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The Single Server Queue with Mixing Dependencies

Author

Listed:
  • Youri Raaijmakers

    (Eindhoven University of Technology)

  • Hansjörg Albrecher

    (University of Lausanne
    Swiss Finance Institute)

  • Onno Boxma

    (Eindhoven University of Technology)

Abstract

We study a single server queue, where a certain type of dependence is introduced between the service times, or between the inter-arrival times, or both between the service times and the inter-arrival times. This dependence arises via mixing, i.e., a parameter pertaining to the distribution of the service times, or of the inter-arrival times, is itself considered to be a random variable. We give a duality result between such queueing models and the corresponding insurance risk models, for which the respective dependence structures have been studied before. For a number of examples we provide exact expressions for the waiting time distribution, and compare these to the ones for the standard M/M/1 queue. We also investigate the effect of dependence and derive first order asymptotics for some of the obtained waiting time tails. Finally, we discuss this dependence concept for the waiting time tail of the G/M/1 queue.

Suggested Citation

  • Youri Raaijmakers & Hansjörg Albrecher & Onno Boxma, 2019. "The Single Server Queue with Mixing Dependencies," Methodology and Computing in Applied Probability, Springer, vol. 21(4), pages 1023-1044, December.
    • Handle: RePEc:spr:metcap:v:21:y:2019:i:4:d:10.1007_s11009-018-9683-7
    DOI: 10.1007/s11009-018-9683-7
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    References listed on IDEAS

    as
    1. Albrecher, Hansjörg & Constantinescu, Corina & Loisel, Stephane, 2011. "Explicit ruin formulas for models with dependence among risks," Insurance: Mathematics and Economics, Elsevier, vol. 48(2), pages 265-270, March.
    2. repec:hal:wpaper:hal-00746251 is not listed on IDEAS
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    4. Badila, E.S. & Boxma, O.J. & Resing, J.A.C., 2015. "Two parallel insurance lines with simultaneous arrivals and risks correlated with inter-arrival times," Insurance: Mathematics and Economics, Elsevier, vol. 61(C), pages 48-61.
    5. Constantinescu, Corina & Hashorva, Enkelejd & Ji, Lanpeng, 2011. "Archimedean copulas in finite and infinite dimensions—with application to ruin problems," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 487-495.
    6. Dutang, C. & Lefèvre, C. & Loisel, S., 2013. "On an asymptotic rule A+B/u for ultimate ruin probabilities under dependence by mixing," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 774-785.
    7. Geurt Jongbloed & Ger Koole, 2001. "Managing uncertainty in call centres using Poisson mixtures," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 17(4), pages 307-318, October.
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