IDEAS home Printed from https://ideas.repec.org/a/gam/jsusta/v14y2022i5p3036-d764462.html
   My bibliography  Save this article

Stability Analysis of Continuous Stochastic Linear Model

Author

Listed:
  • Jun Du

    (Key Laboratory of Transport Industry of Big Data Application Technologies for Comprehensive Transport, Ministry of Transport, Beijing Jiaotong University, Beijing 100044, China)

  • Bin Jia

    (Key Laboratory of Transport Industry of Big Data Application Technologies for Comprehensive Transport, Ministry of Transport, Beijing Jiaotong University, Beijing 100044, China)

  • Shiteng Zheng

    (Key Laboratory of Transport Industry of Big Data Application Technologies for Comprehensive Transport, Ministry of Transport, Beijing Jiaotong University, Beijing 100044, China)

Abstract

Many scholars have conducted research on the traffic oscillations and reproduced the growth pattern by establishing stochastic models and simulations. However, the growth pattern of oscillations caused by uncertainty have not been thoroughly studied. Recently, a frequency domain stability analysis method was proposed to analyze the discrete stochastic model. This paper extends this analysis to a continuous situation based on frequency domain tools (e.g., Laplace transform) by introducing a continuous bandlimited white noise. The analytical expression for the evolution of speed standard deviation has been derived. Our study of a homogeneous case reveals an interesting phenomenon: when | G ( ω ) | ∞ < 1 , the speed variance will converge to a constant value, which only depends on the self-disturbance of vehicles. The simulation results verified that the continuous models and corresponding discrete model tend to be consistent when the discrete time step tends to the infinitesimal. Overall, this paper makes up for the deficiency of previous studies on continuous oscillations in car-following theory and can potentially be used to develop new control strategies to help dampen traffic oscillations.

Suggested Citation

  • Jun Du & Bin Jia & Shiteng Zheng, 2022. "Stability Analysis of Continuous Stochastic Linear Model," Sustainability, MDPI, vol. 14(5), pages 1-13, March.
  • Handle: RePEc:gam:jsusta:v:14:y:2022:i:5:p:3036-:d:764462
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2071-1050/14/5/3036/pdf
    Download Restriction: no

    File URL: https://www.mdpi.com/2071-1050/14/5/3036/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Lang, Lei & Guo, Ning & Jiang, Rui & Zhu, Kong-jin, 2019. "An improved inertia model to reproduce car-following instability," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 526(C).
    2. Chen, Danjue & Laval, Jorge & Zheng, Zuduo & Ahn, Soyoung, 2012. "A behavioral car-following model that captures traffic oscillations," Transportation Research Part B: Methodological, Elsevier, vol. 46(6), pages 744-761.
    3. Zheng, Zuduo & Ahn, Soyoung & Chen, Danjue & Laval, Jorge, 2011. "Applications of wavelet transform for analysis of freeway traffic: Bottlenecks, transient traffic, and traffic oscillations," Transportation Research Part B: Methodological, Elsevier, vol. 45(2), pages 372-384, February.
    4. Coifman, Benjamin & Kim, Seoungbum, 2011. "Extended bottlenecks, the fundamental relationship, and capacity drop on freeways," Transportation Research Part A: Policy and Practice, Elsevier, vol. 45(9), pages 980-991, November.
    5. Arshad Jamal & Muhammad Tauhidur Rahman & Hassan M. Al-Ahmadi & Irfan Ullah & Muhammad Zahid, 2020. "Intelligent Intersection Control for Delay Optimization: Using Meta-Heuristic Search Algorithms," Sustainability, MDPI, vol. 12(5), pages 1-23, March.
    6. Yu Wang & Xiaopeng Li & Junfang Tian & Rui Jiang, 2020. "Stability Analysis of Stochastic Linear Car-Following Models," Transportation Science, INFORMS, vol. 54(1), pages 274-297, January.
    7. Laval, Jorge A. & Daganzo, Carlos F., 2006. "Lane-changing in traffic streams," Transportation Research Part B: Methodological, Elsevier, vol. 40(3), pages 251-264, March.
    8. Shiro Sawada, 2002. "Generalized Optimal Velocity Model For Traffic Flow," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 13(01), pages 1-12.
    9. Nagatani, Takashi, 2009. "Traffic states and fundamental diagram in cellular automaton model of vehicular traffic controlled by signals," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(8), pages 1673-1681.
    10. Taesung Hwang & Yanfeng Ouyang, 2015. "Urban Freight Truck Routing under Stochastic Congestion and Emission Considerations," Sustainability, MDPI, vol. 7(6), pages 1-16, May.
    11. Chen, Danjue & Laval, Jorge A. & Ahn, Soyoung & Zheng, Zuduo, 2012. "Microscopic traffic hysteresis in traffic oscillations: A behavioral perspective," Transportation Research Part B: Methodological, Elsevier, vol. 46(10), pages 1440-1453.
    12. Kai-ching Chu, 1974. "Decentralized Control of High-Speed Vehicular Strings," Transportation Science, INFORMS, vol. 8(4), pages 361-384, November.
    13. Li, Xiaopeng & Ouyang, Yanfeng, 2011. "Characterization of traffic oscillation propagation under nonlinear car-following laws," Transportation Research Part B: Methodological, Elsevier, vol. 45(9), pages 1346-1361.
    14. Mohammed Al-Turki & Arshad Jamal & Hassan M. Al-Ahmadi & Mohammed A. Al-Sughaiyer & Muhammad Zahid, 2020. "On the Potential Impacts of Smart Traffic Control for Delay, Fuel Energy Consumption, and Emissions: An NSGA-II-Based Optimization Case Study from Dhahran, Saudi Arabia," Sustainability, MDPI, vol. 12(18), pages 1-24, September.
    15. G. F. Newell, 1961. "Nonlinear Effects in the Dynamics of Car Following," Operations Research, INFORMS, vol. 9(2), pages 209-229, April.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Li, Xiaopeng & Wang, Xin & Ouyang, Yanfeng, 2012. "Prediction and field validation of traffic oscillation propagation under nonlinear car-following laws," Transportation Research Part B: Methodological, Elsevier, vol. 46(3), pages 409-423.
    2. Li, Xiaopeng & Cui, Jianxun & An, Shi & Parsafard, Mohsen, 2014. "Stop-and-go traffic analysis: Theoretical properties, environmental impacts and oscillation mitigation," Transportation Research Part B: Methodological, Elsevier, vol. 70(C), pages 319-339.
    3. Xu, Tu & Laval, Jorge, 2020. "Statistical inference for two-regime stochastic car-following models," Transportation Research Part B: Methodological, Elsevier, vol. 134(C), pages 210-228.
    4. Yao, Handong & Li, Qianwen & Li, Xiaopeng, 2020. "A study of relationships in traffic oscillation features based on field experiments," Transportation Research Part A: Policy and Practice, Elsevier, vol. 141(C), pages 339-355.
    5. Tian, Junfang & Li, Guangyu & Treiber, Martin & Jiang, Rui & Jia, Ning & Ma, Shoufeng, 2016. "Cellular automaton model simulating spatiotemporal patterns, phase transitions and concave growth pattern of oscillations in traffic flow," Transportation Research Part B: Methodological, Elsevier, vol. 93(PA), pages 560-575.
    6. Chen, Danjue & Ahn, Soyoung & Laval, Jorge & Zheng, Zuduo, 2014. "On the periodicity of traffic oscillations and capacity drop: The role of driver characteristics," Transportation Research Part B: Methodological, Elsevier, vol. 59(C), pages 117-136.
    7. Sun, Jie & Zheng, Zuduo & Sun, Jian, 2020. "The relationship between car following string instability and traffic oscillations in finite-sized platoons and its use in easing congestion via connected and automated vehicles with IDM based control," Transportation Research Part B: Methodological, Elsevier, vol. 142(C), pages 58-83.
    8. Oh, Simon & Yeo, Hwasoo, 2015. "Impact of stop-and-go waves and lane changes on discharge rate in recovery flow," Transportation Research Part B: Methodological, Elsevier, vol. 77(C), pages 88-102.
    9. Xu, Yueru & Zheng, Yuan & Yang, Ying, 2021. "On the movement simulations of electric vehicles: A behavioral model-based approach," Applied Energy, Elsevier, vol. 283(C).
    10. Yu Wang & Xiaopeng Li & Junfang Tian & Rui Jiang, 2020. "Stability Analysis of Stochastic Linear Car-Following Models," Transportation Science, INFORMS, vol. 54(1), pages 274-297, January.
    11. Laval, Jorge A. & Toth, Christopher S. & Zhou, Yi, 2014. "A parsimonious model for the formation of oscillations in car-following models," Transportation Research Part B: Methodological, Elsevier, vol. 70(C), pages 228-238.
    12. He, Zhengbing & Zheng, Liang & Guan, Wei, 2015. "A simple nonparametric car-following model driven by field data," Transportation Research Part B: Methodological, Elsevier, vol. 80(C), pages 185-201.
    13. Chen, Danjue & Laval, Jorge & Zheng, Zuduo & Ahn, Soyoung, 2012. "A behavioral car-following model that captures traffic oscillations," Transportation Research Part B: Methodological, Elsevier, vol. 46(6), pages 744-761.
    14. Zhou, Yang & Ahn, Soyoung & Wang, Meng & Hoogendoorn, Serge, 2020. "Stabilizing mixed vehicular platoons with connected automated vehicles: An H-infinity approach," Transportation Research Part B: Methodological, Elsevier, vol. 132(C), pages 152-170.
    15. Tian, Junfang & Zhu, Chenqiang & Chen, Danjue & Jiang, Rui & Wang, Guanying & Gao, Ziyou, 2021. "Car following behavioral stochasticity analysis and modeling: Perspective from wave travel time," Transportation Research Part B: Methodological, Elsevier, vol. 143(C), pages 160-176.
    16. Han, Youngjun & Ahn, Soyoung, 2018. "Stochastic modeling of breakdown at freeway merge bottleneck and traffic control method using connected automated vehicle," Transportation Research Part B: Methodological, Elsevier, vol. 107(C), pages 146-166.
    17. Song, Tao & Zhu, Wen-Xing, 2020. "Study on state feedback control strategy for car-following system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 558(C).
    18. Zheng, Shi-Teng & Jiang, Rui & Tian, Jun-Fang & Zhang, H.M. & Li, Zhen-Hua & Gao, Lan-Da & Jia, Bin, 2021. "Experimental study on properties of lightly congested flow," Transportation Research Part B: Methodological, Elsevier, vol. 149(C), pages 1-19.
    19. Kai Yuan & Victor L. Knoop & Serge P. Hoogendoorn, 2017. "A Microscopic Investigation Into the Capacity Drop: Impacts of Longitudinal Behavior on the Queue Discharge Rate," Transportation Science, INFORMS, vol. 51(3), pages 852-862, August.
    20. Li, Xiaopeng & Ouyang, Yanfeng, 2011. "Characterization of traffic oscillation propagation under nonlinear car-following laws," Transportation Research Part B: Methodological, Elsevier, vol. 45(9), pages 1346-1361.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jsusta:v:14:y:2022:i:5:p:3036-:d:764462. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.