All-terminal Reliability Research Articles

Uniformly least reliable graphs

The all-terminal reliability (ATR) of an undirected graph G, denoted as R (G, q), is the probability that, when the edges are assigned independent but equal failure probabilities q, 0 < q < 1 (nodes are perfect), the surviving edges induce a spanning connected subgraph of G. A graph G with n nodes and e edges is said to be uniformly least reliable if and only if R(G, q) ≤ R(G′ q) among all connected G′ with the same number of nodes and edges as G and for all failure probabilities 0 < q < 1. In this paper, we characterize uniformly least reliable graphs for e ≥ (n - 1)(n - 2)/2 + 1. We note that, unlike general graphs, for which computing the ATR has been shown to be NP-hard, the ATR of these graphs can be computed in polynomial time, providing an efficient lower bound for the general problem. © 1996 John Wiley & Sons, Inc.

A unified domination approach for reliability analysis of networks with arbitrary logic in vertices

The concept of a probabilistic graph G has been used as a universal model for network reliability. Most of the literature on this subject concentrates on computing various reliability measures (such as k-terminal reliability or all-terminal reliability) which are the probability that vertices of G can communicate with other specified vertices. Primarily, this paper provides a common theoretical framework for addressing k-terminal reliability problems. To this end, it extends the model admitting vertex functions of G to be arbitrary monotone Boolean functions: in this case G is called a monotone (S,t)-graph. In such graphs the signal passability across a node is carried out in accordance with collections of signals delivered on the node inputs from other ones, the collections are subjected to some logic principle realized by a Boolean function. Monotone (S,t)-graphs include all known classes of directed multi-terminal network reliability models. The main result of this paper is the reliability expression for computing the probability of an acyclic monotone (S,t)-graph G being operational. The expression uses the local domination parameters introduced here. That reduces the system level of consideration to the element level, providing a unifying understanding of the combinatorial nature of some results based on the domination theory and developed earlier for ordinary networks.

Generalized activities and K-terminal reliability. II

Tutte's description of the (di)chromatic polynomial of a graph in terms of activities with respect to maximal forests gives rise to a partition of the power-set of the edge-set as a collection of intervals, one corresponding to each maximal forest. This partition is useful in producing simplified expressions of all-terminal reliability. We compare two similar partitions connected with K-terminal reliability, one due to the author and the other due to Colbourn and Pulleyblank.

A Bayes procedure for estimation of current system reliability : R. Calabria, M. Guida and G. Pulcini. IEEE Transactions on Reliability, 41(4), 616 (1992)

One of the most important parameters determining the performance of communication networks is network reliability. The network reliability strongly depends on not only topological layout of the communication networks but also reliability and availability of the communication facilities. The selection of optimal network topology is an NP-hard problem so that computation time of enumeration-based methods grows exponentially with network size. This paper presents a new solution approach based on cross-entropy method, called NCE, to design of communication networks. The design problem is to find a network topology with minimum cost such that all-terminal reliability is not less than a given level of reliability. To investigate the effectiveness of the proposed NCE, comparisons with other heuristic approaches given in the literature for the design problem are carried out in a three-stage experimental study. Computational results show that NCE is an effective heuristic approach to design of reliable networks.

An [formula omitted] algorithm to compute the all-terminal reliability of ( [formula omitted]) free networks : T. Politof, A. Satyanarayana and L. Tung. IEEE Transactions on Reliability, 41(4), 512 (1992)

An [formula omitted] algorithm to compute the all-terminal reliability of ( [formula omitted]) free networks : T. Politof, A. Satyanarayana and L. Tung. IEEE Transactions on Reliability, 41(4), 512 (1992)

Reliability analysis of the double counter-rotating ring with concentrator attachments

The inherently weak reliability behavior of the ring architecture has led network designers to consider various design choices to improve network reliability. We assess the impact of provisions such as node bypass, secondary ring and concentrator trees on network reliability. For this reason, we develop closed-form expressions for the reliability and the mean time-to-failure of the double counter-rotating ring architecture. For our comparisons we adopt the 2-terminal, and the all-terminal reliability criteria. Our network reliability expressions are valid for any time-to-failure distributions of links and nodes. >

Analysis and synthesis problems for network resilience

The resilience of a network is the expected number of node pairs which can communicate in the network, when links fail independently with known probabilities. We examine the analysis and synthesis problems for resilience; many of the results parallel those on all-terminal reliability.

Reliability polynomials can cross twice

An example is given to demonstrate that all-terminal reliability polynomials of networks having the same number of nodes and the same number of links can cross twice as the edge operation probability ranges from 0 to 1. A similar result is shown for two-terminal reliability.

An O(n*log(n)) algorithm to compute the all-terminal reliability of (K/sub 5/, K/sub 2.2.2/) free networks

A graph is (K/sub 5/, K/sub 2.2.2/) free if it has no subgraph contractible to K/sub 5/ or K/sub 2.2.2/. The class of partial 3-trees (also known as Y- Delta graphs) is a proper subset of (K/sub 5/, K/sub 2.2.2/) free graphs. Let G be a network with perfectly reliable points and edges that fail independently with some known probabilities. The all-terminal reliability R(G) of G is the probability that G is connected. Computing R(G) for a general network is NP-hard. This paper presents an O(n log n) algorithm to compute R(G) of any (K/sub 5/, K/sub 2.2.2/) free graphs on n points. The running time of this algorithm is O(n) if G is planar. >

A linear-time algorithm to compute the reliability of planar cube-free networks

A planar graph G=(V,E) is a cube-free graph (CF graph) if it has no subgraph homomorphic to the cube. The cube is the graph whose vertices and edges are the vertices and edges of the three dimensional geometric cube. The all-terminal reliability of a graph G whose edges can fail (with known probabilities) is the probability that G is connected. The problem of computing the all-terminal reliability of a general graph is NP-hard. An O( mod V mod ) time algorithm for computing the all-terminal reliability of CF graphs is presented. >

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.