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Self-Organized Network Flows

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Modelling and Optimisation of Flows on Networks

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2062))

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.

First published in: Networks and Heterogeneous Media 2(2), 193Ú–210 (2007), ©aimSciences.org, reproduction with kind permission.

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Notes

  1. 1.

    The arrival flow A j (t) has previously been denoted by \({Q}_{j}^{\mathrm{arr}}(t)\), the potential arrival flow \(\widehat{{A}}_{j}(t)\) by \({Q}_{j}^{\mathrm{arr,pot}}(t)\), the departure flow O j (t) by \({Q}_{j}^{\mathrm{dep}}(t)\) and the potential departure flow \(\widehat{{O}}_{j}(t)\) by \({Q}_{j}^{\mathrm{dep,pot}}(t)\).

  2. 2.

    Of course, a first-come-first-serve or right-before-left rule will be sufficient at small traffic volumes.

  3. 3.

    If ΔT 1 and ΔT 2 denote the green time periods for the intersecting flows 1 and 2, respectively, the corresponding red time periods for a periodic signal control are ΔT 2 and ΔT 1, to which the switching setup time of duration τ must be added. From formula (23) and with \({O}_{i} =\widehat{ Q}\) we obtain \(\Delta {T}_{1} = (\Delta {T}_{2} + \tau )\widehat{Q}/(\widehat{Q} - {A}_{1})\) and \(\Delta {T}_{2} = (\Delta {T}_{1} + \tau )\widehat{Q}/(\widehat{Q} - {A}_{2})\). Using the definition \({T}^{\mathrm{cyc}} = \Delta {T}_{1} + \tau + \Delta {T}_{2} + \tau \) for the cycle time, we finally arrive at (66).

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. 599–603

    Google Scholar 

  2. V. Astarita, Node and link models for traffic simulation. Math. Comput. Model. 35, 643–656 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. M.K. Banda, M. Herty, A. Klar, Gas flow in pipeline networks. Netw. Heterogeneous Media 1, 41–56 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. G. Bretti, R. Natalini, B. Piccoli, Numerical approximations of a traffic flow model on networks. Netw. Heterogeneous Media 1, 57–84 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. M. Carey, Y.E. Ge, M. McCartney, A whole-link travel-time model with desirable properties. Transp. Sci. 37, 83–96 (2003)

    Article  Google Scholar 

  6. C.F. Daganzo, Requiem for second-order fluid approximations of traffic flow. Transp. Res. B 29, 277–286 (1995)

    Article  Google 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)

    Article  MATH  Google Scholar 

  9. M. Garavello, B. Piccoli, Traffic Flow on Networks (American Institute of Mathematical Sciences, Springfield, 2006)

    MATH  Google Scholar 

  10. S. Goettlich, M. Herty, A. Klar, Modelling and optimization of supply chains on complex networks. Comm. Math. Sci. 4, 315–330 (2006)

    MATH  Google Scholar 

  11. D. Helbing, Traffic and related self-driven many-particle systems. Rev. Mod. Phys. 73, 1067–1141 (2001)

    Article  Google 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)

    Article  MathSciNet  MATH  Google 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. 173–188

    Chapter  Google Scholar 

  14. D. Helbing, P. Molnár, Social force model for pedestrian dynamics. Phys. Rev. E 51, 4282–4286 (1995)

    Article  Google 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)

    Article  Google 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. 239–274

    Chapter  Google 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)

    Article  Google 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)

    Article  Google 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)

    Article  Google 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)

    Article  MathSciNet  MATH  Google Scholar 

  23. M. Herty, S. Moutari, M. Rascle, Optimization criteria for modelling intersections of vehicular traffic flow. NHM 1, 275–294 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. M. Hilliges, W. Weidlich, A phenomenological model for dynamic traffic flow in networks. Transp. Res. B 29, 407–431 (1995)

    Article  Google Scholar 

  25. B. Kerner, The Physics of Traffic (Springer, Berlin, 2004)

    Book  Google 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. 365–386

    Google 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)

    MathSciNet  MATH  Google Scholar 

  28. A.J. Mayne, Some further results in the theory of pedestrians and road traffic. Biometrika 41, 375–389 (1954)

    MathSciNet  Google Scholar 

  29. C.H. Papadimitriou, J.N. Tsitsiklis, The complexity of optimal queuing network control. Math. Oper. Res. 24, 293–305 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  30. K. Peters, U. Parlitz, Hybrid systems forming strange billiards. Int. J. Bifurcat. Chaos 13, 2575–2588 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  31. M. Schönhof, D. Helbing, Empirical features of congested traffic states and their implications for traffic modelling. Transp. Sci. 41, 135–166 (2007)

    Article  Google Scholar 

  32. M. Treiber, A. Hennecke, D. Helbing, Congested traffic states in empirical observations and microscopic simulations. Phys. Rev. E 62, 1805–1824 (2000)

    Article  Google 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)

    Article  Google 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. 8.1–8.47

    Google Scholar 

  35. G.B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974)

    MATH  Google Scholar 

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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.

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Authors and Affiliations

  1. ETH Zurich, UNO D11, Universitätstr. 41, 8092, Zurich, Switzerland

    Dirk Helbing

  2. Institute for Transport and Economics, Dresden University of Technology, Andreas-Schubert-Str. 23, 01062, Dresden, Germany

    Jan Siegmeier & Stefan Lämmer

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Helbing, D., Siegmeier, J., Lämmer, S. (2013). Self-Organized Network Flows. In: Modelling and Optimisation of Flows on Networks. Lecture Notes in Mathematics(), vol 2062. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32160-3_6

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