Multistate Reliability Evaluation of Communication Networks via Multi-Valued Karnaugh Maps and Exhaustive Search

This chapter examines two simple (albeit useful) methods used to evaluate the reliability of two-terminal multistate flow networks. These two methods involve two Karnaugh map versions, namely the Variable-Entered Karnaugh Map (VEKM) and the Multi-Valued Karnaugh Map (MVKM). These two versions are crucial in providing not only the visual insight necessary to write better future software but also adequate means of verifying such software. We assess these two versions of map methods versus the exhaustive search method, which guarantees conceptual clarity at the expense of lack of computational efficiency. Our target is the evaluation of the probability mass function (pmf) in a wide array of cases, in which we consider flow from a source node to a sink node in a capacitated network with a multistate capacity model for the links. Each network link has a varying capacity, which is assumed to exist in a mutually exclusive sense. The reliability of the system is wholly dependent on its ability to successfully transmit at least a certain required system flow from the source (transmitter) to the sink (receiver) station. The max-flow min-cut theorem is critical in obtaining all successful states. To demonstrate the proposed methods applicability, two demonstrative examples are given with ample details. The special case of a binary flow network is also considered in terms of pseudo-Boolean functions.