*(English)*Zbl 0565.60072

The authors consider a cumulative damage shock model. Shocks occur according to a renewal process with intershock times ${X}_{1},{X}_{2},\xb7\xb7$.. 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 $({X}_{n},{Y}_{n})$, $n=1,2,\xb7\xb7\xb7$, are independent and identically distributed but ${X}_{n}$ and ${Y}_{n}$ may be dependent. Failure of the underlying item occurs at ${S}_{z}\equiv inf\{t:{\sum}_{i=1}^{N\left(t\right)}{X}_{i}>z\}$ where z is a fixed breaking threshold and $\{$ N(t), $t\ge 0\}$ is the counting process associated with the renewal process ${\left\{{Y}_{n}\right\}}_{0}^{\infty}\xb7$

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 ${Y}_{n}$, $n=1,2,\xb7\xb7\xb7$, is also studied and analogous results are obtained.

##### MSC:

60K10 | Applications of renewal theory |

90B25 | Reliability, availability, maintenance, inspection, etc. (optimization) |