{
  "id": "1208.5658",
  "version": "v2",
  "published": "2012-08-28T13:29:56.000Z",
  "updated": "2014-11-21T14:26:21.000Z",
  "title": "On modular decompositions of system signatures",
  "authors": [
    "Jean-Luc Marichal",
    "Pierre Mathonet",
    "Fabio Spizzichino"
  ],
  "journal": "Journal of Multivariate Analysis 134 (2015) 19-32",
  "doi": "10.1016/j.jmva.2014.10.002",
  "categories": [
    "math.PR",
    "math.OC"
  ],
  "abstract": "Considering a semicoherent system made up of $n$ components having i.i.d. continuous lifetimes, Samaniego defined its structural signature as the $n$-tuple whose $k$-th coordinate is the probability that the $k$-th component failure causes the system to fail. This $n$-tuple, which depends only on the structure of the system and not on the distribution of the component lifetimes, is a very useful tool in the theoretical analysis of coherent systems. It was shown in two independent recent papers how the structural signature of a system partitioned into two disjoint modules can be computed from the signatures of these modules. In this work we consider the general case of a system partitioned into an arbitrary number of disjoint modules organized in an arbitrary way and we provide a general formula for the signature of the system in terms of the signatures of the modules. The concept of signature was recently extended to the general case of semicoherent systems whose components may have dependent lifetimes. The same definition for the $n$-tuple gives rise to the probability signature, which may depend on both the structure of the system and the probability distribution of the component lifetimes. In this general setting, we show how under a natural condition on the distribution of the lifetimes, the probability signature of the system can be expressed in terms of the probability signatures of the modules. We finally discuss a few situations where this condition holds in the non-i.i.d. and nonexchangeable cases and provide some applications of the main results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-08-28T13:29:56.000Z",
      "abstract": "Considering a coherent system made up of $n$ components having i.i.d. continuous lifetimes, F. Samaniego defined its structural signature as the $n$-tuple whose $k$-th coordinate is the probability that the $k$-th component failure causes the system to fail. This $n$-tuple, which depends only on the structure of the system and not on the distribution of the component lifetimes, is a very useful tool in the theoretical analysis of coherent systems. This concept was recently extended to the general case of semicoherent systems whose components may have dependent lifetimes, where the same definition for the $n$-tuple gives rise to the probability signature, which depends in general on both the structure of the system and the probability distribution of the component lifetimes. The dependence on the latter is encoded via the so-called relative quality function associated with the distribution of the component lifetimes. In this work we consider a system that is partitioned into disjoint modules. Under a natural decomposition property of the relative quality function, we give an explicit and compact expression for the probability signature of the system in terms of the probability signatures of the modules and the structure of the modular decomposition. This formula holds in particular when the lifetimes are i.i.d. or exchangeable, but also in more general cases, thus generalizing results recently obtained in the i.i.d. case and for special modular decompositions.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-11-21T14:26:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "62N05",
      "90B25",
      "94C10"
    ],
    "keywords": [
      "system signatures",
      "component lifetimes",
      "probability signature",
      "relative quality function",
      "coherent system"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1208.5658M"
    }
  }
}