The probability of failure (failure rate) is a key input parameter to integrity monitoring systems used for safety, liability or mission critical applications. A standard approach in the design of Global Positioning System (GPS) integrity monitoring is to utilise the service commitment on the probability of major service failure, often by applying a conservative factor. This paper addresses the question of what factor is appropriate by applying Bayesian inference to real and hypothetical fault histories.

Global Navigation Satellite System (GNSS) anomalies include clock or signal transmission type faults which are punctual (may occur at any time) and incorrect ephemeris data which are broadcast for a nominal two hours. These two types of anomaly, classified as continuous and discrete respectively are addressed. Bounds on the total probability of failure are obtained with given confidence levels subject to well defined hypotheses relating past to future performance. Factors for the GPS service commitment of 10−5 per hour per satellite are obtained within the range two to five with high confidence (up to 1–10−9).

Traditional failure modes and effects analysis (FMEA) methods lack sufficient semantics and structure to provide full traceability between the failure modes and the effects of the complex system. To overcome this limitation, this paper proposes a formal failure knowledge representation model combined with the structural decomposition of the complex system. The model defines the failure modes as the inherent properties of the physical entities at different hierarchical levels, and employs the individual color, unified color, and Boolean matrix of the polychromatic sets to represent the failure modes in terms of their interrelationships and their relations to the physical system. This method is a structure-based modeling technique that provides a simple, yet comprehensive framework to organize the failure modes and their causes and effects more systematically and completely. Using the iterative search process operated on the reasoning matrices, the end effects on the entire system can be achieved automatically, which allows for the consideration of both the single and multiple failures. An example is embedded in the description of the methodology for better understanding. Because of the powerful mathematical modeling capability of the polychromatic sets, the approach presented in this paper makes significant progress in FMEA formalization.