Comparing Renewal Processes, With Application to Reliability Modeling of Highly Reliable Systems

Elementary methods are used to obtain upper and lower bounds for the difference between two renewal functions corresponding to two different life distributions and to obtain various order relationships between them that follow from order relationships on the underlying life distributions. The bounds are used to approximate the value of an unknown renewal function by the value of a known renewal function and to bound the error of approximation by expressions involving the difference between the two underlying life distributions. They are useful mainly in the region where time is small compared with the means of the two distributions, and they therefore supplement the information that can be obtained for large time using the elementary renewal theorem. These results find major application in reliability modeling of highly reliable systems.