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Some reflected autoregressive processes with dependencies

Author

Listed:
  • Ioannis Dimitriou

    (University of Ioannina)

  • Dieter Fiems

    (Ghent University)

Abstract

Motivated by queueing applications, we study various reflected autoregressive processes with dependencies. Among others, we study cases where the interarrival and service times are proportionally dependent on additive and/or subtracting delay, as well as cases where interarrival times depend on whether the service duration of the previous arrival exceeds or not a random threshold. Moreover, we study cases where the autoregressive parameter is constant as well as a discrete or a continuous random variable. More general dependence structures are also discussed. Our primary aim is to investigate a broad class of recursions of autoregressive type for which several independence assumptions are lifted and for which a detailed exact analysis is provided. We provide expressions for the Laplace transform of the waiting time distribution of a customer in the system in terms of an infinite sum of products of known Laplace transforms. An integer-valued reflected autoregressive process that can be used to model a novel retrial queueing system with impatient customers and a general dependence structure is also considered. For such a model, we provide expressions for the probability generating function of the stationary orbit queue length distribution in terms of an infinite sum of products of known generating functions. A first attempt towards a multidimensional setting is also considered.

Suggested Citation

  • Ioannis Dimitriou & Dieter Fiems, 2024. "Some reflected autoregressive processes with dependencies," Queueing Systems: Theory and Applications, Springer, vol. 106(1), pages 67-127, February.
    • Handle: RePEc:spr:queues:v:106:y:2024:i:1:d:10.1007_s11134-023-09899-3
    DOI: 10.1007/s11134-023-09899-3
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    References listed on IDEAS

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    1. Boxma, O. J. & Perry, D., 2001. "A queueing model with dependence between service and interarrival times," European Journal of Operational Research, Elsevier, vol. 128(3), pages 611-624, February.
    2. Onno Boxma & Andreas Löpker & Michel Mandjes & Zbigniew Palmowski, 2021. "A multiplicative version of the Lindley recursion," Queueing Systems: Theory and Applications, Springer, vol. 98(3), pages 225-245, August.
    3. Ivo Adan & Onno Boxma & Jacques Resing, 2022. "Functional equations with multiple recursive terms," Queueing Systems: Theory and Applications, Springer, vol. 102(1), pages 7-23, October.
    4. O. J. Boxma & M. R. H. Mandjes, 2022. "Queueing and risk models with dependencies," Queueing Systems: Theory and Applications, Springer, vol. 102(1), pages 69-86, October.
    Full references (including those not matched with items on IDEAS)

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