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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, 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,···, are independent and identically distributed but X n and Y n may be dependent. Failure of the underlying item occurs at S z inf{t: i=1 N(t) X i >z} where z is a fixed breaking threshold and { N(t), t0} is the counting process associated with the renewal process {Y n } 0 ·

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 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

60K10Applications of renewal theory
90B25Reliability, availability, maintenance, inspection, etc. (optimization)