
Previous Article
Editorial
 NHM Home
 This Issue

Next Article
Gas flow in pipeline networks
HTTP turbulence
1.  Département d'Informatique, ENS, 45 rue d'Ulm, 75005 Paris, France 
2.  Thomson Research 46, quai Alphonse Le Gallo, 92648 Boulogne Cedex, France 
3.  Alcatel Bell, Francis Wellesplein 1, B2018 Antwerp, Belgium 
4.  Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Avenue, Ottawa, Ontario K1N 6M8, Canada 
We study the meanfield model obtained by letting the number of flows go to infinity. This meanfield limit may have two stable regimes: one with out congestion in the link, in which the density of transmission rate can be explicitly described, the other one with periodic congestion epochs, where the intercongestion time can be characterized as the solution of a fixed point equation, that we compute numerically, leading to a density of transmission rate given by as the solution of a Fredholm equation. It is shown that for certain values of the parameters (more precisely when the link capacity per user is not significantly larger than the load per user), each of these two stable regimes can be reached depending on the initial condition. This phenomenon can be seen as an analogue of turbulence in fluid dynamics: for some initial conditions, the transfers progress in a fluid and interactionless way; for others, the connections interact and slow down because of the resulting fluctuations, which in turn perpetuates interaction forever, in spite of the fact that the load per user is less than the capacity per user. We prove that this phenomenon is present in the Tahoe case and both the numerical method that we develop and simulations suggest that it is also be present in the Reno case. It translates into a bistability phenomenon for the finite population model within this range of parameters.
[1] 
Zhen Li, Jicheng Liu. Synchronization for stochastic differential equations with nonlinear multiplicative noise in the mean square sense. Discrete & Continuous Dynamical Systems  B, 2017, 22 (11) : 128. doi: 10.3934/dcdsb.2019103 
[2] 
Shu Zhang, Jian Xu. Timevarying delayed feedback control for an internet congestion control model. Discrete & Continuous Dynamical Systems  B, 2011, 16 (2) : 653668. doi: 10.3934/dcdsb.2011.16.653 
[3] 
Yuri B. Gaididei, Carlos Gorria, Rainer Berkemer, Peter L. Christiansen, Atsushi Kawamoto, Mads P. Sørensen, Jens Starke. Stochastic control of traffic patterns. Networks & Heterogeneous Media, 2013, 8 (1) : 261273. doi: 10.3934/nhm.2013.8.261 
[4] 
Shu Zhang, Yuan Yuan. The Filippov equilibrium and sliding motion in an internet congestion control model. Discrete & Continuous Dynamical Systems  B, 2017, 22 (3) : 11891206. doi: 10.3934/dcdsb.2017058 
[5] 
Simone Göttlich, Ute Ziegler. Traffic light control: A case study. Discrete & Continuous Dynamical Systems  S, 2014, 7 (3) : 483501. doi: 10.3934/dcdss.2014.7.483 
[6] 
Lijuan Wang, Qishu Yan. Optimal control problem for exact synchronization of parabolic system. Mathematical Control & Related Fields, 2019, 9 (3) : 411424. doi: 10.3934/mcrf.2019019 
[7] 
Mohamed Ouzahra. Controllability of the semilinear wave equation governed by a multiplicative control. Evolution Equations & Control Theory, 2019, 8 (4) : 669686. doi: 10.3934/eect.2019039 
[8] 
Oliver Kolb, Simone Göttlich, Paola Goatin. Capacity drop and traffic control for a second order traffic model. Networks & Heterogeneous Media, 2017, 12 (4) : 663681. doi: 10.3934/nhm.2017027 
[9] 
Yinfei Li, Shuping Chen. Optimal traffic signal control for an $M\times N$ traffic network. Journal of Industrial & Management Optimization, 2008, 4 (4) : 661672. doi: 10.3934/jimo.2008.4.661 
[10] 
Fethallah Benmansour, Guillaume Carlier, Gabriel Peyré, Filippo Santambrogio. Numerical approximation of continuous traffic congestion equilibria. Networks & Heterogeneous Media, 2009, 4 (3) : 605623. doi: 10.3934/nhm.2009.4.605 
[11] 
Gang Qian, Deren Han, Hongjin He. Congestion control with pricing in the absence of demand and cost functions: An improved trial and error method. Journal of Industrial & Management Optimization, 2010, 6 (1) : 103121. doi: 10.3934/jimo.2010.6.103 
[12] 
Jingzhen Liu, Ka Fai Cedric Yiu, Alain Bensoussan. Ergodic control for a mean reverting inventory model. Journal of Industrial & Management Optimization, 2018, 14 (3) : 857876. doi: 10.3934/jimo.2017079 
[13] 
Michael Herty, Lorenzo Pareschi, Sonja Steffensen. Meanfield control and Riccati equations. Networks & Heterogeneous Media, 2015, 10 (3) : 699715. doi: 10.3934/nhm.2015.10.699 
[14] 
Shahad Alazzawi, Jicheng Liu, Xianming Liu. Convergence rate of synchronization of systems with additive noise. Discrete & Continuous Dynamical Systems  B, 2017, 22 (2) : 227245. doi: 10.3934/dcdsb.2017012 
[15] 
J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the SakawaShindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391406. doi: 10.3934/mcrf.2016008 
[16] 
Alexandre M. Bayen, Hélène Frankowska, JeanPatrick Lebacque, Benedetto Piccoli, H. Michael Zhang. Special issue on Mathematics of Traffic Flow Modeling, Estimation and Control. Networks & Heterogeneous Media, 2013, 8 (3) : iii. doi: 10.3934/nhm.2013.8.3i 
[17] 
Lino J. AlvarezVázquez, Néstor GarcíaChan, Aurea Martínez, Miguel E. VázquezMéndez. Optimal control of urban air pollution related to traffic flow in road networks. Mathematical Control & Related Fields, 2018, 8 (1) : 177193. doi: 10.3934/mcrf.2018008 
[18] 
Jingmei Zhou, Xiangmo Zhao, Xin Cheng, Zhigang Xu. Visualization analysis of traffic congestion based on floating car data. Discrete & Continuous Dynamical Systems  S, 2015, 8 (6) : 14231433. doi: 10.3934/dcdss.2015.8.1423 
[19] 
Yves Achdou, Mathieu Laurière. On the system of partial differential equations arising in mean field type control. Discrete & Continuous Dynamical Systems  A, 2015, 35 (9) : 38793900. doi: 10.3934/dcds.2015.35.3879 
[20] 
Yanqing Hu, Zaiming Liu, Jinbiao Wu. Optimal impulse control of a meanreverting inventory with quadratic costs. Journal of Industrial & Management Optimization, 2018, 14 (4) : 16851700. doi: 10.3934/jimo.2018027 
2018 Impact Factor: 0.871
Tools
Metrics
Other articles
by authors
[Back to Top]