Abstract
Uniqueness is an important characteristic of network user equilibrium. Uniqueness in dynamic user equilibrium (DUE) has been shown to exist under certain conditions, whereas a case with multiple Wardrop equilibria has been found in a twofold symmetric network with two bottlenecks. This setting is very special, and therefore, it is worthwhile to investigate the factors that are essential to obtain non-unique solutions in this case in order to discuss the implication of this special example for real network problems. The asymmetric structure of the network, stochastic user equilibrium (SUE), and whole-link model were employed to check whether these factors affect the existence of multiple equilibria. It was shown that placing bottlenecks with equivalent capacities causes the singularity of the linear equation system that describes Wardrop’s first principle in a particular network structure called as a loopy network. Then, it was shown that this singularity causes the existence of multiple equilibria. On the other hand, this singularity was not found in SUE or the whole-link cases, and therefore, non-unique solutions were not found in them. It was also shown that the singularity does not appear to be the only source of non-uniqueness by presenting another example in which at least three equilibrium solutions exists when either the bottleneck model or the whole-link model is employed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Carey M (2001) Dynamic traffic assignment with more flexible modelling within links. Networks Spat Econ 1(3–4):349–375
Friesz TL, Bernstein D, Smith TE, Tobin RL, Wie BW (1993) A variational inequality formulation of the dynamic network user equilibrium problem. Oper Res 41(1):179–191
Friesz T, Bernstein D, Suo Z, Tobin R (2001) Dynamic network user equilibrium with state-dependent time lags. Networks Spat Econ 1(3–4):319–347
Iryo T (2011) Multiple equilibria in a dynamic traffic network. Transp Res B 45(6):867–879
Iryo T (2013) Properties of dynamic user equilibrium solution: existence, uniqueness, stability, and robust solution methodology. Transportmetrica B Transp Dyn 1(1):52–67
Kuwahara M (1990) Some aspects of the dynamic equilibrium assignment in oversaturated networks. JSCE J Infrastruct Plan Manag 419/IV-13:123–126, in Japanese
Mounce R (2001) Non-monotonicity in dynamic traffic assignment networks. In: Proceedings of the 33rd Universities Transport Study Group Annual Conference. University of Oxford, UK
Mounce R (2007) Existence of equilibrium in a continuous dynamic queueing model for traffic networks. In: Heydecker BG (ed) Mathematics in transport. Elsevier, Oxford, pp 219–229
Mounce R, Smith MJ (2007) Uniqueness of equilibrium in steady state and dynamic traffic networks. In: Allsop RE, Bell MGH, Heydecker BG (eds) Transportation and traffic theory 2007. Elsevier, Oxford, pp 281–299
Smith MJ (1984) The stability of a dynamic model of traffic assignment—an application of a method of Lyapunov. Transp Sci 18(3):245–252
Smith MJ, Ghali M (1990) Dynamic traffic assignment and dynamic traffic control. In: Masaki K (ed) Transportation and traffic theory: Proceedings of the 11th International Symposium on Transportation and Traffic Theory. Elsevier Science Publishing, New York, pp 273–290
Acknowledgments
This study was financially supported by the Japan Society for the Promotion of Science Grant-in-Aid #22686048.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Iryo, T. Investigating Factors for Existence of Multiple Equilibria in Dynamic Traffic Network. Netw Spat Econ 15, 599–616 (2015). https://doi.org/10.1007/s11067-013-9206-6
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11067-013-9206-6