The rank of the current lifetime
The rank \(R_ t\) of the current lifetime of a component is denoted as \(R_ t= \#\{1\leq i\leq N_ t\): \(X_ i> X_{N_ t+1}\}\), where \(\{X_ i\}\) denotes the lifetimes (interarrival times) of a standard renewal process \(\{N_ t\), \(t\geq 0\}\). The asymptotic distribution of the relative rank \(R_ t/N_ t\) as \(t\to\infty\) is obtained. This result is shown to exhibit another feature of the inspection paradox.
scientific article; zbMATH DE number 633494