Tolerance Limits for $k$-Out-of- $n$ Systems With Exponentially Distributed Component Lifetimes

This paper deals with the problem of determining statistical tolerance limits for the time-to-failure of a
k
-out-of-
n
:F system from component life data. The lifetimes of the components are assumed to be
s
-independently and identically distributed exponential variates. On the basis of a random sample of components, exact β-content, and β-expectation
s
-tolerance limits for the system lifetime are provided, which are valid even when certain percentages of the smallest, and the largest component failure times are missing. The proportion of conforming systems in a large batch is assessed by constructing an appropriate
s
-tolerance limit. The results presented are extended to several distributions, including the Weibull case with an unknown scale parameter. An example concerning a system of water pumps for cooling a reactor is used for illustration. The proposed method of obtaining
s
-tolerance limits can be readily applied, and requires only minimal computational effort. Our perspective is particularly attractive when system test data are not available, or are very scarce. In many circumstances, component testing is also advantageous in terms of time and cost savings.