On Threshold Optimization in Fault Tolerant Systems

Fault tolerant systems are considered, where a nominal system is monitored by a fault detection algorithm, and the nominal system is switched to a backup system in case of a detected fault. Conventional fault detection is in the classical setting a trade-off between detection probability and false alarm probability. For the considered fault tolerant system, a system failure occurs either when the nominal system gets a fault that is not detected, or when the fault detector signals an alarm and the backup system breaks down. This means that the trade-off for threshold setting is different and depends on the overall conditions, and the characterization and understanding of this trade-off is important. It is shown that the probability of system failure can be expressed in a general form based on the probability of false alarm and detection power, and based on this form the influence ratio is introduced. This ratio includes all information about the supervised system and the backup system that is needed for the threshold optimization problem. It is shown that the influence ratio has a geometrical interpretation as the gradient of the receiver operating characteristics (ROC) curve at the optimal point, and furthermore, it is the threshold for the optimal test quantity in important cases.