Probability Models for Sequential-Stage System Reliability Growth via Failure Mode Removal

This paper provides guidance for the planners of a test of any system that operates in sequential stages: only if the first stage functions properly (e.g., a vehicle's starter motor rotates adequately) can the second stage be activated (ignition system performs) and hence tested, followed by a third stage (engine starts and propels vehicle), with further stages such as wheels, and steering, and finally brakes eventually brought to test. Each sequential stage may fail to operate because its design, manufacture, or usage has faults or defects that may give rise to failure. Testing of all stages in the entire system in appropriate environments allows failures at the various stages to reveal defects, which are targets for removal. Early stages' fault activations thus postpone exposure of later stages to test. It is clear that only by allowing the entire system to be tested end-to-end, through all stages, and to observe several total system successes can one be assured that the integrated system is relatively free of defects and is likely to perform well if fielded. The methodology of the paper permits a test planner to hypothesize the numbers of (design) faults present in each stage, and the stagewise probability of a fault activation, leading to a system failure at that stage, given survival to that stage. If the test item fails at some stage, then rectification ("fix") of the design occurs, and the fault is (likely) removed. Failure at that stage is hence less likely on future tests, allowing later stages to be activated, tested, and fixed. So reliability grows. To allow many Test and Fix (TAF) cycles is obviously impractical. A stopping criterion proposed by E. A. Seglie that suggests test stopping as soon as an uninterrupted run/sequence of r (e.g., 5) consecutive system successes has been achieved is studied quantitatively here. It is shown how to calculate the probability of eventual field success if the design is frozen and the system fielded after such a sequential stopping criterion is achieved. The mean test length is also calculated. Many other calculations are possible, based on formulas presented.