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Product-Form Queueing Networks with Batches

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Computer Performance Engineering (EPEW 2018)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 11178))

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Abstract

A Markovian queue, with both batch arrivals and batch departures, is first shown to have a geometric queue length probability distribution at equilibrium under certain conditions. From this a product-form solution follows directly for networks of such queues at equilibrium, by application of the reversed compound agent theorem (RCAT). The method is illustrated using small batches of sizes 1 and 2, as well as geometric sizes.

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Notes

  1. 1.

    This is because the special departures are active transitions, with the empty queue as their only possible destination-state, whereas RCAT requires all states to be possible destinations [8]. Similarly, the special arrivals are passive but enabled only in the empty queue, i.e. not in every state, as required by RCAT. One could consider the special departures as passive and the special arrivals as active, provided they could occur in, or lead to, every state, respectively. However, a special arrival transition from the empty state to itself would then be required – i.e. an active “invisible transition”. This would lead to an increased rate of special departures in the synchronising queue, changing the model’s specification. Worse still, the special arrival rates would have to be carefully chosen (geometrically) so as to ensure constant reversed rates. Similarly, an invisible, passive, special departure transition would also be needed on the empty state, allowing spontaneous special arrivals at the synchronising queue, which again would probably not be wanted.

  2. 2.

    We exclude feedback from a node to itself, so that \(j \ne i\) or, equivalently, we can define \(p_{ikjl}=0\) whenever \(i=j\).

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Author information

Authors and Affiliations

  1. Department of Computing, Imperial College London, London, UK

    P. G. Harrison

Authors
  1. P. G. Harrison
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Correspondence to P. G. Harrison .

Editor information

Editors and Affiliations

  1. eScience Center, Amsterdam, The Netherlands

    Rena Bakhshi

  2. University Paris-Saclay, Gif-sur-Yvette, France

    Paolo Ballarini

  3. Université Paris-Est Créteil, Créteil, France

    Benoît Barbot

  4. Telecom SudParis, Evry, France

    Hind Castel-Taleb

  5. Westfälische Wilhelms Universität Münster, Münster, Germany

    Anne Remke

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Harrison, P.G. (2018). Product-Form Queueing Networks with Batches. In: Bakhshi, R., Ballarini, P., Barbot, B., Castel-Taleb, H., Remke, A. (eds) Computer Performance Engineering. EPEW 2018. Lecture Notes in Computer Science(), vol 11178. Springer, Cham. https://doi.org/10.1007/978-3-030-02227-3_17

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  • DOI: https://doi.org/10.1007/978-3-030-02227-3_17

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-02226-6

  • Online ISBN: 978-3-030-02227-3

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