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Pollaczek contour integrals for the fixed-cycle traffic-light queue

Author

Listed:
  • M. A. A. Boon

    (Eindhoven University of Technology)

  • A. J. E. M. Janssen

    (Eindhoven University of Technology)

  • J. S. H. Leeuwaarden

    (Eindhoven University of Technology)

  • R. W. Timmerman

    (Eindhoven University of Technology)

Abstract

The fixed-cycle traffic-light (FCTL) queue is the standard model for intersections with static signaling, where vehicles arrive, form a queue and depart during cycles controlled by a traffic light. Classical analysis of the FCTL queue based on transform methods requires a computationally challenging step of finding the complex-valued roots of some characteristic equation. Building on the recent work of Oblakova et al. (Exact expected delay and distribution for the fixed-cycle traffic-light model and similar systems in explicit form, 2016), we obtain a contour-integral expression, reminiscent of Pollaczek integrals for bulk-service queues, for the probability generating function of the steady-state FCTL queue. We also show that similar contour integrals arise for generalizations of the FCTL queue introduced in Oblakova et al. (2016) that relax some of the classical assumptions. Our results allow us to compute the queue-length distribution and all its moments using algorithms that rely on contour integrals and avoid root-finding procedures.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:queues:v:91:y:2019:i:1:d:10.1007_s11134-018-9595-9
    DOI: 10.1007/s11134-018-9595-9
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. Warren B. Powell & Pierre Humblet, 1986. "The Bulk Service Queue with a General Control Strategy: Theoretical Analysis and a New Computational Procedure," Operations Research, INFORMS, vol. 34(2), pages 267-275, April.
    4. 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.
    5. A. J. E. M. Janssen & J. S. H. Van Leeuwaarden, 2008. "Back to the roots of the M/D/s queue and the works of Erlang, Crommelin and Pollaczek," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 62(3), pages 299-313, August.
    6. Warren B. Powell, 1985. "Analysis of Vehicle Holding and Cancellation Strategies in Bulk Arrival, Bulk Service Queues," Transportation Science, INFORMS, vol. 19(4), pages 352-377, November.
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    Cited by:

    1. Boon, Marko & Janssen, Guido & van Leeuwaarden, Johan & Timmerman, Rik, 2023. "Optimal capacity allocation for heavy-traffic fixed-cycle traffic-light queues and intersections," Transportation Research Part B: Methodological, Elsevier, vol. 167(C), pages 79-98.
    2. 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.

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