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Stability Analysis of a Multiclass Retrial System with Coupled Orbit Queues

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Computer Performance Engineering (EPEW 2017)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 10497))

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Abstract

In this work we consider a single-server system accepting N types of retrial customers, which arrive according to independent Poisson streams. In case of blocking, type-i customer, \(i=1,2,...,N\) is routed to a separate type-i orbit queue of infinite capacity. Customers from the orbit queues try to access the server according to the constant retrial policy. We consider coupled orbit queues. More precisely, the orbit queue i retransmits a blocked customer of type-i to the main service station after an exponentially distributed time with rate \(\mu _{i}\), when at least one other orbit queue is non-empty. Otherwise, if all other orbit queues are empty, the orbit queue i changes its retransmission rate from \(\mu _{i}\) to \(\mu _{i}^{*}\). Such a scheme arises in the modeling of cooperative cognitive wireless networks, in which a node is aware of the status of other nodes, and accordingly, adjusts its retransmission parameters in order to exploit the idle periods of the other nodes. Using the regenerative approach we obtain the necessary conditions of the ergodicity of our system, and show that these conditions have a clear probabilistic interpretation. We also suggest a sufficient stability condition. Simulation experiments show that the obtained conditions delimit the stability domain with remarkable accuracy.

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Notes

  1. 1.

    In this work the terms “orbit” and “orbit queue” are identical.

  2. 2.

    We mention first the general case of non-zero initial conditions, in which case \(T_1\not = T\). For such a case, positive recurrence means both \(ET<\infty \) and \(T_1<\infty \) w.p. 1. For zero initial state \(T_1=_{st}T\), and thus in order to prove the positive recurrence, it only remains to show that \(ET<\infty \). See also some comments in Remark 1.

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Acknowledgments

The research of EM is supported by Russian Foundation for Basic Research, projects 15-07-02341, 15-07-02354, 15-07-02360.

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Authors and Affiliations

  1. Karelian Research Centre RAS, Institute of Applied Mathematical Research, Petrozavodsk State University, Petrozavodsk, Russian Federation

    Evsey Morozov

  2. Department of Mathematics, University of Patras, 26500, Patras, Greece

    Ioannis Dimitriou

Authors
  1. Evsey Morozov
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  2. Ioannis Dimitriou
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Correspondence to Ioannis Dimitriou .

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Editors and Affiliations

  1. Bristol, Germany

    Philipp Reinecke

  2. University of L’Aquila, L’Aquila, Italy

    Antinisca Di Marco

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Morozov, E., Dimitriou, I. (2017). Stability Analysis of a Multiclass Retrial System with Coupled Orbit Queues. In: Reinecke, P., Di Marco, A. (eds) Computer Performance Engineering. EPEW 2017. Lecture Notes in Computer Science(), vol 10497. Springer, Cham. https://doi.org/10.1007/978-3-319-66583-2_6

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  • DOI: https://doi.org/10.1007/978-3-319-66583-2_6

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  • Online ISBN: 978-3-319-66583-2

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