{
  "abstract": "Decentralized networks can learn collision-free periodic transmission schedules, but acquisition can consume much of a finite operating horizon. We characterize acquisition in a reservation protocol for a fixed population of $n$ nodes and a period of $Γ_n\\ge n$ slots. Initially, each node attempts independently in each slot with probability proportional to $1/n$. A success reserves a slot that the node reuses every $Γ_n$ slots, retaining it after collisions. Nodes use only their own acknowledgments and small local memory. An exact finite-population Markov reduction, independent of the arrangement of reserved slots, characterizes the time until every node has a distinct reservation. As $n\\to\\infty$, mean acquisition takes $O(n^2)$ slots when $Γ_n=n$ and $O(n\\log n)$ slots when $Γ_n\\sim(1+θ)n$, for fixed $θ>0$. With a fixed number $Γ_n-n$ of spare slots, the standard deviation of acquisition time has the same asymptotic order as its mean. If that number diverges while $Γ_n/n$ remains bounded, the ratio of standard deviation to mean tends to zero. A diverging sublinear number of spare slots yields $o(n^2)$ mean acquisition while long-run utilization, the fraction $n/Γ_n$ of slots with successful transmissions, tends to one. Simulations validate these laws and compare acquisition and delivered packets per slot with existing access protocols.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09934",
  "author_total": 1,
  "authors": [
    {
      "name": "Plínio Santini Dester"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/87twgyyhky1f3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09934",
  "fulltext_url": null,
  "id": "87twgyyhky1f3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09934",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1705|Age of Information Optimization"
    ]
  ],
  "openalex": "W7224127140",
  "pdf_url": "https://arxiv.org/pdf/2609.09934",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/87twgyyhky1f3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2609.09934",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09934"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Collision-Free Schedule Acquisition through Slotted ALOHA: Scaling Laws and Fluctuations",
  "title_text": "Collision-Free Schedule Acquisition through Slotted ALOHA: Scaling Laws and Fluctuations",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/87twgyyhky1f3",
  "venue": "arXiv",
  "versions": []
}