On a Gibbs Measure Representation for Complex Load-sharing Parallel Systems

Complex load-sharing parallel systems incorporate dependencies between components through a load-sharing rule. Such systems were used to model the strength of fibrous composites where component fiber failures occur in cycles having two phases. As the load on the system increases, a single component fails (Phase I), which then causes a cascade of component failures (Phase II) due to the load transfer as these Phase II failures occur. Under monotone loadsharing the failures have a simple Markovian structure. The current state of the system only depends on the latest Phase I failure and the corresponding Phase II failures have a Gibbs measure (GM) representation when the component strengths are independent. The GM representation formalism indicates that log odds ratios for the strength distribution are the fundamental quantities in determining the potentials in the GM for the state of the system. Formulas for the potentials are particularly simple through the use of generalizations of the log-logistic and the log-logistic distributions for the component strength distributions. These simplified formulas can be used to understand the system structural reliability.