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Functional equations with multiple recursive terms

Author

Listed:
  • Ivo Adan

    (Eindhoven University of Technology)

  • Onno Boxma

    (Eindhoven University of Technology)

  • Jacques Resing

    (Eindhoven University of Technology)

Abstract

In this paper, we study a functional equation for generating functions of the form $$f(z) = g(z) \sum _{i=1}^M p_i f(\alpha _i(z)) + K(z)$$ f ( z ) = g ( z ) ∑ i = 1 M p i f ( α i ( z ) ) + K ( z ) , viz. a recursion with multiple recursive terms. We derive and analyze the solution of this equation for the case that the $$\alpha _i(z)$$ α i ( z ) are commutative contraction mappings. The results are applied to a wide range of queueing, autoregressive and branching processes.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:queues:v:102:y:2022:i:1:d:10.1007_s11134-022-09861-9
    DOI: 10.1007/s11134-022-09861-9
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    References listed on IDEAS

    as
    1. Ivo Adan & Brett Hathaway & Vidyadhar G. Kulkarni, 2019. "On first-come, first-served queues with two classes of impatient customers," Queueing Systems: Theory and Applications, Springer, vol. 91(1), pages 113-142, February.
    2. Onno Boxma & Michel Mandjes, 2021. "Shot-noise queueing models," Queueing Systems: Theory and Applications, Springer, vol. 99(1), pages 121-159, October.
    3. Christian Weiß, 2008. "Thinning operations for modeling time series of counts—a survey," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 92(3), pages 319-341, August.
    4. Onno Boxma & Offer Kella & Uri Yechiali, 2016. "An ASIP model with general gate opening intervals," Queueing Systems: Theory and Applications, Springer, vol. 84(1), pages 1-20, October.
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    Cited by:

    1. Onno Boxma & David Perry & Wolfgang Stadje, 2022. "Peer-to-Peer Lending: a Growth-Collapse Model and its Steady-State Analysis," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 96(2), pages 233-258, October.
    2. 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.

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