The failure rate in reliability: Approximations and bounds
This is a partly expository article dedicated to the memory of R. L. Dobrushin. It is concerned with a random process \((\eta_s: s\geq 0)\) describing the dynamics of a system with a state space partitioned into two subsets \(\mathcal M\) and \(\mathcal P\), the latter denoting failed states. Let \(\sigma=\inf\{s:\eta_s\in{\mathcal P}\}\) be the first failure time. The paper is concerned with comparing different failure rate functions, including the hazard function \(\lambda(t)\) of \(\sigma\) and the Vesely failure rate \(\lambda_V(t)=\lim_{\Delta\to 0}\Delta^{-1}P(\eta_{t+\Delta}\in {\mathcal P}\mid \eta_t\in{\mathcal M})\), \(t\geq 0\), especially for large \(t\). Explicit results are obtained by supposing that the dynamics of \((\eta_s)\), up to its first entrance to \(\mathcal P\), are represented by a regenerative process on \(\mathcal M\), which is killed at a self-modulated rate.
scientific article; zbMATH DE number 988087