An optimum testing algorithm for some symmetric coherent systems

A symmetric coherent system (or k-out-of-n system) is a system composed of n components CO₁, CO₂ …, CO_n, each component existing in either a working or failing state. Such a system is in a working state if and only if k or more of its components are working, where 1 ? k ? n. It is assumed that the components can only be tested individually, and every test gives perfect information as to whether the tested component is working or failing. Let P_i, be the a priori probability that the component CO_i is working and C_i, be the cost of testing component CO_i. An optimal (minimum total expected cost) testing algorithm is an algorithm to determine the condition of a given symmetric coherent system by testing some of its components individually. In general, such an algorithm is a sequential process, that is, the next component to be tested is a function of the outcomes of the tests already applied. Every (optimal) testing algorithm corresponds to a (optimal) feasible testing policy which is basically a binary rooted tree with some component assigned to each node. In this paper an algorithm is presented for constructing an optimal feasible testing policy for symmetric coherent systems, where $\text{C}{}_{\mathrm{i}}\text{P}{}_{\mathrm{i}}\text{≠}\text{C}{}_{\mathrm{j}}\text{P}{}_{\mathrm{j}}$ and $\text{C}{}_{\mathrm{i}}\text{(1 ? P}{}_{\mathrm{i}}\text{)}\text{≠}\text{C}{}_{\mathrm{j}}\text{(1 ? P}{}_{\mathrm{j}}\text{)}$ whenever i ≠ j. This algorithm can be implemented as an optimal testing algorithm with polynomial complexity. Moreover, it is proven that any optimal testing algorithm corresponds to some feasible testing policy which can be generated by this algorithm.     

Finding coherent cyclic orders in strong digraphs

A cyclic order in the vertex set of a digraph is said to be coherent if any arc is contained in a directed cycle whose winding number is one. This notion plays a key role in the proof by Bessy and Thomassé (2004) of a conjecture of Gallai (1964) on covering the vertex set by directed cycles. This paper presents an efficient algorithm for finding a coherent cyclic order in a strongly connected digraph, based on a theorem of Knuth (1974). With the aid of ear decomposition, the algorithm runs in O(nm) time, where n is the number of vertices and m is the number of arcs. This is as fast as testing if a given cyclic order is coherent.     

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

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.     

Knuth’s Coherent Presentations of Plactic Monoids of Type A

We construct finite coherent presentations of plactic monoids of type A. Such coherent presentations express a system of generators and relations for the monoid extended in a coherent way to give a family of generators of the relations amongst the relations. Such extended presentations are used for representations of monoids, in particular, it is a way to describe actions of monoids on categories. Moreover, a coherent presentation provides the first step in the computation of a categorical cofibrant replacement of a monoid. Our construction is based on a rewriting method introduced by Squier that computes a coherent presentation from a convergent one. We compute a finite coherent presentation of a plactic monoid from its column presentation and we reduce it to a Tietze equivalent one having Knuth’s generators.     

Stochastic Properties of Components in a Used Coherent System

Some coherent systems are such that failure of the system does not mean that all components fail. This paper investigates the stochastic behavior and reliability properties of the residual lifetime of live components in coherent systems under the assumption that the system fails at time t. We also investigate the stochastic properties of inactivity time of failed components in coherent systems where failure of some components does not cause the failure of the complete system.     

失效单调关联系统中可存活元件的剩余寿命的混合表示

本文中研究的是由$n$个独立同分布元件构成的单调关联系统,当该单调关联系统失效时得到了系统中可存活元件的剩余寿命的可靠性函数的混合表示.基于signature的概念,对两个系统的剩余寿命进行了随机比较.     

多态单调关联系统的模型特性研究

本文引入向量水平函数和向量水平集的概念。这两个概念的意义在于：（１）利用向量水平集的性质推断单调系统的路集性质；（２）利用向量水平集的性质判断多态系统是否关联；（３）利用向量水平集的性质可心简化多态单调关联系统模型类的分析，本文先给出一般性结果，然后分析了二部件三态系统。     

Stochastic comparisons of residual lifetimes and inactivity times of coherent systems with dependent identically distributed components

In this paper we compare the residual lifetime of a used coherent system of age t>0$t>0$ with the lifetime of the similar coherent system made up of used components of age t. Here ‘similar’ means that the system has the same structure and the component lifetimes have the same dependence (joint reliability copula). Some comparison results are obtained for the likelihood ratio order, failure rate order, reversed failure rate order and the usual stochastic order. Similar results are reported for comparing inactivity time of a coherent system with lifetime of similar coherent system having component lifetimes same as inactivity times of failed components.     

Generalized oscillator and its coherent states

We construct a system (a generalized oscillator) that is similar to the oscillator and is related to a system of orthogonal polynomials on the real axis. We define coherent states in the Fock space associated with the generalized oscillator. In the example of the generalized oscillator related to the Gegenbauer polynomials, we prove the (super)completeness of these coherent states, i.e., we construct a measure determining a partition of unity. We present a formula that allows calculating the Mandel parameter for the constructed coherent states. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 3, pp. 363–380, December, 2007.     

New reliability bounds for coherent systems

In 1970, Esary and Proschan proposed simple formulae for the system reliability lower bound and system reliability upper bound. Their formulae of reliability bounds have been classic and have been incorporated into almost all recent textbooks on reliability. In this paper, we decompose a coherent system into several consecutive-k-out-of-n : F(G) systems, and then based upon their exact formulae for system reliabilities, we develop new formulae for both reliability lower bound and reliability upper bound for the coherent system. In addition, we show that the new proposed reliability bounds are superior to those of Esary and Proschan for all coherent systems when the minimal cut/path sets have elements in common. Numerical results are reported, compared and discussed for various systems.     

Measures of Component Importance in Nonrepairable and Repairable Multistate Strongly Coherent Systems

In Natvig and Gåsemyr (Methodol Comput Appl Probab 11:603–620, 2009) dynamic and stationary measures of importance of a component in a binary system were considered. To arrive at explicit results the performance processes of the components were assumed to be independent and the system to be coherent. Especially the Barlow–Proschan and the Natvig measures were treated in detail and a series of new results and approaches were given. For the case of components not undergoing repair it was shown that both measures are sensible. Reasonable measures of component importance for repairable systems represent a challenge. A basic idea here is also to take a so-called dual term into account. For a binary coherent system, according to the extended Barlow–Proschan measure a component is important if there are high probabilities both that its failure is the cause of system failure and that its repair is the cause of system repair. Even with this extension results for the stationary Barlow–Proschan measure are not satisfactory. For a binary coherent system, according to the extended Natvig measure a component is important if both by failing it strongly reduces the expected system uptime and by being repaired it strongly reduces the expected system downtime. With this extension the results for the stationary Natvig measure seem very sensible. In the present paper most of these results are generalized to multistate strongly coherent systems. For such systems little has been published until now on measures of component importance even in the nonrepairable case.     

Some results on information properties of coherent systems

This paper considers information properties of coherent systems when component lifetimes are independent and identically distributed. Some results on the entropy of coherent systems in terms of ordering properties of component distributions are proposed. Moreover, various sufficient conditions are given under which the entropy order among systems as well as the corresponding dual systems hold. Specifically, it is proved that under some conditions, the entropy order among component lifetimes is preserved under coherent system formations. The findings are based on system signatures as a useful measure from comparison purposes. Furthermore, some results on the system's entropy are derived when lifetimes of components are dependent and identically distributed. Several illustrative examples are also given.     

Ultrametric space of free coherent states

Free coherent states for a system with two degrees of freedom are defined. It is shown that for a set of coherent states corresponding to an eigenvalue of the annihilation operator, a 2-adic parameter naturally appears. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 2, pp. 334–336, February, 1997.     

Applications of average and projected systems to the study of coherent systems

In this paper, we introduce the concepts of average and projected systems associated to a coherent (parent) system. We analyze several aspects of these notions and show that they can be useful tools in studying the performance of coherent systems with non-exchangeable components. We show that the average and projected systems are especially useful in studying the tail behavior of reliability, hazard rate and mean residual life functions of the parent system and also in obtaining the tail best systems (under different criteria) by permuting the components at the system structure. Moreover, they can be useful in assessing how the asymmetry of the joint distribution of the component lifetimes (with respect to permutations of the components in the system structure) affects the system performance.     

一致风险度量和锥优化分析

一致风险理论的公理系统为风险分析建立了坚实的基础,然而它背后的数学却和凸优化理论思想密切相关,特别是对偶理论. 本文在有限维空间中,利用锥优化的对偶定理给出了一致风险度量的一般表达式的简单证明. 分析了可接受集的概念在一致风险度量中的中心作用,根据锥优化的对偶关系,探索了常用风险度量的性质. 尽管可接受集的大小能够表达风险控制的强弱,但是我们不知道如何定量地表示. 本文提出用相对熵控制风险度量松紧度的方法和意义. 另外,根据一致风险度量的灵活的结构,给出了无套利条件的一种放松,这一结果可用于不完全市场中的期权定价问题.     

Categorification, term rewriting and the Knuth-Bendix procedure

An axiomatization of a finitary, equational universal algebra by a convergent term rewrite system gives rise to a finite, coherent categorification of the algebra.     

On system reliability in stress–strength setup

In this paper we study stress–strength reliability for a general coherent system. The exact expression as well as bounds and approximations for system reliability are presented. We also illustrate the estimation procedure for exponential stress–strength distributions.     

Penultimate Approximations in Statistics of Extremes and Reliability of Large Coherent Systems

In reliability theory any coherent system can be represented as either a series-parallel or a parallel-series system. Its lifetime can thus be written as the minimum of maxima or the maximum of minima. For large-scale coherent systems it is sensible to assume that the number of system components goes to infinity. Then, the possible non-degenerate extreme value laws either for maxima or for minima are eligible candidates for the system reliability or at least for the finding of adequate lower and upper bounds for the reliability. The identification of the possible limit laws for the system reliability of homogeneous series-parallel (or parallel-series) systems has already been done under different frameworks. However, it is well-known that in most situations such non-degenerate limit laws are better approximated by an adequate penultimate distribution. Dealing with regular and homogeneous parallel-series systems, we assess both theoretically and through Monte-Carlo simulations the gain in accuracy when a penultimate approximation is used instead of the ultimate one.     

Orderings of coherent systems with randomized dependent components

Consider a general coherent system with independent or dependent components, and assume that the components are randomly chosen from two different stocks, with the components of the first stock having better reliability than the others. Then here we provide sufficient conditions on the component’s lifetimes and on the random numbers of components chosen from the two stocks in order to improve the reliability of the whole system according to different stochastic orders. We also discuss several examples in which such conditions are satisfied and an application to the study of the optimal random allocation of components in series and parallel systems. As a novelty, our study includes the case of coherent systems with dependent components by using basic mathematical tools (and copula theory).     

Reliability and expectation bounds for coherent systems with exchangeable components

Sharp upper and lower bounds are obtained for the reliability functions and the expectations of lifetimes of coherent systems based on dependent exchangeable absolutely continuous components with a given marginal distribution function, by use of the concept of Samaniego's signature. We first show that the distribution of any coherent system based on exchangeable components with absolutely continuous joint distribution is a convex combination of distributions of order statistics (equivalent to the k-out-of-n systems) with the weights identical with the values of the Samaniego signature of the system. This extends the Samaniego representation valid for the case of independent and identically distributed components. Combining the representation with optimal bounds on linear combinations of distribution functions of order statistics from dependent identically distributed samples, we derive the corresponding reliability and expectation bounds, dependent on the signature of the system and marginal distribution of dependent components. We also present the sequences of exchangeable absolutely continuous joint distributions of components which attain the bounds in limit. As an application, we obtain the reliability bounds for all the coherent systems with three and four exchangeable components, expressed in terms of the parent marginal reliability function and specify the respective expectation bounds for exchangeable exponential components, comparing them with the lifetime expectations of systems with independent and identically distributed exponential components.