{
  "id": "2004.09574",
  "version": "v1",
  "published": "2020-04-20T19:03:34.000Z",
  "updated": "2020-04-20T19:03:34.000Z",
  "title": "A Note on Load Balancing in Many-Server Heavy-Traffic Regime",
  "authors": [
    "Xingyu Zhou",
    "Ness Shroff"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:2003.06454",
  "categories": [
    "math.PR",
    "cs.DC"
  ],
  "abstract": "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 $N^{1-\\alpha}$ with $\\alpha > 1$. 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 $\\sigma_{\\Sigma}^2$ and $\\nu_{\\Sigma}^2$ as $N\\to \\infty$, then for any $\\alpha > 3$, the distribution of the sum queue length scaled by $N^{1-\\alpha}$ converges to an exponential random variable with rate $\\frac{\\sigma_{\\Sigma}^2 + \\nu_{\\Sigma}^2}{2}$. (2) If the second moments linearly increase with $N$ with coefficients $\\sigma_a^2$ and $\\nu_s^2$, then for any $\\alpha > 2$, the distribution of the sum queue length scaled by $N^{-\\alpha}$ converges to an exponential random variable with rate $\\frac{\\sigma_a^2 + \\nu_s^2}{2}$. (3) If the second moments quadratically increase with $N$ with coefficients $\\tilde{\\sigma}_a^2$ and $\\tilde{\\nu}_s^2$, then for any $\\alpha > 1$, the distribution of the sum queue length scaled by $N^{-\\alpha-1}$ converges to an exponential random variable with rate $\\frac{\\tilde{\\sigma}_a^2 + \\tilde{\\nu}_s^2}{2}$. All the results are simple applications of our previously developed framework of Stein's method for heavy-traffic analysis in [9].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-20T19:03:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "many-server heavy-traffic regime",
      "load balancing",
      "sum queue length",
      "exponential random variable",
      "steins method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}