{
  "abstract": "Two-sided threshold crossings arise in discrete monitoring systems whenever intervention is triggered by departure from an operating band. This paper develops a finite-state first-passage framework based on a biased random walk with symmetric absorbing barriers. To remove parity gaps of the raw hitting time, we analyse a parity-corrected lifetime through even- and odd-state Markov skeletons. Adjacent-row likelihood-ratio inequalities verify the monotonicity condition and establish increasing failure rate and new-better-than-used properties. These ageing properties yield log-concave survival probabilities and geometric persistence bounds. Substochastic-matrix formulas are derived for survival, hazard, mean lifetime, state-conditioned remaining useful life, and finite-horizon crossing risk. A renewal-cost criterion then converts the first-passage distribution into an optimal preventive-intervention schedule. Model parameters are obtained by matching the drift and variance of observed increments to a two-point random-walk approximation, with a rolling-window extension for nonstationary regimes. Numerical experiments compare exact calculations with Monte Carlo diagnostics, moment-matched inverse-Gaussian and Weibull benchmarks, serially correlated increments, calibration perturbations, and fixed-interval policies. A synthetic remote-patient-monitoring example illustrates how discrete physiological deviations can be mapped to transparent risk scores and personalized review schedules. The framework provides an auditable link between discrete stochastic dynamics, remaining-lifetime prediction, and condition-based intervention, while separating mathematical validation from clinical validation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13000",
  "author_total": 1,
  "authors": [
    {
      "name": "Ye Liang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0yncz27xfkgyt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13000",
  "fulltext_url": null,
  "id": "0yncz27xfkgyt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13000",
  "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",
      "math.AP"
    ],
    [
      "openalex",
      "2213|Reliability and Maintenance Optimization"
    ]
  ],
  "openalex": "W7212712409",
  "pdf_url": "https://arxiv.org/pdf/2609.13000",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0yncz27xfkgyt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.13000",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13000"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Parity-Skeleton First-Passage Dynamics and Optimal Intervention in Two-Sided Discrete Monitoring Systems",
  "title_text": "Parity-Skeleton First-Passage Dynamics and Optimal Intervention in Two-Sided Discrete Monitoring Systems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/0yncz27xfkgyt",
  "venue": "arXiv",
  "versions": []
}