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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

60K10 Applications of renewal theory (reliability, demand theory, etc.)
90B25 Reliability, availability, maintenance, inspection in operations research
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