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Zbl 0565.60072
Sumita, Ushio; Shanthikumar, J.George
A class of correlated cumulative shock models.
(English)
[J] Adv. Appl. Probab. 17, 347-366 (1985). ISSN 0001-8678

The authors consider a cumulative damage shock model. Shocks occur according to a renewal process with intershock times $X\sb 1,X\sb 2,..$.. At the time of the n-th epoch, $\sum\sp{n}\sb{j=1}X\sb i$, the magnitude of the damage is $Y\sb n$. It is assumed that the pairs $(X\sb n,Y\sb n)$, $n=1, 2,...$, are independent and identically distributed but $X\sb n$ and $Y\sb n$ may be dependent. Failure of the underlying item occurs at $S\sb z\equiv \inf \{t: \sum\sp{N(t)}\sb{i=1}X\sb i>z\}$ where z is a fixed breaking threshold and $\{$ N(t), $t\ge 0\}$ is the counting process associated with the renewal process $\{Y\sb n\}\sb 0\sp{\infty}.$ \par The authors obtain the Laplace transform, the distribution function and the moments of $S\sb z$. They also find conditions which imply that $S\sb z$ is NBU, NBUE and HNBUE. The limiting distributions of $S\sb z$ (normalized) as $z\to \infty$ and a strong law of large numbers for $S\sb z$ are also given. The case in which $X\sb n$ and $Y\sb{n+1}$ are dependent but not $X\sb n$ and $Y\sb n$, $n=1, 2,...$, is also studied and analogous results are obtained.
[M.Shaked]
MSC 2000:
*60K10 Appl. of renewal theory
90B25 Reliability, etc.

Keywords: cumulative damage shock model; renewal process; Laplace transform; strong law of large numbers

Cited in: Zbl 0836.60094

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