Error bounds for exponential approximation of large-system reliability
The paper considers several exponential approximations and bounds on the distribution of a sum \(T= \sum_{i=1}^N X_i\) of i.i.d. random variables randomly stopped by an independent geometric random variable on the positive integers. In the context of reliability, \(T\) is the cumulative sojourn time spent in the regenerative state (that corresponds to beginning with an equipment that is as good as new), and is often viewed as a pessimistic approximation to the total system lifetime when successive repair times to restore a failed equipment to the regenerative state are included. Apart from being applicable in other contexts such as queueing and risk theory, it is worth noting -- although not mentioned by the author -- that the random variable \(T\) is also the lifetime of an equipment under a renewal process shock model, when the number of shocks to failure is geometric. After a review of exponential bounds to the compound geometric distribution of \(T\), known in the literature, the author develops a structural measure of closeness to exponential distributions in the spirit of the well known Stein-Chen method of Poisson approximation to the distribution of discrete random variables. A Stein-type equation characterizing exponential distributions is developed, and then used to derive upper and lower bounds to the tail of the distribution of \(T\) via the exponential distribution with the same mean as the expected value of \(T\).
scientific article; zbMATH DE number 5144811
scientific article