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Mode Locking and Chaos in a Deterministic Queueing Model with Feedback


  • Christian Haxholdt

    () (Department of Statistics, Copenhagen Business School, Solbjerg Plads 3, DK-2000, Frederiksberg, Denmark)

  • Erik R. Larsen

    () (Cass Business School, City University, 106 Bunhill Row, London EC1Y 8TZ, United Kingdom)

  • Ann van Ackere

    () (HEC, Université de Lausanne, BFSH1, 1015 Dorigny-Lausanne, Switzerland)


We consider a simple, deterministic queueing system with feedback, which exhibits the phenomena of sustained oscillation, mode locking, quasi-periodic behaviour, and chaos. This implies that a fully deterministic queueing system can exhibit seemingly unpredictable behaviour. We ignore variability, and focus on two forms of feedback: (i) the service rate increases as queue length increases, and (ii) the arrival rate depends on customers' perception of past waiting times. We model a customer's decision to seek service as a two stage process: (i) deciding whether or not to use a facility, and (ii) deciding the frequency of visit (daily, weekly, monthly, etc.). This frequency is initially constant, and later on replaced by a deterministic, time-dependent pattern. Although highly stylised, this model captures the essential features of many real-life systems whose average arrival rate varies over time. Reducing the amplitude of cycles in demand makes the system more predictable and thus easier to manage. Although we represent this model as a queue of customers waiting for service, the model can be interpreted more generally as any situation where an increase in demand lowers the quality of service.

Suggested Citation

  • Christian Haxholdt & Erik R. Larsen & Ann van Ackere, 2003. "Mode Locking and Chaos in a Deterministic Queueing Model with Feedback," Management Science, INFORMS, vol. 49(6), pages 816-830, June.
  • Handle: RePEc:inm:ormnsc:v:49:y:2003:i:6:p:816-830

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    References listed on IDEAS

    1. Larsen, Erik Reimer & Morecroft, John D. W. & Thomsen, Jesper Skovhus & Mosekilde, Erik, 1993. "Devil's staircase and chaos from macroeconomic mode interaction," Journal of Economic Dynamics and Control, Elsevier, vol. 17(5-6), pages 759-769.
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    7. Naor, P, 1969. "The Regulation of Queue Size by Levying Tolls," Econometrica, Econometric Society, vol. 37(1), pages 15-24, January.
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    Cited by:

    1. Gallay, Olivier & Hongler, Max-Olivier, 2009. "Circulation of autonomous agents in production and service networks," International Journal of Production Economics, Elsevier, vol. 120(2), pages 378-388, August.
    2. van Ackere, Ann & Haxholdt, Christian & Larsen, Erik R., 2013. "Dynamic capacity adjustments with reactive customers," Omega, Elsevier, vol. 41(4), pages 689-705.


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