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A class of correlated cumulative shock models. (English) Zbl 0565.60072

The authors consider a cumulative damage shock model. Shocks occur according to a renewal process with intershock times ${X}_{1},{X}_{2},··$.. At the time of the n-th epoch, ${\sum }_{j=1}^{n}{X}_{i}$, the magnitude of the damage is ${Y}_{n}$. It is assumed that the pairs $\left({X}_{n},{Y}_{n}\right)$, $n=1,2,···$, are independent and identically distributed but ${X}_{n}$ and ${Y}_{n}$ may be dependent. Failure of the underlying item occurs at ${S}_{z}\equiv inf\left\{t:{\sum }_{i=1}^{N\left(t\right)}{X}_{i}>z\right\}$ where z is a fixed breaking threshold and $\left\{$ N(t), $t\ge 0\right\}$ is the counting process associated with the renewal process ${\left\{{Y}_{n}\right\}}_{0}^{\infty }·$

The authors obtain the Laplace transform, the distribution function and the moments of ${S}_{z}$. They also find conditions which imply that ${S}_{z}$ is NBU, NBUE and HNBUE. The limiting distributions of ${S}_{z}$ (normalized) as $z\to \infty$ and a strong law of large numbers for ${S}_{z}$ are also given. The case in which ${X}_{n}$ and ${Y}_{n+1}$ are dependent but not ${X}_{n}$ and ${Y}_{n}$, $n=1,2,···$, is also studied and analogous results are obtained.

Reviewer: M.Shaked

##### MSC:
 60K10 Applications of renewal theory 90B25 Reliability, availability, maintenance, inspection, etc. (optimization)