MARKOV CHAIN METHOD FOR COMPUTING THE RELIABILITY OF HAMMOCK NETWORKS

In this paper, we develop a new method for evaluating the reliability polynomial of a hammock network. The method is based on a homogeneous absorbing Markov chain and provides the exact reliability for networks of width less than 5 and arbitrary length. Moreover, it produces a lower bound for the reliability polynomial for networks of width greater than or equal to 5. To investigate how sharp this lower bound is, we compare our method with other approximation methods and it proves to be the most accurate in terms of absolute as well as relative error. Using the fundamental matrix, we also calculate the average time to absorption, which provides the mean length of a network that is expected to work.

Full Hermite interpolation of the reliability of a hammock network

Although the hammock networks were introduced more than sixty years ago, there is no general formula of the associated reliability polynomial. Using the full Hermite interpolation polynomial, we propose an approximation for the reliability polynomial of a hammock network of arbitrary size. In the second part of the paper, we provide combinatorial formulas for the first two non-zero coefficients of the reliability polynomial.

A Recursive Formula for the Reliability of a r-Uniform Complete Hypergraph and Its Applications

The reliability polynomial R(S,p) of a finite graph or hypergraph S=(V,E) gives the probability that the operational edges or hyperedges of S induce a connected spanning subgraph or subhypergraph, respectively, assuming that all (hyper)edges of S fail independently with an identical probability q=1-p. In this paper, we investigate the probability that the hyperedges of a hypergraph with randomly failing hyperedges induce a connected spanning subhypergraph. The computation of the reliability for (hyper)graphs is an NP-hard problem. We provide recurrence relations for the reliability of r-uniform complete hypergraphs with hyperedge failure. Consequently, we determine and calculate the number of connected spanning subhypergraphs with given size in the r-uniform complete hypergraphs.

Graphs models and algorithms for reliability assessment of coherent and non-coherent systems

In this article, we propose new models and algorithms for the reliability assessment of systems relying on concepts of graphs theory. These developments exploit the order relation on the set of system components’ states which is graphically represented by the Hasse diagram. The monotony property of the reliability structure function of coherent systems allows us to obtain an ordered graph from the Hasse diagram. This ordered graph represents all the system states and it can be obtained from only the knowledge of the system tie-sets. First of all, this model gives a new way for the research of a minimal disjoint Boolean polynomial, and, second, it is able to directly find the system reliability without resorting to an intermediate Boolean polynomial. Browsing the paths from the minimal tie-sets to the maxima of the ordered graph and using a weight associated with each node, we are able to propose a new algorithm to directly obtain the reliability polynomial by the research of sub-graphs representing eligible monomials. This approach is then extended to non-coherent systems thanks to the introduction of a new concept of terminal tie-sets. These algorithms are applied to some case studies, for both coherent and non-coherent real systems, and the results compared with those computed using standard reliability block diagram or fault tree models validate the proposed approach. Formal definitions of used graphs and of developed algorithms are also given, making their software implementation easy and efficient.