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Asymptotic results on infinite tandem queueing networks
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  • Published: November 2000

Asymptotic results on infinite tandem queueing networks

  • F. Baccelli1,
  • A. Borovkov2 &
  • J. Mairesse3 

Probability Theory and Related Fields volume 118, pages 365–405 (2000)Cite this article

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  • 23 Citations

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Abstract.

We consider an infinite tandem queueing network consisting of ·/GI/1/∞ stations with i.i.d. service times. We investigate the asymptotic behavior of t(n, k), the inter-arrival times between customers n and n + 1 at station k, and that of w(n, k), the waiting time of customer n at station k. We establish a duality property by which w(n, k) and the “idle times”y(n, k) play symmetrical roles. This duality structure, interesting by itself, is also instrumental in proving some of the ergodic results. We consider two versions of the model: the quadrant and the half-plane. In the quadrant version, the sequences of boundary conditions {w(0,k), k∈ℕ} and {t(n, 0), n∈ℕ}, are given. In the half-plane version, the sequence {t(n, 0), n∈ℕ} is given. Under appropriate assumptions on the boundary conditions and on the services, we obtain ergodic results for both versions of the model. For the quadrant version, we prove the existence of temporally ergodic evolutions and of spatially ergodic ones. Furthermore, the process {t(n, k), n∈ℕ} converges weakly with k to a limiting distribution, which is invariant for the queueing operator. In the more difficult half plane problem, the aim is to obtain evolutions which are both temporally and spatially ergodic. We prove that 1/n∑ k=1 n w(0, k) converges almost surely and in L 1 to a finite constant. This constitutes a first step in trying to prove that {t(n,k), n∈ℤ} converges weakly with k to an invariant limiting distribution.

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Authors and Affiliations

  1. INRIA-ENS, Département de Mathématiques et dInformatique, 45 rue dUlm, Paris 75005, France., , , , , , FR

    F. Baccelli

  2. Institute of Mathematics, Novosibirsk, Russia., , , , , , RU

    A. Borovkov

  3. LIAFA, CNRS-Université Paris VII, case 7014, 2 place Jussieu, 75251 Paris Cedex 05, France. e-mail: mairesse@liafa.jussieu.fr, , , , , , FR

    J. Mairesse

Authors
  1. F. Baccelli
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  2. A. Borovkov
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  3. J. Mairesse
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Received: 23 March 1999 / Revised version: 5 January 2000 / Published online: 12 October 2000

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Cite this article

Baccelli, F., Borovkov, A. & Mairesse, J. Asymptotic results on infinite tandem queueing networks. Probab Theory Relat Fields 118, 365–405 (2000). https://doi.org/10.1007/PL00008747

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  • Issue Date: November 2000

  • DOI: https://doi.org/10.1007/PL00008747

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Keywords

  • Boundary Condition
  • Asymptotic Behavior
  • Service Time
  • Plane Problem
  • Half Plane
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