Beyond Scaling

This paper provides a recipe for deriving calculable approximation errors of mean-field models in heavy-traffic with the focus on the well-known load balancing algorithm --- power-of-two-choices (Po2). The recipe combines Stein's method for linearized mean-field models and State Space Concentration (SSC) based on geometric tail bounds. In particular, our approach divides the state space into two regions, a neighborhood near the mean-field equilibrium and the complement of that. We first use a tail bound to show that the steady-state probability being outside the neighborhood is small. Then, we use a linearized mean-field model and Stein's method to characterize the generator difference, which provides the dominant term of the approximation error. From the dominant term, we are able to obtain an asymptotically-tight bound and a nonasymptotic upper bound, both are calculable bounds, not order-wise scaling results like most results in the literature. Finally, we compare the theoretical bounds with numerical evaluations to show the effectiveness of our results. We note that the simulation results show that both bounds are valid even for small size systems such as a system with only ten servers.