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Some multivariate distributions derived from a non-fatal shock model. (English) Zbl 0646.60092
Consider a nonhomogeneous Poisson process with continuous mean function $$\Lambda$$. Suppose that at the k-th epoch of the process, at time $$S_ k$$, there is a shock which causes the simultaneous failure of the components $$j\in J\subset \{1$$,..., $$n\}$$ with probability $$p_ J(S_ k)$$, $$k=1$$,.... Let $$T_ j$$ be the failure time of component j, $$j=1$$,..., n. Denote $$F\equiv 1-e^{-\Lambda}.$$
The main result of the paper is the following. Suppose $$p_ J(t)$$ is nondecreasing in $$t\geq 0$$ for each nonempty set $$J\subset \{1$$,..., $$n\}$$. If F is IFR, IFRA or NBU then $$(T_ 1$$,..., $$T_ n)$$ is multivariate IFR [in the sense of the author, ibid. 22, 197-204 (1985; Zbl 0566.62039)], IFRA [in the sense of H. W. Block and the author, Ann Prob. 8, 793-801 (1980; Zbl 0455.62078)], or NBU [in the sense of A. W. Marshall and the reviewer, Ann. Prob. 10, 259-264 (1982; Zbl 0481.62077)].
Reviewer: M.Shaked

##### MSC:
 60K10 Applications of renewal theory (reliability, demand theory, etc.) 90B25 Reliability, availability, maintenance, inspection in operations research
##### Citations:
Zbl 0566.62039; Zbl 0455.62078; Zbl 0481.62077
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