{
  "id": "1406.1781",
  "version": "v2",
  "published": "2014-06-06T19:42:10.000Z",
  "updated": "2014-12-29T17:55:28.000Z",
  "title": "Renyi's Parking Problem Revisited",
  "authors": [
    "Matthew P. Clay",
    "Nandor J. Simanyi"
  ],
  "comment": "Final version; to appear in Proceedeings of \"Probability and Dynamics at IM-UFRJ\", 13 pages, 6 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "R\\'enyi's parking problem (or $1D$ sequential interval packing problem) dates back to 1958, when R\\'enyi studied the following random process: Consider an interval $I$ of length $x$, and sequentially and randomly pack disjoint unit intervals in $I$ until the remaining space prevents placing any new segment. The expected value of the measure of the covered part of $I$ is $M(x)$, so that the ratio $M(x)/x$ is the expected filling density of the random process. Following recent work by Gargano {\\it et al.} \\cite{GWML(2005)}, we studied the discretized version of the above process by considering the packing of the $1D$ discrete lattice interval $\\{1,2,...,n+2k-1\\}$ with disjoint blocks of $(k+1)$ integers but, as opposed to the mentioned \\cite{GWML(2005)} result, our exclusion process is symmetric, hence more natural. Furthermore, we were able to obtain useful recursion formulas for the expected number of $r$-gaps ($0\\le r\\le k$) between neighboring blocks. We also provided very fast converging series and extensive computer simulations for these expected numbers, so that the limiting filling density of the long line segment (as $n\\to \\infty$) is R\\'enyi's famous parking constant, $0.7475979203...$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-06T19:42:10.000Z",
      "abstract": "R\\'enyi's parking problem (or $1D$ sequential interval packing problem) dates back to 1958, when R\\'enyi studied the following random process: Consider an interval $I$ of length $x$, and sequentially and randomly pack disjoint unit intervals in $I$ until the remaining space prevents placing any new segment. The expected value of the measure of the covered part of $I$ is $M(x)$, so that the ratio $M(x)/x$ is the expected filling density of the random process. Following recent work by Gargano {\\it et al.} \\cite{GWML(2005)}, we studied the discretized version of the above process by considering the packing of the $1D$ discrete lattice interval $\\{1,2,\\dots ,n+2k-1\\}$ with disjoint blocks of $(k+1)$ integers but, as opposed to the mentioned \\cite{GWML(2005)} result, our exclusion process is symmetric, hence more natural. Furthermore, we were able to obtain useful recursion formulas for the expected number of $r$-gaps ($0\\le r\\le k$) between neighboring blocks. We also provided very fast converging series and extensive computer simulations for these expected numbers, so that the limiting filling density of the long line segment (as $n\\to \\infty$) is R\\'enyi's famous parking constant, $0.7475979203\\dots$.",
      "comment": "12 pages, 6 figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-12-29T17:55:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60D05"
    ],
    "keywords": [
      "renyis parking problem",
      "random process",
      "randomly pack disjoint unit intervals",
      "sequential interval packing problem",
      "filling density"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.1781C"
    }
  }
}