IDEAS home Printed from https://ideas.repec.org/a/spr/annopr/v239y2016i2d10.1007_s10479-014-1664-9.html
   My bibliography  Save this article

Mean-field analysis of hybrid Markov population models with time-inhomogeneous rates

Author

Listed:
  • Anton Stefanek

    (Imperial College London)

  • Richard A. Hayden

    (Imperial College London)

  • Jeremy T. Bradley

    (Imperial College London)

Abstract

We consider a hybrid extension of population continuous time Markov chains (PCTMC)—a class of Markov processes capturing interactions between large groups of identically behaved agents. We augment the discrete state space of a PCTMC with continuous variables that evolve as integrals over the population vector and that can simultaneously provide feedback to the rates of transitions in the PCTMC. Additionally, we include time-inhomogeneous rate parameters, which can be used to incorporate real measurement data into the models. We extend mean-field techniques for PCTMCs and show how to derive a system of integral equations that approximate the evolution of means and higher-order moments of populations and continuous variables in a hybrid PCTMC. We prove first- and second-order convergence results that justify the approximations. We use a moment closure based on the normal distribution which improves the accuracy of the moment approximation in case of proportional control where transition rates depend on the amount a continuous variable is above or below a fixed threshold. We demonstrate how this framework is suitable for modelling feedback from globally-accumulated quantities in a large scale system, such as energy consumption, total cost or temperature in a data centre. We present a model of a many server system with temperature management and external workload that varies with time. We show how to use real data to represent the workload within the framework. We use stochastic simulation to validate the example and an earlier example of a hypothetical heterogeneous computing cluster.

Suggested Citation

  • Anton Stefanek & Richard A. Hayden & Jeremy T. Bradley, 2016. "Mean-field analysis of hybrid Markov population models with time-inhomogeneous rates," Annals of Operations Research, Springer, vol. 239(2), pages 667-693, April.
  • Handle: RePEc:spr:annopr:v:239:y:2016:i:2:d:10.1007_s10479-014-1664-9
    DOI: 10.1007/s10479-014-1664-9
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10479-014-1664-9
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1007/s10479-014-1664-9?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Horton, Graham & Kulkarni, Vidyadhar G. & Nicol, David M. & Trivedi, Kishor S., 1998. "Fluid stochastic Petri nets: Theory, applications, and solution techniques," European Journal of Operational Research, Elsevier, vol. 105(1), pages 184-201, February.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Jianing Wu & Shaoze Yan & RX Gao, 2014. "Modeling and analysis of failure propagation of mechanical system with multi-operation states using high-level Petri net," Journal of Risk and Reliability, , vol. 228(4), pages 347-361, August.
    2. Samuelson, Aviva & Haigh, Andrew & O'Reilly, Małgorzata M. & Bean, Nigel G., 2017. "Stochastic model for maintenance in continuously deteriorating systems," European Journal of Operational Research, Elsevier, vol. 259(3), pages 1169-1179.
    3. Jianing Wu & Shaoze Yan, 2014. "An approach to system reliability prediction for mechanical equipment using fuzzy reasoning Petri net," Journal of Risk and Reliability, , vol. 228(1), pages 39-51, February.
    4. Macchi, Marco & Kristjanpoller, Fredy & Garetti, Marco & Arata, Adolfo & Fumagalli, Luca, 2012. "Introducing buffer inventories in the RBD analysis of process production systems," Reliability Engineering and System Safety, Elsevier, vol. 104(C), pages 84-95.
    5. M. Gribaudo & D. Manini & B. Sericola & M. Telek, 2008. "Second order fluid models with general boundary behaviour," Annals of Operations Research, Springer, vol. 160(1), pages 69-82, April.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:annopr:v:239:y:2016:i:2:d:10.1007_s10479-014-1664-9. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.