Ordering series and parallel compositions

Abstract

In this article we analyze the reliability of networks constructed as repetitive compositions of simple identical structures (building blocks). More precisely, we limit our analyses to two-terminal networks composed of only two motifs (two basic building blocks): two devices in series, and two devices in parallel. Therefore, for a given number of devices (which is a power of two) we build all the possible compositions of series and parallel networks of two devices. For all the resulting two-terminal networks, we compute exactly their associated reliability polynomials. These two-terminal networks are ranked according to partial orders over the set of all compositions. These partial orders have reasonably simple algebraic descriptions and can reveal interesting properties, and in particular they allow us to order the resulting two-terminal networks with respect to reliability.