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Robust queueing theory: an initial study using imprecise probabilities

Author

Listed:
  • Stavros Lopatatzidis

    (Ghent University)

  • Jasper Bock

    (Ghent University)

  • Gert Cooman

    (Ghent University)

  • Stijn Vuyst

    (Ghent University)

  • Joris Walraevens

    (Ghent University)

Abstract

We study the robustness of performance predictions of discrete-time finite-capacity queues by applying the framework of imprecise probabilities. More concretely, we consider the Geo/Geo/1/L model with probabilities of arrival and departure that are no longer fixed, but are allowed to vary within given intervals. We distinguish between two concepts of independence in this framework, namely repetition independence and epistemic irrelevance. In the first approach, we assume the existence of time-homogeneous probabilities for arrival and departure, which leads us to consider a collection of stationary queues. In the second, the stationarity assumption is dropped and we allow the arrival and departure probabilities to vary from time point to time point; they may even depend on the complete history of queue lengths. We calculate bounds on the expected queue length, the probability of a particular queue length and the probability of turning on the server. For the expected queue length, both approaches coincide. For the other performance measures, we observe and discuss various differences between the bounds obtained for these two approaches. One of our observations is that ergodicity may break down due to imprecision: bounds on expected time averages of certain functions on the state space are not necessarily equal to the bounds on the expectation of that function at random instants in a steady-state queue.

Suggested Citation

  • Stavros Lopatatzidis & Jasper Bock & Gert Cooman & Stijn Vuyst & Joris Walraevens, 2016. "Robust queueing theory: an initial study using imprecise probabilities," Queueing Systems: Theory and Applications, Springer, vol. 82(1), pages 75-101, February.
    • Handle: RePEc:spr:queues:v:82:y:2016:i:1:d:10.1007_s11134-015-9458-6
    DOI: 10.1007/s11134-015-9458-6
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    References listed on IDEAS

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    1. Rubinstein, Reuven Y., 1997. "Optimization of computer simulation models with rare events," European Journal of Operational Research, Elsevier, vol. 99(1), pages 89-112, May.
    2. Xia, Li & Cao, Xi-Ren, 2012. "Performance optimization of queueing systems with perturbation realization," European Journal of Operational Research, Elsevier, vol. 218(2), pages 293-304.
    3. D. Škulj & R. Hable, 2013. "Coefficients of ergodicity for Markov chains with uncertain parameters," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 76(1), pages 107-133, January.
    4. Couso, Ines & Moral, Serafin & Walley, Peter, 2000. "A survey of concepts of independence for imprecise probabilities," Risk, Decision and Policy, Cambridge University Press, vol. 5(2), pages 165-181, June.
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    Cited by:

    1. Kristoffel C. Pandiangan & Palti Maruli Tua Sitorus, 2017. "Analysis of Queue System to Improve the Quality of Service in GraPARI Telkomsel Banda Aceh," International Journal of Business and Economic Affairs (IJBEA), Sana N. Maswadeh, vol. 2(4), pages 220-226.
    2. Marla, Lavanya & Rikun, Alexander & Stauffer, Gautier & Pratsini, Eleni, 2020. "Robust modeling and planning: Insights from three industrial applications," Operations Research Perspectives, Elsevier, vol. 7(C).

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