: In a phased mission the relevant system configuration (block diagram or fault tree) changes during consecutive time periods (phases). Many systems are required to perform phased missions. A classic example is a space vehicle. A reliability analysis for a phased mission encounters complexities not present with just one phase, but can be transformed into an analysis of a synthetic single phase case. The transformation has a potential for direct application, or can be used to study various computational algorithms and approximations.

https://apps.dtic.mil/sti/pdfs/ADA008290.pdf

Reliability analysis of phased missions.

The concept of a “module”, i.e., a package of components of a system which can be removed and replaced as a whole, has been long in use in systems design and analysis. In this paper a formal definition of this concept is given and its properties are studied.Most of the results are obtained for “coherent” systems, i.e., for systems whose performance improves as the performance of their components improves. For such systems the relationship between modules and minimal paths can be fully clarified, and the results obtained lead to a criterion for deciding whether or not a given set of components constitutes a module of a given system. It is shown that a coherent system always has uniquely determined maximal modules which have the property that either all are disjoint or no two of them are disjoint. These maximal modules determine uniquely a decomposition of the system into disjoint modular factors. Furthermore, a theorem on the union of modules (the “three modules theorem”), which has been previously known f...

Modules of Coherent Binary Systems

: For the class of coherent binary systems the inequality h'(p) > or = h(p)(1 - h(p))/(p(1-p)) has been previously proven, where p = component reliability, h(p) = system reliability; hence h'(p) = rate of change of system reliability as a function of component reliability. In the present study essential improvements of this inequality are obtained under the assumption that n = number of components, and one or both of the following parameters are known: the 'length' of the system, i.e. the smallest number of components such that their functioning assures the functioning of the system, and the 'width' of the system, i.e. the smallest number of components such that if they fail the system must fail. The usefulness of these results is due to the fact that in practice n, length, and width are often known although the system is so complex that a study of some of its details for reliability purposes may be prohibitive. An immediate application of such inequalities is that when it is known that for component reliability P0 the system has attained reliability h(P sub 0) = P sub 0, then a lower bound can be given for the system reliability h(p) for all P > or = P sub 0. (Author)

/pdf/some-inequalities-for-reliability-functions-2y9bu9vepz.pdf

Some inequalities for reliability functions

Binary coherent system theory has played an important part in reliability. Its extension to (‘degradable’ or ‘multistate’ or) multinary systems has recently been considered in various papers, through various definitions. This paper lays the foundations of a unified theory for coherent systems by first giving unified arguments to apply and to investigate further binary and multinary systems. Monotone binary systems are introduced and examined by generalizing classic deterministic and probabilistic results. Applications of monotone coherence to the multinary case are proposed in a companion paper with a unified viewpoint on multinary coherent systems. As an indication, monotone constraints are defined with a partition of the component set and some total orderings imposed on the elements of the concerned partition. The discrete partition retrieves the classic theory of (free) binary coherent systems; some constraints defined from component levels lead to multinary coherent systems; some other constraints apply to systems submitted to some ‘common stresses’, e.g. the organizing system of a monotone coherent decomposition.

A unified theory for coherent systems in reliability I. A generalization for some classic theory: Monotone binary coherent systems

: Some known results concerning modules of coherent systems from reliability are given with new proofs. The relationship of coherent systems to blocking systems is pointed out, as well as an interpretation of modules for blocking systems.

http://www.dtic.mil/cgi-bin/GetTRDoc?AD=AD0693968&Location=U2&doc=GetTRDoc.pdf

Modules of coherent systems and their relationship to blocking systems

The reliability at time t of a system in sustained operation is often taken to be the probability that it functions continuously during the time interval [0, t]. The standard computation of system reliability finds the probability that the system functions at time t in terms of the probabilities that its components function at time t. This procedure is relevant only if the system, and its components, have lives (roughly speaking, a device has a life if it functions continuously until some time of failure, and remains failed thereafter). We show that if each component of a coherent system has a life, then the system has a life (again roughly, a system is coherent if its performance is not impaired by an improvement in the performance of its components). Our principal results are that, under reasonable conditions, converses are true. If a system has a life, only a very weak condition is needed to show that the system is coherent. With a somewhat stronger condition, we show that not only must the system be c...

System Structure and the Existence of a System Life

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System Structure and the Existence of a System Life