Mean-Field analysis of markov models with reward feedback

Authors:
Anton Stefanek;Richard A. Hayden;Mark Mac Gonagle;Jeremy T. Bradley
Affiliations:
Department of Computing, Imperial College London, London, UK;Department of Computing, Imperial College London, London, UK;Department of Computing, Imperial College London, London, UK;Department of Computing, Imperial College London, London, UK
Venue:
ASMTA'12 Proceedings of the 19th international conference on Analytical and Stochastic Modeling Techniques and Applications
Year:
2012

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Abstract

We extend the population continuous time Markov chain formalism so that the state space is augmented with continuous variables accumulated over time as functions of component populations. System feedback can be expressed using accumulations that in turn can influence the Markov chain behaviour via functional transition rates. We show how to obtain mean-field differential equations capturing means and higher-order moments of the discrete populations and continuous accumulation variables. We also provide first- and second-order convergence results and suggest a novel normal moment closure that can greatly improve the accuracy of means and higher moments. We demonstrate how such a framework is suitable for modelling feedback from globally-accumulated quantities such as energy consumption, cost or temperature. Finally, we present a worked example modelling a hypothetical heterogeneous computing cluster and its interaction with air conditioning units.