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Mean field approximation is a powerful technique to study the performance oflarge stochastic systems represented as $n$ interacting objects. Applicationsinclude load balancing models, epidemic spreading, cache replacement policies,or large-scale data centers. Mean field approximation is asymptotically exactfor systems composed of $n$ homogeneous objects under mild conditions. In thispaper, we study what happens when objects are heterogeneous. This can representservers with different speeds or contents with different popularities. Wedefine an interaction model that allows obtaining asymptotic convergenceresults for stochastic systems with heterogeneous object behavior, and showthat the error of the mean field approximation is of order $O(1/n)$. Moreimportantly, we show how to adapt the refined mean field approximation,developed by Gast et al. 2019, and show that the error of this approximation isreduced to $O(1/n^2)$. To illustrate the applicability of our result, wepresent two examples. The first addresses a list-based cache replacement modelRANDOM($m$), which is an extension of the RANDOM policy. The second is aheterogeneous supermarket model. These examples show that the proposedapproximations are computationally tractable and very accurate. They also showthat for moderate system sizes ($n\approx30$) the refined mean fieldapproximation tends to be more accurate than simulations for any reasonablesimulation time.

Mean Field and Refined Mean Field Approximations for Heterogeneous Systems: It Works!