Starting-wave and Optimal Density in a Queue

  • Akiyasu Tomoeda
  • Daichi Yanagisawa
  • Takashi Imamura
  • Katsuhiro Nishinari
Conference paper


We have investigated the fundamental relation between the density and the propagation speed of starting-wave, which is a wave of people’s successive reaction in a relaxation process of a queue, by both our mathematical model built on the stochastic cellular automata and experimental measurements. The analysis of our mathematical model implies that the relation is well approximated by power law a = αρ β (β ≠ 1) and the experimental results verify this feature. Moreover, when the starting-wave is characterized by power law (β ≠ 1), we have revealed the existence of optimal density, where the required time which is sum of the waiting time until the starting-wave reaches and the travel time of last pedestrians in a queue to pass the head position of the initial queue is minimized. This optimal density inevitably plays a significant role to achieve a smooth movement of crowds and vehicles in a queue.


Jamology Stochastic cellular automaton Queuing system Starting-wave Optimization problem 



We thank Kozo Keikaku Engineering Inc. in Japan and the members of the Meiji University Global COE Program “Formation and Development of Mathematical Sciences Based on Modeling and Analysis” for the assistance of the experiments, which is described in Sect. 3. We also acknowledge the support of Japan Society for the Promotion of Science and Japan Science and Technology Agency.


  1. 1.
    D. Chowdhury, L. Santen and A. Schadschneider, Phys. Rep. 329, 199 (2000).Google Scholar
  2. 2.
    D. Helbing, Rev. Mod. Phys. 73, 1067 (2001).Google Scholar
  3. 3.
    A. Schadschneider, D. Chowdhury and K. Nishinari, STOCHASTIC TRANSPORT IN COMPLEX SYSTEMS FROM MOLECULES TO VEHICLES, Elsevier, (2010).Google Scholar
  4. 4.
    A. Tomoeda, D. Yanagisawa and K. Nishinari, fifth International Conference on Pedestrian and Evacuation Dynamics, Springer.Google Scholar
  5. 5.
    B. Derrida, Phys. Rep., 301, 65 (1998).Google Scholar
  6. 6.
    N. Rajewsky, L. Santen, A. Schadschneider and M. Schreckenberg, J. Stat. Phys., 92, 151 (1998).Google Scholar
  7. 7.
    G. M. Schutz, Phase Transitions and Critical Phenomena, 19 (Acad. Press, 2001), G. M. Schutz, J. Phys. A: Math. Gen., 36, R339 (2003).Google Scholar
  8. 8.
    F. Spitzer, Adv. Math., 5, 246 (1970).Google Scholar
  9. 9.
    M. R. Evans and T. Hanney, J. Phys. A: Math. Gen., 38, R195 (2005).Google Scholar
  10. 10.
    M. Kanai, J. Phys. A, 40, 7127 (2007).Google Scholar
  11. 11.
    G. Bolch, S. Greiner, H. de Meer and K. S. Trivedi, Queueing Networks and Markov Chains, Wiley-Interscience, U.S.A., 1998.Google Scholar
  12. 12.
    C. Arita, Phys. Rev. E, 80, 051119 (2009).Google Scholar
  13. 13.
    C. Arita and D. Yanagisawa, J. Stat. Phys., 141, 829 (2010).Google Scholar
  14. 14.
    C. Arita and A. Schadschneider, Phys. Rev. E, 83, 051128 (2011).Google Scholar
  15. 15.
    A. Seyfried, B. Steffen, W. Klingsch, and M. Boltes. J. Stat. Mech., 10002 (2005).Google Scholar
  16. 16.
    K. Joahnsson, Commun. Math. Phys., 209, 437 (2000).Google Scholar
  17. 17.
    T. Imamura and T. Sasamoto, J. Stat. Phys., 128, 799 (2007).Google Scholar
  18. 18.
    A. Borodin, P.L. Ferrari, and T. Sasamoto, Commun. Math. Phys. 283, 417 (2008).Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Akiyasu Tomoeda
    • 1
    • 2
  • Daichi Yanagisawa
    • 3
  • Takashi Imamura
    • 4
  • Katsuhiro Nishinari
    • 4
    • 5
  1. 1.Meiji Institute for Advanced Study of Mathematical SciencesMeiji UniversityKawasakiJapan
  2. 2.JST CRESTKawasakiJapan
  3. 3.College of ScienceIbaraki UniversityMitoJapan
  4. 4.Research Center for Advanced Science and TechnologyThe University of TokyoMeguro-kuJapan
  5. 5.Department of Aeronautics and Astronautics, School of EngineeringThe University of TokyoBunkyo-kuJapan

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