zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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

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