{
  "id": "2112.01893",
  "version": "v2",
  "published": "2021-12-03T13:10:33.000Z",
  "updated": "2022-07-22T15:15:00.000Z",
  "title": "Aggregation of network traffic and anisotropic scaling of random fields",
  "authors": [
    "Remigijus Leipus",
    "Vytautė Pilipauskaitė",
    "Donatas Surgailis"
  ],
  "comment": "46 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We discuss joint spatial-temporal scaling limits of sums $A_{\\lambda,\\gamma}$ (indexed by $(x,y) \\in \\mathbb{R}^2_+$) of large number $O(\\lambda^{\\gamma})$ of independent copies of integrated input process $X = \\{X(t), t \\in \\mathbb{R}\\}$ at time scale $\\lambda$, for any given $\\gamma>0$. We consider two classes of inputs $X$: (I) Poisson shot-noise with (random) pulse process, and (II) regenerative process with random pulse process and regeneration times following a heavy-tailed stationary renewal process. The above classes include several queueing and network traffic models for which joint spatial-temporal limits were previously discussed in the literature. In both cases (I) and (II) we find simple conditions on the input process in order that normalized random fields $A_{\\lambda,\\gamma}$ tend to an $\\alpha$-stable L\\'evy sheet $(1< \\alpha <2)$ if $ \\gamma < \\gamma_0$, and to a fractional Brownian sheet if $\\gamma > \\gamma_0$, for some $\\gamma_0>0$. We also prove an `intermediate' limit for $\\gamma = \\gamma_0$. Our results extend previous works Mikosch et al. (2002), Gaigalas, Kaj (2003) and other papers to more general and new input processes.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-07-22T15:15:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G18",
      "60G51",
      "60G52",
      "60G60",
      "60G22",
      "60F05",
      "60K25"
    ],
    "keywords": [
      "random fields",
      "anisotropic scaling",
      "input process",
      "aggregation",
      "joint spatial-temporal scaling limits"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}