805.

Treiber, M. and Kesting, A. (2013). Traffic Flow Dynamics; Data, Models and Simulation.

Springer.

Treiber, M., Kesting, A., and Helbing, D. (2006). Delays, inaccuracies and anticipation

in microscopic traffic models. Physica A: Statistical Mechanics and its Applications,

(1):71–88.

Van der Gun, J. P., Pel, A. J., and Van Arem, B. (2015). A general activity-based methodology

for simulating multimodal transportation networks during emergencies. In ICEM

: 3rd International Conference on EvacuationModeling andManagement, Tainan,

Taiwan, 1-3 June 2015.

Van Lint, J., Bertini, R. L., and Hoogendoorn, S. P. (2014). Data fusion solutions to compute

performance measures for urban arterials. In Symposium Celebrating 50 years

of Traffic Flow Theory 2014 FTF SummerMeeting, Portland (USA), 11-13 August, 2014.

TRB.

van Lint, J. W., Hoogendoorn, S. P., and Schreuder, M. (2008). Fastlane: New multiclass

first-order traffic flow model. Transportation Research Record: Journal of the Transportation

Research Board, 2088(1):177–187.

van Wageningen-Kessels, F., van Lint, H., Hoogendoorn, S., and Vuik, K. (2010). Lagrangian

formulation of multiclass kinematic wave model. Transportation Research

Record: Journal of the Transportation Research Board, (2188):29–36.

Van Wageningen-Kessels, F. L. M. (2013). Multi-class continuum traffic flow models:

Analysis and simulation methods. Ph.D. thesis, Delft University of Technology.

Wiedemann, R. (1974). Stimulation des Strassenverkehrsflusses. Heft 8 der schriftenreihe

des ifv, Universitâ"šÃ„â^žt Karlsruhe.

Wu, N. (2002). A new approach for modeling of Fundamental Diagrams. Transportation

Research Part A: Policy and Practice, 36(10):867–884. Doi: DOI: 10.1016/S0965-

(01)00043-X.

Yperman, I. (2007). The link transmission model for dynamic network loading.

Yperman, I., Logghe, S., Tampère, C. M. J., and Immers, B. (2006). Multicommodity link

transmission model for dynamic network loading. In Proceedings of the 85th Annual

Meeting of the Transportation Research Board.

Yuan, K., Knoop, V., and Hoogendoorn, S. (2015a). Capacity drop: A relation between the

speed in congestion and the queue discharge rate. In Proceedings of the 94th Annual

Meeting of the Transportation Research Board.

Yuan, Y. (2013). Lagranian Multi-Class traffic State Estimation. Ph.D. thesis, Delft University

of Technology.

Yuan, Y., Van Wageningen-Kessels, F., Van Lint, H., and Hoogendoorn, S. (2015b). Proceedings

of Traffic and Granular Flow 2011, chapter Two modeling and discretization

choices for Lagrangian multi-class first-order traffic flow model and their related (dis-

)advantages. Springer.

Zhang, H.M. (1999). A mathematical theory of traffic hysteresis. Transportation Research

Part B:Methodological, 33(1):1 – 23.

## Published

## Categories

## License

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.