A Note on Load Balancing in Many-Server Heavy-Traffic Regime
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In this note, we apply Stein's method to analyze the performance of general load balancing schemes in the many-server heavy-traffic regime. In particular, consider a load balancing system of N servers and the distance of arrival rate to the capacity region is given by N^1-α with α > 1. We are interested in the performance as N goes to infinity under a large class of policies. We establish different asymptotics under different scalings and conditions. Specifically, (i) if the second moments of total arrival and total service processes approach to constants σ_Σ^2 and ν_Σ^2 as N→∞, then for any α > 3, the distribution of the sum queue length scaled by N^1-α converges to an exponential random variable with rate σ_Σ^2 + ν_Σ^2/2. (2) If the second moments linearly increase with N with coefficients σ_a^2 and ν_s^2, then for any α > 2, the distribution of the sum queue length scaled by N^-α converges to an exponential random variable with rate σ_a^2 + ν_s^2/2. (3) If the second moments quadratically increase with N with coefficients σ̃_a^2 and ν̃_s^2, then for any α > 1, the distribution of the sum queue length scaled by N^-α-1 converges to an exponential random variable with rate σ̃_a^2 + ν̃_s^2/2. All the results are simple applications of our previously developed framework of Stein's method for heavy-traffic analysis in [9].
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