: The vast majority of reliability analyses assume that components and system are in either of two states: functioning or failed. The present paper develops basic theory for the study of systems of components in which any of a finite number of states may occur, representing at one extreme perfect functioning and at the other extreme complete failure. Axioms are laid down extending the standard notion of a coherent system to the new notion of a multistate coherent system. For such systems deterministic and probabilistic properties are obtained for system performance which are analogous to well-known results for coherent system reliability.

Multistate Coherent Systems.

In this paper an axiomatic development of multistate systems is presented. Three types of coherence based on the strength of the relevancy axiom are studied. The strongest of these has been investigated previously by El-Neweihi, Proschan, and Sethuraman [3]. One of the weaker types of coherence permits wider applicability to real life situations without sacrificing any of the mathematical results obtained by El-Neweihi, Proschan, and Sethuraman. The concept of system performance is formalized through expected utility and the effect of component improvement on system performance is studied using a generalization of Birnbaum's reliability importance. RELIABILITY; MULTISTATE COMPONENTS; MULTISTATE COHERENT SYSTEMS; COMPONENT IMPORTANCE

Multistate reliability models

Discrete function theory, which extends switching function theory and multiple-valued logic function theory, is introduced into multistate system analysis. Some theoretical conclusions and algorithms which play key roles in multistate system analysis are presented. The concepts of s-coherence and duality in binary-state system analysis are generalized. The set of minimal upper (maximum lower) vectors for level j, which play the role of min path (cut) set, is introduced to represent the states of a monotonic multistate system. Two approaches to computing state probability of multistate systems are given, one is based on inclusion-exclusion, the other is based on enumeration. Binary-state fault-tree is extended to multistate fault-tree. A computer code (MSTA1) has been programmed and is used to evaluate a multistate fault-tree. Multistate fault-tree and the computer code have been applied to paper-making industry; the results are consistent with the field data.

On Multistate System Analysis

This paper presents two efficient algorithms for reliability evaluation of monotone multistate systems with s-independent multistate components. The algorithms are based on the Doulliez & Jamoulle decomposition method. Algorithm 1 requires the minimal paths to be known; Algorithm 2 requires the minimal cuts to be known (the state of the system need not be specified for each vector of component states). Computer programs for implementing the algorithms are given. Computational-times are presented, and compared with the ``Inclusion-Exclusion Method'' and the ``State Enumeration Method''. The results demonstrate clearly the superiority of the algorithms to the two other methods.

Reliability Evaluation of Multistate Systems with Multistate Components

Many real-world systems such as electric power transmission and distribution systems, transportation systems, and manufacturing systems can be regarded as flow networks whose arcs have independent, finite, and multivalued random capacities. Such a flow network is indeed a multistate system with multistate components and so its reliability for the system demand d, i.e., the probability that the maximal flow is no less than d, can be computed in terms of minimal path vectors to level d (named d-MPs here). The main objective of this paper was to present a simple algorithm to generate all d-MPs of such a system for each system capacity level d in terms of minimal pathsets. Analysis of our algorithm and comparison to Xue's algorithm shows that our method has the following advantages: (1) the family of d-MP candidates that it generates is smaller in size and so d-MPs can be generated more efficiently, (2) it is expressed more intuitively and so easier to understand, and (3) whenever applied in a series-parallel case, both algorithms are essentially the same, but in a non series-parallel case, Xue's algorithm needs the extra work to transform the system into a series-parallel in advance. Two examples are illustrated to show how all d-MPs are generated by our algorithm and then the reliability of one example is computed.

On reliability evaluation of a capacitated-flow network in terms of minimal pathsets

The theory of binary coherent systems is generalized for multi-state components. The system state is defined to be the state of the “worst” component in the “best” min path, or equivalently, the state of the “best” component in the “worst” min cut. Many of the results for the binary case can be computed for multi-state systems using the binary structure and reliability function concepts. Monotonicity results are now valid with respect to stochastic ordering of component probability vectors.

Coherent Systems with Multi-State Components

: In considering life test experiments from a behavioristic Bayes point of view, we may speculate on the effect of model misspecification on the posterior mean under alternative stopping rules. Of course, under the life distribution model, we expect the mean of the posterior to equal the mean of the prior. However, if we think the model may have been misspecified, we would no longer expect this equality. We show that for Bayes estimators of mean life under the exponential model and general priors on mean life, we expect, based on a preposterior analysis, that the mean of the posterior will increase with sample size when the true model has an increasing failure rate and certain fixed stopping rules are used. Also, we show that for Bayes estimators of mean life based on the natural conjugate prior for the exponential model and complete samples, the Bayes risk is less when the coefficient of variations is less than for the exponential model. Recommendations relative to use of life test sampling plans based on an exponential model are made. (Author)

Alexander S. Wu

Papers

Coherent Systems with Multi-State Components

Preposterior analysis of Bayes estimators of mean life