Reliability Polynomial Research Articles

A Set System Polynomial with Colouring and Reliability Applications

In order to relate the chromatic and all-terminal reliability polynomials, a simple two-variable polynomial is introduced. The latter polynomial is defined on a set system; as a result, many similarities between the chromatic and all-terminal reliability polynomials can be derived. In fact, within this general framework it is often possible to generalize from one of these two graph polynomials in order to get new results for the other.

Algebraic Methods Applied to Network Reliability Problems

An algebraic structure underlying network reliability problems is presented for determining the 2-terminal reliability of directed networks. An iterative algorithm is derived from this algebraic perspective to solve the $(s,j)$-terminal reliability problem simultaneously for all nodes j. In addition to providing an exact answer (in the form of a reliability polynomial), the algorithm also yields a nondecreasing sequence of approximate solutions guaranteed to be lower bounds on the exact solution. Empirical results, presented for two different implementations of the algorithm, show that useful approximate solutions can be obtained in a reasonable amount of computation time.

Generalized S-Shapedness of Reliability Functions

Reliability functions have the property that if they intersect the diagonal y = x it is done in an S-shaped form (Moore and Shannon). In this paper a wider class of polynomials is presented, the members of which form a dense grid on the unit square. The S-shapedness intersection property holds whenever a reliability function intersects a polynomial in this set. The members of this set are all reliability polynomials. The diagonal y = x is also a member of this set, thus a generalization of Moore-Shannon result is achieved.

On an Invariant of Graphs and the Reliability Polynomial

A combinatorial invariant called the parity of a graph is introduced. An earlier concept of signed domination of a graph, relevant to computing certain reliability measures on an undirected graph, ...

Polyhedral Combinatorics and Network Reliability

This paper studies the reliability of systems comprised of variables which must satisfy a set of linear equalities and nonnegativity constraints, and which are subject to random failure. A major example, which will be given special emphasis, is the reachability (source-to-all connectedness reliability) problem for stochastic networks. We show how properties of convex polyhedra, and their network counterparts, are useful in determining properties of the reliability polynomial for these systems. In particular, we show how recent results by Stanley and by Billera and Lee concerning the face numbers of convex polytopes and polyhedra can be applied to provide good upper and lower bounds on reliability in linear systems, and we apply these bounds in the network examples.

Reliability polynomials of computer communication networks

The expression for the overall reliability of a computer communication network (CCN) when all edges have equal probability of being up is called the reliability polynomial of the CCN. For a complicated network, the overall reliability of the CCN can be approximated by the truncated polynomial. A characterization of the reliability polynomial and its approximation are give in this paper.

Bounds on the Reliability Polynomial for Shellable Independence Systems

The reliability polynomial associated with an independence system is $g ( p ) = \sum_{k = 0}^n f_k p^k ( 1 - p )^{n - k} $, where $f_k $ is the number of independent sets of cardinality k and n is the cardinality of the ground set. An independence system $( T,\Gamma )$ is shellable if all maximal independent sets have the same cardinality and if there exists an ordered partition of the set of independent sets into intervals $\{ [ F_i ,G_i ] \}_{i = 1}^I $ (an interval $ [ F ,G ] = \{ F^\prime :F \subseteq F^\prime \subseteq G \}$) where for all $n^\prime $, $n^\prime \leqq I$, $G_{n^\prime } $, is a maximal independent set and ($T, \cup _{i=1}^{n^\prime } [ F_i ,G_i ]$) is an independence system. For the class of shellable independence systems, tight upper and lower bounds are given on $g ( p )$, when the number of maximal independent sets and the number of minimum cardinality dependent sets are fixed. These results can be applied to obtain bounds on the reachability measure, which is the probability that...