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Bounds and Approximations for the Fixed-Cycle Traffic-Light Queue

Author

Listed:
  • M. S. van den Broek

    (Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands)

  • J. S. H. van Leeuwaarden

    (EURANDOM, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands)

  • I. J. B. F. Adan

    (Eindhoven University of Technology and EURANDOM, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands)

  • O. J. Boxma

    (Eindhoven University of Technology and EURANDOM, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands)

Abstract

This paper deals with the fixed-cycle traffic-light (FCTL) queue, where vehicles arrive at an intersection controlled by a traffic light and form a queue. The traffic light alternates between green and red periods, and delayed vehicles are assumed to depart during the green period at equal time intervals. The key performance characteristic in the FCTL queue is the so-called mean overflow, defined as the mean queue length at the end of a green period.An exact solution for the mean overflow is available, but it has been considered to be of little practical value because it requires some numerical procedures. Therefore, most of the literature on the FCTL queue is about deriving approximations for the mean overflow. In deriving these approximations, most authors first approximate the FCTL queue by a bulk-service queue, approximate the mean overflow in the bulk-service queue, and use this as an approximation for the mean overflow in the FCTL queue. So far no quantitative comparison of both models has been given. We compare both models and assess the quality of the approximation for various settings of the parameter values. In this comparison and throughout the paper we do not restrict ourselves to Poisson arrivals, but consider a more general arrival process instead.We discuss the numerical issues that need to be resolved to calculate the exact expression for the mean overflow in both queues and show that clear computational schemes are available. Next, we present several bounds and approximations of the mean overflow that do not require numerical procedures. In particular, we derive a new approximation based on the heavy traffic limit and a scaling argument. We compare the new bounds and approximation with the existing ones. We elaborate on the impact of several parameters, like the length of the green and red period and the variance of the arrival distribution. Each of these parameters turns out to be crucial.

Suggested Citation

  • M. S. van den Broek & J. S. H. van Leeuwaarden & I. J. B. F. Adan & O. J. Boxma, 2006. "Bounds and Approximations for the Fixed-Cycle Traffic-Light Queue," Transportation Science, INFORMS, vol. 40(4), pages 484-496, November.
    • Handle: RePEc:inm:ortrsc:v:40:y:2006:i:4:p:484-496
    DOI: 10.1287/trsc.1050.0146
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    References listed on IDEAS

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    1. D. Denteneer & A.J.E.M. Janssen & J.S.H. van Leeuwaarden, 2005. "Moment inequalities for the discrete-time bulk service queue," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 61(1), pages 85-108, March.
    2. J. S. H. van Leeuwaarden, 2006. "Delay Analysis for the Fixed-Cycle Traffic-Light Queue," Transportation Science, INFORMS, vol. 40(2), pages 189-199, May.
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    Cited by:

    1. Gao, Yuhong & Qu, Zhaowei & Song, Xianmin & Yun, Zhenyu & Xia, Yingji, 2021. "A novel relationship model between signal timing, queue length and travel speed," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 583(C).
    2. Hofleitner, Aude & Herring, Ryan & Bayen, Alexandre, 2012. "Arterial travel time forecast with streaming data: A hybrid approach of flow modeling and machine learning," Transportation Research Part B: Methodological, Elsevier, vol. 46(9), pages 1097-1122.
    3. A. Oblakova & A. Al Hanbali & R. J. Boucherie & J. C. W. Ommeren & W. H. M. Zijm, 2019. "An exact root-free method for the expected queue length for a class of discrete-time queueing systems," Queueing Systems: Theory and Applications, Springer, vol. 92(3), pages 257-292, August.
    4. Yang, Qiaoli & Shi, Zhongke, 2018. "Effects of the design of waiting areas on the dynamic behavior of queues at signalized intersections," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 509(C), pages 181-195.
    5. Yang, Qiaoli & Shi, Zhongke & Yu, Shaowei & Zhou, Jie, 2018. "Analytical evaluation of the use of left-turn phasing for single left-turn lane only," Transportation Research Part B: Methodological, Elsevier, vol. 111(C), pages 266-303.
    6. M. A. A. Boon & A. J. E. M. Janssen & J. S. H. Leeuwaarden & R. W. Timmerman, 2019. "Pollaczek contour integrals for the fixed-cycle traffic-light queue," Queueing Systems: Theory and Applications, Springer, vol. 91(1), pages 89-111, February.
    7. Comert, Gurcan, 2016. "Queue length estimation from probe vehicles at isolated intersections: Estimators for primary parameters," European Journal of Operational Research, Elsevier, vol. 252(2), pages 502-521.
    8. Yang, Qiaoli & Shi, Zhongke, 2021. "The queue dynamics of protected/permissive left turns at pre-timed signalized intersections," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 562(C).
    9. G. Celik & S. C. Borst & P. A. Whiting & E. Modiano, 2016. "Dynamic scheduling with reconfiguration delays," Queueing Systems: Theory and Applications, Springer, vol. 83(1), pages 87-129, June.
    10. Boon, Marko A.A. & van Leeuwaarden, Johan S.H., 2018. "Networks of fixed-cycle intersections," Transportation Research Part B: Methodological, Elsevier, vol. 117(PA), pages 254-271.
    11. Yang, Qiaoli & Shi, Zhongke, 2018. "The evolution process of queues at signalized intersections under batch arrivals," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 505(C), pages 413-425.
    12. Comert, Gurcan, 2013. "Effect of stop line detection in queue length estimation at traffic signals from probe vehicles data," European Journal of Operational Research, Elsevier, vol. 226(1), pages 67-76.
    13. António Pacheco & Maria Lurdes Simões Simões & Paula Milheiro-Oliveira, 2017. "Queues with Server Vacations as a Model for Pretimed Signalized Urban Traffic," Transportation Science, INFORMS, vol. 51(3), pages 841-851, August.

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