A logarithmic lower estimate for the mathematical expectation of the first failure moment of a higher-reliable system with protection in a nonstationary regime. (Ukrainian, English) Zbl 1026.60101

Kushnir, O. O.

Teor. Jmovirn. Mat. Stat. 65, 104-109 (2001); translation in Theory Probab. Math. Stat. 65, 115-122 (2002).

Summary: Let \((\xi_n)\), \((\eta_n)\), \((\zeta_n)\), \(n \geq 1\), be three sequences of independent and nonnegative random variables with distribution functions \(F\), \(G\) and \(J\), respectively. Let \(T_0 =0\), \(S_0=-s\) \((s>0)\), \(T_n' =T_n +\zeta_n\), \(T_{n+1} =T_n' +\eta_n\), \(S_n=S_{n-1}+\xi_n\), \(n \geq 1.\) Let \(\tau_s\) be the moment of the first hit of the sequence \((S_k)_{k\geq 1}\) in the random segment \([T_n, T_n'],n\geq 1\), i.e. \(\tau=\inf_{k\geq 1}\left\{S_k: S_k\in\bigcup_{n=1}^{\infty} [T_n, T_n']\right\}.\) Denote by \(b_s=E\{\tau/S_0=-s\}\) the conditional expectation of the random variable \(\tau\). Under some rather strong assumptions on the distribution functions \(F\), \(G\) and \(J\) (existence of exponential moments, in particular) the author finds estimates from below for the conditional expectation \(b_s\). For more details see the articles by the author [Theory Probab. Math. Stat. 55, 121-128 (1997); translation from Teor. Jmovirn. Mat. Stat. 55, 117-124 (1996; Zbl 0923.60092) and ibid. 61, 95-100 (2000) resp. ibid. 61, 91-96 (2000; Zbl 0985.60079)].

MSC:

60K05	Renewal theory
60K10	Applications of renewal theory (reliability, demand theory, etc.)

Keywords:

higher-reliable system with protection; non-stationary regime; mean time to failure; lower estimate

Citations:

Zbl 0923.60092; Zbl 0985.60079