{
  "id": "2405.11510",
  "version": "v1",
  "published": "2024-05-19T10:25:52.000Z",
  "updated": "2024-05-19T10:25:52.000Z",
  "title": "On some open problems in reliability theory",
  "authors": [
    "Geni Gupur"
  ],
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "We study a stochastic scheduling on an unreliable machine with general up-times and general set-up times which is described by a group of partial differential equations with Dirac-delta functions in the boundary and initial conditions. In special case that the random processing rate of job $i,$ the random up-time rate of job $i$ and the random repair rate of job $i$ are constants, we determine the explicit expression of its time-dependent solution and give the asymptotic behavior of its time-dependent solution. Our result implies that $C_0-$semigroup theory is not suitable for this model. In general case, we determine the Laplace transform of its time-dependent solution. Next, we convert the model into an abstract Cauchy problem whose underlying operator is an evolution family. Finally, we leave some open problems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-19T10:25:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "90B25",
      "35A01"
    ],
    "keywords": [
      "open problems",
      "reliability theory",
      "time-dependent solution",
      "general set-up times",
      "partial differential equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}