一类多状态连续n中取k(G)系统的可靠性计算

为了计算多状态连续厅中取后（G）系统的可靠性,引入4个定理,将满足引理的多状态系统转换为二元状态系统。分别推导了多状态线形连续k／n（G）系统和环形连续k／n（G）系统的可靠性计算公式。证明了固定k值增加一个新部件,若部件可靠性独立同分布,线形和环形系统可靠性均增加;若部件可靠性独立但不同分布,环形系统存在一个极值,新增加部件可靠性大于这个极值时得到的新系统可靠性增加,反之系统可靠性下降。     

Reliability of 2-dimensional consecutive-k-out-of-n:F systems

The authors derive lower and upper reliability bounds for the two-dimensional consecutive k-out-of-n:F system (Salvia Lasher, 1990) with independent, but not necessarily identically distributed, components. A Weibull limit theory is proven for system time-to-failure for i.i.d. components     

Reliability of a connected-(r1, s1)-or-(r2, s2)-or-..-or-(rk, sk)-out-of-(m,n): F lattice system

A (linear or circular) connected-(r₁, s₁)-or-(r₂, s₂)-or-^{. .}-or-(r_k, s_k)-out-of-(m, n): F lattice system is the (linear or circular) (m, n)-lattice system if the system fails whenever all components in a connected-(r₁, s₁)-submatrix or all components in a connected-(r₂, s₂)-submatrix or ^{. .} or all components in a connected-(r_k, s_k)-submatrix fail. This paper presents a recursive algorithm for the reliability of the (linear or circular) connected-(r₁, s₁)-or-(r₂, s₂)or-^{. .}-or-(r_k,s_k)-out- of-(m, n):F lattice system. The recursive algorithm requires time and time in the linear case and the circular case, respectively Furthermore, we can reduce the more computing time in the statistically independent and identically distributed case or considering some special systems. Especially, the closed formula is given for the reliability of the linear connected-(2, 1)-or-(1, 2)-out-of-(m, 2): F lattice system in the statistically independent and identically distributed case.     

Failure Probability of Strict Consecutive-k-out-of-n:F Systems

A system with n components in sequence is a strict consecutive-k-out-of-n:F system if and only if it fails when at least k consecutive components are failed, but isolated strings of component failures of length less than k do not occur. This paper gives the failure probability function of a strict linear consecutive-k-out-of-n:F system in a closed form. The calculation of the failure probability of a strict circular consecutive-k-out-of-n:F system is reduced to the linear case.     

Efficient algorithms for k-out-of-n andconsecutive-weighted-k-out-of-n:F system

A new reliability model, consecutive-weighted-k-out-of-n:F system, is proposed and an O(n) algorithm is provided to evaluate its reliability. An O(n·min(n,k)) algorithm is also presented for the circular case of this model. The authors design an O(n) parallel algorithm using k processors to compute the reliability of k-out-of-n systems, that achieves linear speedup     

An Efficient Algorithm for Computing the Reliability of Consecutive-$k$-Out-Of-$n$:F Systems

Many algorithms for computing the reliability of linear or circular consecutive-k-out-of-n:F systems appeared in this Transactions. The best complexity estimate obtained for solving this problem is O(k³ log(n/k)) operations in the case of i.i.d. components. Using fast algorithms for computing a selected term of a linear recurrence with constant coefficients, we provide an algorithm having arithmetic complexity O(k log (k) log(log(k)) log(n)+k^omega) where 2相似文献   

Lifetime of Combined $k$-out-of-$n$ , and Consecutive $k_{c}$ -out-of-$n$ Systems

A combined k-out-of-n:F(G) & consecutive k_c -out-of-n :F(G) system fails (functions) iff at least k components fail (function), or at least fcc consecutive components fail (function). Explicit formulas are given for the lifetime distribution of these combined systems whenever the lifetimes of components are exchangeable, and have an absolutely continuous joint distribution. The lifetime distributions of the aforementioned systems are represented as a linear combination of distributions of order statistics by using the concept of Samaniego's signature. Formulas for the mean lifetimes are given. Some numerical results are also presented.     

Bounds for reliability of consecutivek-within-m-out-of-n:F systems

Upper and lower bounds for the reliability of a (linear or circular) consecutive k-within-m-out-of-n:F system with unequal component-failure probabilities are provided. Numerical calculations indicate that, for systems with components of good enough reliability, these bounds quite adequately estimate system reliability. The estimate is easy to calculate, having computational complexity O(m²×n). For identically distributed components, a Weibull limit theorem for system time-to-failure is proved     

Reliability evaluation of combined k-out-of-n:F,consecutive-k-out-of-n:F and linear connected-(r, s)-out-of-(m, n):Fsystem structures

Based on a real industrial application, three new system reliability models are proposed: combined k-out-of-n:F and consecutive-k _c-out-of-n:F system; combined k-out-of-m·n:F and linear connected-(r,s)-out-of-(m,n):F system; and combined k-out-of-m·n:F consecutive-k_c-out-of-n:F and linear connected-(r,s)-out-of-(m,n):F system. Reliability evaluation algorithms are provided for these models. The computation times of the algorithms for these models are, respectively: O(n·k), O(k·n·2 ^m·s^m-r+2), O(k·n·(2k_c )^m·s^m-r+1). The algorithms are used for system reliability evaluation of furnace systems. The concept of the combined k-out-of-n:F and 1-dimensional and 2-dimensional consecutive-k-out-of-n:F systems can be extended to other variations of the consecutive-k-out-of-n:F systems, e.g., the consecutive-k-out-of-n:G system and 1-dimensional and 2-dimensional r-within-k-out-of-n:F systems. The concept of Markov chain imbeddable (MIS) systems is another excellent tool that can be used for analysis of such combined system structures     

Reliability Analysis of the k-out-of-n:F System

A class of repairable systems known as k-out-of-n:F systems, 1 ? k ? n, consists of n units in parallel redundancy which are serviced by a single repairman; system failure occurs when k units are simultaneously inoperable for the first time. In this paper, assuming constant failure rates and general repair distributions, reliability characteristics of the k-out-of-n:F system are treated using two different methods. In Part I, a conditional transform approach is applied to the 2-out-of-n:F system. Transforms of distributions are obtained for T (the time to system failure), the time spent on repairs during (0, T) and the free time of the repairman during (0, T). In Part II, the supplementary variable technique is used to investigate time to failure characteristics of the k-out-of-n:F system for k = 2 and k = 3. A model of an airport limousine service illustrates the use of the results.     

An algorithm for computing the reliability of weighted-k-out-of-nsystems

This paper constructs a new k-out-of-n model, viz, a weighted-k-out-of-n system, which has n components, each with its own positive integer weight (total system weight=w), such that the system is good (failed) if the total weight of good (failed) components is at least k. The reliability of the weighted-k-out-of-n:G system is the complement of the unreliability of a weighted-(w-k+1)-out-of-n:F system. Without loss of generality, the authors discuss the weighted-k-out-of-n:G system only. The k-out-of-n:G system is a special case of the weighted-k-out-of-n:G system wherein the weight of each component is 1. An efficient algorithm is given to evaluate the reliability of the weighted-k-out-of-n:G system. The time complexity of this algorithm is O(n.k)     

The mean residual life function of a k-out-of-n structure at the system level

In the study of the reliability of technical systems, k-out-of-n systems play an important role. In the present paper, we consider a k-out-of-n system consisting of n identical components with independent lifetimes having a common distribution function F. Under the condition that, at time t, all the components of the system are working, we propose a new definition for the mean residual life (MRL) function of the system, and obtain several properties of that system.     

Ternary State Circular Sequential k-out-of-n Congestion System

A ternary state circular sequential k-out-of-n congestion (TSCSknC) system is presented. The system is an extension of the circular sequential k-out-of-n congestion (CSknC) system which consists of two connection states: a) congestion (server busy), and b) successful. In contrast, a TSCSknC system considers three connection states: i) congestion, ii) break down, and iii) successful. It finds applications in some reliable systems to prevent single-point failures, such as the ones used in (k,n) secret key sharing systems. The system further assumes that each of the n servers has known connection probabilities in congestion, break-down, and successful states. These n servers are arranged in a circle, and are made with connection attempts sequentially round after round. If a server is not congested, the connection can be either successful, or a failure. Previously connected servers are blocked from reconnecting if they were in either states ii), or iii). Congested servers are attempted repeatedly until k servers are connected successfully, or (n-k+1) servers have a break-down status. In other words, the system works when k servers are successfully connected, but fails when (n-k+1) servers are in the break-down state. In this paper, we present the recursive, and marginal formulas for the system successful probability, the system failure probability, as well as the average stop length, i.e. number of connections needed to terminate the system to a successful or failure state, and its computational complexity.     

Fast optimal diagnosis procedures for k-out-of-n:G systems

A k-out-of-n:G system consists of a set of components, where each component is either faulty or fault-free. The system is working if at least k components are fault-free. The problem of finding an optimal diagnosis procedure for a given k-out-of-n:G system has been considered in several research fields including medical diagnosis, redundant-system testing, and searching data-files. A polynomial-time algorithm for this problem was presented first by Salloum, and later by Salloum and Breuer, and independently by Ben-Dov. This paper implements the Salloum-Breuer-Ben-Dov algorithm, leading to an optimal diagnosis procedure that can determine the state of any given system in O(n·log(n)) time complexity and O(n) space complexity. The efficiency is achieved by using a generalized radix sorting procedure that uses a heap data structure. For some k-out-of-n:G systems, including those with equal testing costs for all components, the components along the leftmost and rightmost paths in the optimal diagnostic tree uniquely determine the other components in the tree. This property is used to devise a faster optimal diagnosis procedure than the one for the general k-out-of-n:G system. With regard to complexity, these procedures are the best solutions for the problem under consideration. This conjecture is supported by the fact that all these procedures require a sorting operation which has O(n·log(n)) as a lower bound on its time complexity     

Optimal arrangement of components in a consecutive k-out-of-r-from n:F system

A consecutive k-out-of-r-from n:F system is an ordered linear arrangement of n components that fails if and only if at least k in a “window” of r consecutive components fail. Suppose that all components are interchangeable and that component failures are s-independent. Component failure probabilities need not be equal. In this paper the ordering of the components in order to minimize the probability of system failure is examined. All values of k,r,n for which an optimal configuration can be determined, without knowledge of the component failure probabilities, are given.     

Performance evaluation of generalized multi-state k-out-of-n systems

The generalized multi-state k-out-of-n:G system model defined by Huang provides more flexibilities for modeling of multi-state systems. However, the performance evaluation algorithm they proposed for such systems is not efficient, and it is applicable only when the k/sub i/ values follow a monotonic pattern. In this paper, we defined the concept of generalized multi-state k-out-of-n:F systems. There is an equivalent generalized multi-state k-out-of-n:G system with respect to each generalized multi-state k-out-of-n:F system, and vice versa. The form of minimal cut vector for generalized multi-state k-out-of-n:F systems is presented. An efficient recursive algorithm based on minimal cut vectors is developed to evaluate the state distributions of a generalized multi-state k-out-of-n:F system. Thus, a generalized multi-state k-out-of-n:G system can first be transformed to the equivalent generalized multi-state k-out-of-n:F system, and then be evaluated using the proposed recursive algorithm. Numerical examples are given to illustrate the effectiveness and efficiencies of the proposed recursive algorithms.     

Optimal Consecutive-k-out-of-n:F Component Sequencing

A consecutive-k-out-of-n:F system is an ordered linear arrangement of n components that fails if and only if at least k consecutive components fail. Suppose that all components are interchangeable, that component failures are s-independent, and that component failure probabilities need not be equal. When k = 2, a certain ordering of the components minimizes the probability of system failure regardless of the particular component failure probabilities. This paper characterizes all other values of k and n for which such an optimal configuration can be determined without knowledge of the component failure probabilities.     

Reliability of consecutive-k-out-of-n:F systemswith nonidentical component reliabilities

A system with n components in sequence is a consecutive- k-out-of-n:F system if it fails whenever k consecutive components are failed. Under the supposition that component failures need not be independent and that component failure probabilities need not be equal, a topological formula is presented for the exact system reliability of linear and circular consecutive-k -out-of-n:F networks. The number of terms in the reliability formula is O(n⁴) in the linear case and O(n⁵) in the circular case     

Bounds for reliability of k-within connected-(r, s)-out-of-(m, n) failure systems

A k-within linear connected-(r, s)-out-of-(m, n) failure system is a two-dimensional grid whose components are ordered like the elements of an (m, n)-matrix. A k-within circular connected-(r, s)-out-of (m, n) failure system consists of the intersection points of m circles centered at the same point with n rays starting from that point and crossing the circles. The components of both systems either operate or fail. By definition, a k-within (linear or circular) connected-(r, s)-out-of-(m, n) failure system fails if at least one (r, s)-submatrix contains k or more failed components. These systems are used as mathematical models for design and operation of many engineering systems. For systems with statistically independent and identically distributed components, a lower and upper bound of system reliability are derived using improved Bonferroni inequalities. These bounds are easy to compute and provide good estimates for system reliability. New bounds for the reliability of other connected-(r, s)-out-of-(m, n) failure systems existing in the current literature are also obtained. Several failure systems with various values of the parameters k, r, s, m, n and p are used as numerical examples for comparison and illustrative purposes.     

Optimal Consecutive-2-out-of-n:F Component Sequencing

A consecutive-k-out-of-n:F system is an ordered linear arrangement of n components that fails if and only if at least k consecutive components fail. When the components are not necessarily equally likely to fail, the problem of interest is to assign components to positions in the system in a way that minimizes the probability of system failure. This paper shows that when k = 2 and component failures are s-independent, the optimal configuration can be determined without knowledge of the exact particular component-failure probabilities, but with knowledge of the component ranks (in terms of failure probability).