Numerical analysis of stochastic PDEs in traffic flow: Investigating density-flow relations
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References
- AKAMATSU, T. (1996). Cyclic Flows, Markov Process and Stochastic Traffic Assignment.
Transportation Research Part B: Methodological , 30(5), pp. 369–386. - ALPEROVICH, T. and SOPASAKIS, A. (2008). Stochastic Description of Traffic Flow.
Journal of Statistical Physics , 133(6), pp. 1083–1105. - BENJAMIN, S. C., JOHNSON, N. F., and HUI, P. M. (1996). Cellular Automata Models of Traffic Flow along a Highway Containing a Junction.
Journal of Physics A: Mathematical and General , 29, pp. 3119–3127. - BHUTTO, I.A., BHUTTO, A.A., KHOKHAR, R.B., SOOMRO, M.A., and SHAIKH, F. (2023). The effect of uniform and exponential streams on Magnetohydrodynamic flows of viscous fluids.
VFAST Transactions on Mathematics , 11(1), pp. 121–140. - BONZANI, I. and MUSSONE, L. (2002). Stochastic Modelling of Traffic Flow.
Mathematical and Computer Modelling , 36(1-2), pp. 109–119. - CALVERT, S., TAALE, H., SNELDER, M., and HOOGENDOORN, S. (2015). Vehicle specific behaviour in macroscopic traffic modelling through stochastic advection invariant.
Transportation Research Procedia , 10, pp. 71–81. - FAN, T., WONG, S. C., ZHANG, Z., and DU, J. (2023). A dynamically bi-orthogonal solution method for a stochastic Lighthill-Whitham-Richards traffic flow model.
Computer-Aided Civil and Infrastructure Engineering , 38(11), pp. 1447–1461. - HO, I. W. H., LEUNG, K. K., and POLAK, J. W. (2010). Stochastic Model and Connectivity Dynamics for VANETs in Signalized Road Systems.
IEEE/ACM Transactions on Networking , 19(1), pp. 195–208. - CHENG, Q., LIN, Y., ZHOU, X. S., and LIU, Z. (2024). Analytical formulation for explaining the variations in traffic states: A fundamental diagram modeling perspective with stochastic parameters.
European Journal of Operational Research , 312(1), pp. 182–197. - CHU, K. C., SAIGAL, R., and SAITOU, K. (2016). Stochastic Lagrangian traffic flow modeling and real-time traffic prediction.
IEEE International Conference on Automation Science and Engineering (CASE) , pp. 213–218. IEEE. - JABARI, S., ZHENG, F., LIU, H., and FILIPOVSKA, M. (2018). Stochastic Lagrangian modeling of traffic dynamics. In
Proceedings of the 97th Annual Meeting of the Transportation Research Board , pp. 1–14. - JABARI, S. E. and LIU, H. X. (2012). A stochastic model of traffic flow: Theoretical foundations.
Transportation Research Part B: Methodological , 46(1), pp. 156–174. - JIANG, R. and WU, Q. S. (2002). Cellular Automata Models for Synchronized Traffic Flow.
Journal of Physics A: Mathematical and General , 36(2), pp. 381–390. - KENDZIORRA, A., WAGNER, P., and TOLEDO, T. (2016). A stochastic car following model.
Transportation Research Procedia , 15, pp. 198–207. - KERNER, B. S., KLENOV, S. L., and WOLF, D. E. (2002). Cellular Automata Approach to Three-Phase Traffic Theory.
Journal of Physics A: Mathematical and General , 35(47), pp. 9971–0013. - KHOKHAR, R.B., BHUTTO, A.A., SIDDIQUI, N.F., SHAIKH, F., and BHUTTO, I.A. (2023). Numerical analysis of flow rates, porous media, and Reynolds numbers affecting the combining and separating of Newtonian fluid flows.
VFAST Transactions on Mathematics , 11(1), pp. 217–236. - KNOSPE, W., SANTEN, L., SCHADSCHNEIDER, A., and SCHRECKENBERG, M. (2000). Towards a Realistic Microscopic Description of Highway Traffic.
Journal of Physics A: Mathematical and General , 33(48), pp. L477–L485. - LEE, S., NGODUY, D., and KEYVAN-EKBATANI, M. (2019). Integrated deep learning and stochastic car-following model for traffic dynamics on multi-lane freeways.
Transportation Research Part C: Emerging Technologies , 106, pp. 360–377. - LI, J., CHEN, Q. Y., WANG, H. Z., and NI, D. H. (2012). Analysis of LWR Model with Fundamental Diagram Subject to Uncertainties.
Transportmetrica , 8(6), pp. 387–405. - LIM, S. S., VOS, T., FLAXMAN, A. D., DANAEI, G., SHIBUYA, K., ADAIR-ROHANI, H. and PELIZZARI, P. M. (2012). A comparative risk assessment of burden of disease and injury attributable to 67 risk factors and risk factor clusters in 21 regions, 1990–2010: a systematic analysis for the Global Burden of Disease Study 2010.
The Lancet , 380(9859), pp. 2224–2260. - LIN, S., PAN, T. L., LAM, W. H. K., ZHONG, R. X., and DE SCHUTTER, B. (2018). Stochastic link flow model for signalized traffic networks with uncertainty in demand.
IFAC-PapersOnLine , 51(9), pp. 458–463. - LUO, X., LI, D., and ZHANG, S. (2019). Traffic flow prediction during the holidays based on DFT and SVR.
Journal of Sensors , 2019. - MARTÍNEZ, I. and JIN, W. (2020). Stochastic LWR model with heterogeneous vehicles: Theory and application for autonomous vehicles.
Transportation Research Procedia , 47, pp. 155–162. - NAGEL, K. and SCHRECKENBERG, M. (1992). A Cellular Automaton Model for Freeway Traffic.
Journal de Physique I , 2(12), pp. 2221–2229. - NI, DAIHENG, HSIEH, H. K., and JIANG, T. (2018). Modeling phase diagrams as stochastic processes with application in vehicular traffic flow.
Applied Mathematical Modelling , 53, pp. 106–117. - NISHINARI, K., TREIBER, M., and HELBING, D. (2003). Interpreting the Wide Scattering of Synchronized Traffic Data by Time Gap Statistics.
Physical Review E , 68(6), p. 067101. - SHAO, H., LAM, W. H., and TAM, M. L. (2006). A Reliability-Based Stochastic Traffic Assignment Model for Network with Multiple User Classes Under Uncertainty in Demand.
Networks and Spatial Economics , 6(3), pp. 173–204. - SUMALEE, A., ZHONG, R. X., PAN, T. L., and SZETO, W. Y. (2011). Stochastic Cell Transmission Model (SCTM): A Stochastic Dynamic Traffic Model for Traffic State Surveillance and Assignment.
Transportation Research Part B: Methodological , 45(3), pp. 507–533. - TAKAYASU, M. and TAKAYASU, H. (1993). 1/f Noise in a Traffic Model.
Fractals , 1(4), pp. 860–866. - THONHOFER, E. and JAKUBEK, S. (2018). Investigation of Stochastic Variation of Parameters for a Macroscopic Traffic Model.
Journal of Intelligent Transportation Systems , 22(6), pp. 547–564. - TIGHT, M., TIMMS, P., BANISTER, D., BOWMAKER, J., COPAS, J., DAY, A. and WATLING, D. (2011). Visions for a walking and cycling focussed urban transport system.
Journal of Transport Geography , 19(6), pp. 1580–1589. - TREIBER, M. and HELBING, D. (2003). Memory Effects in Microscopic Traffic Models and Wide Scattering in Flow-Density Data.
Physical Review E , 68(4), p. 046119. - TREIBER, M., KESTING, A., and HELBING, D. (2006). Understanding Widely Scattered Traffic Flows, the Capacity Drop, and Platoons as Effects of Variance-Driven Time Gaps.
Physical Review E , 74(1), p. 016123. - WADA, K., MARTÍNEZ, I., and JIN, W. L. (2020). Continuum car-following model of capacity drop at sag and tunnel bottlenecks.
Transportation Research Part C: Emerging Technologies , 113, pp. 260–276. - ZAMITH, M., LEAL-TOLEDO, R. C. P., CLUA, E., TOLEDO, E. M., and DE MAGALHÃES, G. V. (2015). A new stochastic cellular automata model for traffic flow simulation with drivers’ behavior prediction.
Journal of Computational Science , 9, pp. 51–56. - ZHENG, S. T., JIANG, R., JIA, B., TIAN, J., and GAO, Z. (2020). Impact of stochasticity on traffic flow dynamics in macroscopic continuum models.
Transportation Research Record , 2674(10), pp. 690–704. - ZHENG, F., LIU, C., LIU, X., JABARI, S. E., and LU, L. (2020). Analyzing the impact of automated vehicles on uncertainty and stability of the mixed traffic flow.
Transportation Research Part C: Emerging Technologies , 112, pp. 203–219. - XU, T. and LAVAL, J. A. (2019). Analysis of a two-regime stochastic car-following model: Explaining capacity drop and oscillation instabilities.
Transportation Research Record , 2673(10), pp. 610–619.
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