Modelling and Optimisation of Flows on Networks pp 335-355 | Cite as

# Self-Organized Network Flows

## Abstract

A model for traffic flow in street networks or material flows in supply networks is presented, that takes into account the conservation of cars or materials and other significant features of traffic flows such as jam formation, spillovers, and load-dependent transportation times. Furthermore, conflicts or coordination problems of intersecting or merging flows are considered as well. Making assumptions regarding the permeability of the intersection as a function of the conflicting flows and the queue lengths, we find self-organized oscillations in the flows similar to the operation of traffic lights.

## Keywords

Traffic Flow Shock Front Queue Length Traffic Light Road Section## Notes

### Acknowledgements

The authors are grateful for partial financial support by the German Research Foundation (research projects He 2789/5-1, 8-1) and by the “Cooperative Center for Communication Networks Data Analysis”, a NAP project sponsored by the Hungarian National Office of Research and Technology under grant No. KCKHA005.

## References

- 1.V. Astarita, Flow propagation description in dynamic network loading models, in
*Proceedings of the IV International Conference on Applications of Advanced Technologies in Transportation Engineering (AATT)*, ed. by Y.J. Stephanedes, F. Filippi (Capri, Italy, 1995), pp. 599–603Google Scholar - 2.V. Astarita, Node and link models for traffic simulation. Math. Comput. Model.
**35**, 643–656 (2002)MathSciNetzbMATHCrossRefGoogle Scholar - 3.M.K. Banda, M. Herty, A. Klar, Gas flow in pipeline networks. Netw. Heterogeneous Media
**1**, 41–56 (2006)MathSciNetzbMATHCrossRefGoogle Scholar - 4.G. Bretti, R. Natalini, B. Piccoli, Numerical approximations of a traffic flow model on networks. Netw. Heterogeneous Media
**1**, 57–84 (2006)MathSciNetzbMATHCrossRefGoogle Scholar - 5.M. Carey, Y.E. Ge, M. McCartney, A whole-link travel-time model with desirable properties. Transp. Sci.
**37**, 83–96 (2003)CrossRefGoogle Scholar - 6.C.F. Daganzo, Requiem for second-order fluid approximations of traffic flow. Transp. Res. B
**29**, 277–286 (1995)CrossRefGoogle Scholar - 7.C.F. Daganzo, The cell transmission model, Part II: Network traffic. Transp. Res. B
**29**, 79–93 (1995)Google Scholar - 8.B. De Schutter, Optimizing acyclic traffic signal switching sequences through an extended linear complementarity problem formulation. Eur. J. Oper. Res.
**139**, 400–415 (2002)zbMATHCrossRefGoogle Scholar - 9.M. Garavello, B. Piccoli,
*Traffic Flow on Networks*(American Institute of Mathematical Sciences, Springfield, 2006)zbMATHGoogle Scholar - 10.S. Goettlich, M. Herty, A. Klar, Modelling and optimization of supply chains on complex networks. Comm. Math. Sci.
**4**, 315–330 (2006)zbMATHGoogle Scholar - 11.D. Helbing, Traffic and related self-driven many-particle systems. Rev. Mod. Phys.
**73**, 1067–1141 (2001)CrossRefGoogle Scholar - 12.D. Helbing, A section-based queueing-theoretical traffic model for congestion and travel time analysis in networks. J. Phys. Math. Gen.
**36**, L593–L598 (2003)MathSciNetzbMATHCrossRefGoogle Scholar - 13.D. Helbing, Production, supply, and traffic systems: A unified description, in
*Traffic and Granular Flow ’03*, ed. by S.P. Hoogendoorn, S. Luding, P.H.L. Bovy, M. Schreckenberg, D.E. Wolf (Springer, Berlin, 2005), pp. 173–188CrossRefGoogle Scholar - 14.D. Helbing, P. Molnár, Social force model for pedestrian dynamics. Phys. Rev. E
**51**, 4282–4286 (1995)CrossRefGoogle Scholar - 15.D. Helbing, S. Lämmer, T. Seidel, P. Seba, T. Platkowski, Physics, stability and dynamics of supply networks. Phys. Rev. E
**70**, 066116 (2004)CrossRefGoogle Scholar - 16.D. Helbing, S. Lämmer, J.-P. Lebacque, Self-organized control of irregular or perturbed network traffic, in
*Optimal Control and Dynamic Games*, ed. by C. Deissenberg, R.F. Hartl (Springer, Dordrecht, 2005), pp. 239–274CrossRefGoogle Scholar - 17.D. Helbing, L. Buzna, A. Johansson, T. Werner, Self-organized pedestrian crowd dynamics: Experiments, simulations, and design solutions. Transp. Sci.
**39**, 1–24 (2005)CrossRefGoogle Scholar - 18.D. Helbing, R. Jiang, M. Treiber, Analytical investigation of oscillations in intersecting flows of pedestrian and vehicle traffic. Phys. Rev. E
**72**, 046130 (2005)CrossRefGoogle Scholar - 19.D. Helbing, A. Johansson, J. Mathiesen, M.H. Jensen, A. Hansen, Analytical approach to continuous and intermittent bottleneck flows. Phys. Rev. Lett.
**97**, 168001 (2006)CrossRefGoogle Scholar - 20.D. Helbing, T. Seidel, S. Lämmer, K. Peters, Self-organization principles in supply networks and production systems. in
*Econophysics and Sociophysics*, ed. by B.K. Chakrabarti, A. Chakraborti, A. Chatterjee (Wiley, New York, 2006)Google Scholar - 21.M. Herty, A. Klar, Modeling, simulation, and optimization of traffic flow networks. SIAM Appl. Math.
**64**, 565–582 (2003)Google Scholar - 22.M. Herty, A. Klar, Simplified dynamics and optimization of large scale traffic flow networks. Math. Mod. Meth. Appl. Sci.
**14**, 579–601 (2004)MathSciNetzbMATHCrossRefGoogle Scholar - 23.M. Herty, S. Moutari, M. Rascle, Optimization criteria for modelling intersections of vehicular traffic flow. NHM
**1**, 275–294 (2006)MathSciNetzbMATHCrossRefGoogle Scholar - 24.M. Hilliges, W. Weidlich, A phenomenological model for dynamic traffic flow in networks. Transp. Res. B
**29**, 407–431 (1995)CrossRefGoogle Scholar - 25.B. Kerner,
*The Physics of Traffic*(Springer, Berlin, 2004)CrossRefGoogle Scholar - 26.J.-P. Lebacque, M.M. Khoshyaran, First-order macroscopic traffic flow models: Intersection modeling, network modeling, in
*16th International Symposium on Transportation and Traffic Theory*, ed. by H.S. Mahmasani (Elsevier, Amsterdam, 2005), pp. 365–386Google Scholar - 27.M.J. Lighthill, G.B. Whitham, On kinematic waves: II. A theory of traffic on long crowded roads. Proc. R. Soc. Lond. A
**229**, 317–345 (1955)MathSciNetzbMATHGoogle Scholar - 28.A.J. Mayne, Some further results in the theory of pedestrians and road traffic. Biometrika
**41**, 375–389 (1954)MathSciNetGoogle Scholar - 29.C.H. Papadimitriou, J.N. Tsitsiklis, The complexity of optimal queuing network control. Math. Oper. Res.
**24**, 293–305 (1999)MathSciNetzbMATHCrossRefGoogle Scholar - 30.K. Peters, U. Parlitz, Hybrid systems forming strange billiards. Int. J. Bifurcat. Chaos
**13**, 2575–2588 (2003)MathSciNetzbMATHCrossRefGoogle Scholar - 31.M. Schönhof, D. Helbing, Empirical features of congested traffic states and their implications for traffic modelling. Transp. Sci.
**41**, 135–166 (2007)CrossRefGoogle Scholar - 32.M. Treiber, A. Hennecke, D. Helbing, Congested traffic states in empirical observations and microscopic simulations. Phys. Rev. E
**62**, 1805–1824 (2000)CrossRefGoogle Scholar - 33.R.J. Troutbeck, Average delay at an unsignalized intersection with two major each having a dichotomized headway distribution. Transp. Sci.
**20**, 272–286 (1986)CrossRefGoogle Scholar - 34.R.J. Troutbeck, W. Brilon, Unsignalized intersection theory, in
*Traffic Flow Theory: A State-of-the-Art Report*, ed. by N. Gartner, H. Mahmassani, C.H. Messer, H. Lieu, R. Cunard, A.K. Rathi (Transportation Research Board, Washington, 1997), pp. 8.1–8.47Google Scholar - 35.G.B. Whitham,
*Linear and Nonlinear Waves*(Wiley, New York, 1974)zbMATHGoogle Scholar