{
  "id": "1407.3837",
  "version": "v2",
  "published": "2014-07-14T22:22:05.000Z",
  "updated": "2015-10-28T10:27:01.000Z",
  "title": "Diffusion limits for shortest remaining processing time queues under nonstandard spatial scaling",
  "authors": [
    "Amber L. Puha"
  ],
  "comment": "Published at http://dx.doi.org/10.1214/14-AAP1076 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Applied Probability 2015, Vol. 25, No. 6, 3381-3404",
  "doi": "10.1214/14-AAP1076",
  "categories": [
    "math.PR"
  ],
  "abstract": "We develop a heavy traffic diffusion limit theorem under nonstandard spatial scaling for the queue length process in a single server queue employing shortest remaining processing time (SRPT). For processing time distributions with unbounded support, it has been shown that standard diffusion scaling yields an identically zero limit. We specify an alternative spatial scaling that produces a nonzero limit. Our model allows for renewal arrivals and i.i.d. processing times satisfying a rapid variation condition. We add a corrective spatial scale factor to standard diffusion scaling, and specify conditions under which the sequence of unconventionally scaled queue length processes converges in distribution to the same nonzero reflected Brownian motion to which the sequence of conventionally scaled workload processes converges. Consequently, this corrective spatial scale factor characterizes the order of magnitude difference between the queue length and workload processes of SRPT queues in heavy traffic. It is determined by the processing time distribution such that the rate at which it tends to infinity depends on the rate at which the tail of the processing time distribution tends to zero. For Weibull processing time distributions, we restate this result in a manner that makes the resulting state space collapse more apparent.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-14T22:22:05.000Z",
      "title": "Diffusion Limits for Shortest Remaining Processing Time Queues under Nonstandard Spatial Scaling",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-10-28T10:27:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K25",
      "68M20",
      "90B22"
    ],
    "keywords": [
      "shortest remaining processing time queues",
      "nonstandard spatial scaling",
      "diffusion limit",
      "processing time distribution",
      "spatial scale factor"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.3837P"
    }
  }
}