Checking Correctness of a Symbolic Reliability Expression for a Capacitated Network

Abstract

Checking a symbolic reliability expression for a flow network is useful for detecting faults in hand derivations and for debugging computer programs. This checking can be achieved in a systematic way, though it may be a formidable task. Three exhaustive tests are given when a reliability system or network has a flow constraint. These tests apply to unreliability and reliability expressions for non-coherent as well as coherent systems, and to cases when both nodes and branches are unreliable. Further properties of reliability expressions derived through various methods are discussed. All the tests and other pertinent results are proved and illustrated by examples.

A Review of Flow-Capacitated Networks: Algorithms, Techniques and Applications

“…In this case, the network capacity is replaced by a pseudo-Boolean capacity function that depends on component capacities and successes. Many of concepts and procedures considered in this paper have direct extensions for such a probabilistic case [21,22,25,34,36,41,[68][69][70]…”

Network Analysis via Pseudo-Boolean Functions and Boole-Shannon Expansion

“…A Boolean function, also known as a Switching function, is a type of mapping that takes as input a combination of 𝑛 binary digits (either 0 or 1) and produces a single output digit that is also binary (either 0 or 1), {0, 1} 𝑛 → {0, 1}. In other words, 𝑆(𝑿) represents a unique combination of 0's and 1's for every possible combination of 𝑛 binary digit [3,7,[9][10][11]13]. However, a pseudo-switching (pseudo-Boolean) function 𝐶(𝑿) is a mapping {0, 1} 𝑛 → 𝑅 where 𝑅 is the field of real numbers, i.e., 𝐶(𝑿) assigns a real number to each of the 2 𝑛 possible 𝑿 values.…”

“…However, a pseudo-switching (pseudo-Boolean) function 𝐶(𝑿) is a mapping {0, 1} 𝑛 → 𝑅 where 𝑅 is the field of real numbers, i.e., 𝐶(𝑿) assigns a real number to each of the 2 𝑛 possible 𝑿 values. For binary capacitated networks and other applications, pseudo-Boolean functions play an essential role [2,3,7,[9][10][11]13]. This section discusses pseudo-switching functions briefly and provides some ideas on their utility in the analysis of binary capacitated networks [13].…”

“…Many novel approaches for studying a capacitated network have been developed over the past few decades [1][2][3][4][5][6][7][8][9][10][11][12][13]. A broad class of these approaches is based on the repeated application of Bayesian decomposition to the network graph [14][15][16], or equivalently, of the Boole-Shannon expansion [17] to the switching (Boolean) function of the network success or network failure [18][19][20][21].…”

“…The Total Probability Theorem [30][31][32] in the probability domain and the Factoring Theorem in the "Graph Domain" [15] are both equivalent to the Boole-Shannon Expansion in the Boolean domain [21]. The Variable-Entered Karnaugh Map (VEKM) serves as a powerful visual tool for implementing this expansion [33,[7][8][9][10][11].…”