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Mean-field Analysis of Large Scale Markov Fluid Models with Fluid Dependent and Time-Inhomogeneous Rates

Anton Stefanek, Richard Hayden, Jeremy T. Bradley

Technical Report
Annals of Operations Research

We consider a subset of Markov fluid models where the discrete background process is a Population Continuous Time Markov Chain (PCTMC) - a Markov process capturing interactions between large groups of identically behaved components. We allow the transition and fluid evolution rates to depend on the fluid level. Additionally, we include time-inhomogeneous rate parameters, which can be used to incorporate real measurement data into the models.

We extend the mean-field techniques for PCTMCs and show how to derive a system of ordinary differential equations (ODEs) that approximate the evolution of means and higher-order moments of populations and fluid levels in Markov fluid models with PCTMC background processes. 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 the rates depend on a truncation of the fluid level.

We demonstrate how such a framework is suitable for modelling feedback from globally-accumulated quantities such as energy consumption, cost or temperature. We present a worked example of a hypothetical heterogeneous computing cluster and its interaction with air conditioning units. We also show a model of a multi-server queue with temperature management and external workload that varies with time.


Accepted subject to minor corrections to AOR September 2013

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