{
  "id": "1812.04775",
  "version": "v1",
  "published": "2018-12-12T01:56:17.000Z",
  "updated": "2018-12-12T01:56:17.000Z",
  "title": "Replace-after-Fixed-or-Random-Time Extensions of the Poisson Process",
  "authors": [
    "James E. Marengo",
    "Joseph G. Voelkel",
    "David L. Farnsworth",
    "Kimberlee S. M. Keithley"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We analyze extensions of the Poisson process in which any interarrival time that exceeds a fixed value $r$ is counted as an interarrival of duration $r$. In the engineering application that initiated this work, one part is tested at a time, and $N(t)$ is the number of parts that, by time $t$, have either failed, or if they have reached age $r$ while still functioning, have been replaced. We refer to $\\{N(t), t \\geq 0\\}$ as a replace-after-fixed-time process. We extend this idea to the case where the replacement time for the process is itself random, and refer to the resulting doubly stochastic process as a replace-after-random-time process.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-12T01:56:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K40",
      "60G99"
    ],
    "keywords": [
      "poisson process",
      "replace-after-fixed-or-random-time extensions",
      "analyze extensions",
      "interarrival time",
      "replace-after-random-time process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}