D-spectrum and Reliability of a Binary System With Ternary Components

We consider a monotone binary system with ternary components. ”Ternary” means that each component can be in one of three states: up, middle (mid) and down. Handling such systems is a hard task, even if a part of the components have no mid state. Nevertheless, the permutation Monte Carlo methods, that proved very useful for dealing with binary components, can be efficiently used also for ternary monotone systems. It turns out that for ”ternary” system there also exists a combinatorial invariant by means of which it becomes possible to count the number C(r;x) of system failure sets which have a given number r and x of components in up and down states, respectively. This invariant is called ternary D-spectrum and 0 † This research was supported by the Australian Research Council under grant number CE140100049. it is an analogue of the D-spectrum (or signature) of a system with binary components. Its value is the knowledge of system failure or path set properties which do not depend on stochastic mechanism governing component failures. In case of independent and identical components, knowing D-spectrum makes it easy to calculate system UP or DOWN probability for a variety of UP/DOWN definitions suitable for systems of many types, like communication networks, flow and supply networks. etc.