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Asymptotics for the late arrivals problem

Author

Listed:
  • Carlo Lancia

    (Mathematical Institute Leiden University)

  • Gianluca Guadagni

    (University of Virginia)

  • Sokol Ndreca

    (Universidade Federal de Minas Gerais)

  • Benedetto Scoppola

    (Università degli Studi di Roma Tor Vergata)

Abstract

We study a discrete time queueing system where deterministic arrivals have i.i.d. exponential delays $$\xi _{i}$$ ξ i . We describe the model as a bivariate Markov chain, prove its ergodicity and study the joint equilibrium distribution. We write a functional equation for the bivariate generating function, finding the solution on a subset of its domain. This solution allows us to prove that the equilibrium distribution of the chain decays super-exponentially fast in the quarter plane. We exploit the latter result and discuss the numerical computation of the solution through a simple yet effective approximation scheme in a wide region of the parameters. Finally, we compare the features of this queueing model with the standard M / D / 1 system, showing that the congestion turns out to be very different when the traffic intensity is close to 1.

Suggested Citation

  • Carlo Lancia & Gianluca Guadagni & Sokol Ndreca & Benedetto Scoppola, 2018. "Asymptotics for the late arrivals problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 88(3), pages 475-493, December.
    • Handle: RePEc:spr:mathme:v:88:y:2018:i:3:d:10.1007_s00186-018-0643-3
    DOI: 10.1007/s00186-018-0643-3
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    References listed on IDEAS

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    Cited by:

    1. Antonio Di Crescenzo & Barbara Martinucci & Paola Paraggio, 2023. "Vessels Arrival Process and its Application to the SHIP/M/ $$\infty$$ ∞ Queue," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-33, March.

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