Consecutive Components Research Articles

Mean time-to-failure for a linear-consecutive-k-out-of-n:F system

The authors present new simple formulas for the mean time-to-failure of a liner consecutive-k-out-of-n:F system. The assumptions are: the components and the system either fail or operate; all component lifetimes are s-independent and identically distributed (i.i.d.); and the system fails if and only if at least k consecutive components fail. The special classes of Weibull and exponential life distribution are studied. >

A consecutive-k-out-of-n:G system: the mirror image of a consecutive-k-out-of-n:F system

A consecutive-k-out-of-n:F (consecutive-k-out-of-n:G) system consists of an ordered sequence of n components such that the system is failed (good) if and only if at least k consecutive components in the system are failed (good). In the present work, the relationship between the consecutive-k-out-of-n:F system and the consecutive-k-out-of-n:G system is studied, theorems for such systems are developed, and available results for one type of system are applied to the other. The topics include system reliability, reliability bounds, component reliability importance, and optimal system design. A case study illustrates reliability analysis and optimal design of a train operation system. An optimal configuration rule is suggested by use of the Birnbaum importance index.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Limit distribution for a consecuttve-k-out-of-n: F system

A consecutive-k-out-of-n: F system consists of n components ordered on a line. Each component, and the system as a whole, has two states: it is either functional or failed. The system will fail if and only if at least k consecutive components fail. The components are not necessarily equal and we assume that components' failures are stochastically independent. Using a result of Barbour and Eagleson (1984) we find a bound for the distance of the distribution of system's lifetime from the Weibull distribution. Subsequently, using this bound limit theorems are derived under quite general conditions.

Limit distribution for a consecuttve-k-out-of-n: F system

A consecutive-k-out-of-n: F system consists of n components ordered on a line. Each component, and the system as a whole, has two states: it is either functional or failed. The system will fail if and only if at least k consecutive components fail. The components are not necessarily equal and we assume that components' failures are stochastically independent. Using a result of Barbour and Eagleson (1984) we find a bound for the distance of the distribution of system's lifetime from the Weibull distribution. Subsequently, using this bound limit theorems are derived under quite general conditions.

Invariant permutations for consecutive k-out-of-n cycles: F. K. Hwang. IEEE Trans. Reliab.38(1), 65 (1989)

Purpose: Solve an optimal assignment problem Special math needed for explanations: Probability Special math nedded to use results: None Results useful to: Reliability theoreticians Abstract - Consecutive-k-out-of-n cycles have been pro- posed as topologies for k-loop computer networks and describe a circular system of n components where the system fails if and only if any k consecutive components all fail. Suppose that the com- ponents are interchangeable. Then the question arises as to which permutation maximizes the system reliability, assuming that the components have unequal reliabilities. If there exists an optimal permutation which depends on the ordering, but not the values, of the component reliabilities, then we call the system (and the permutation) invariant. The circular system is not invariant ex- cept fork = 1, 2, n - 2, n - 1, n.

Reliability of consecutive-k-out-of-n:F systems with 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(n/sup 4/) in the linear case and O(n/sup 5/) in the circular case.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Invariant permutations for consecutive k-out-of-n cycles

Consecutive-k-out-of-n cycles are proposed as topologies for k-loop computer networks and describe a circular system of n components where the system fails if and only if any k consecutive components all fail. Suppose that the components are interchangeable. The the question arises as to which permutation maximizes the system reliability, assuming that the components have unequal reliabilities. If there exists on optimal permutation which depends on the ordering, but not the values, of the component reliabilities, then the system (and the permutation) is called invariant. The circular system is found to be not invariant except for k=1, 2, n-2, n-1, and n. >

A weibull limit for the reliability of a consecutive k-within-m-out-of-n system

A consecutive k-within-m-out-of-n system consists of n identical and stochastically independent components arranged on a line. The system will fail if and only if within m consecutive components, there are at least k failures. Let Tn be the system's lifetime. Then, under quite general conditions we prove that there is a positive constant a such that the random variable n1/kaTn converges to a Weibull distribution as n →∞.

A weibull limit for the reliability of a consecutive k-within-m-out-of-n system

A consecutive k-within-m-out-of-n system consists of n identical and stochastically independent components arranged on a line. The system will fail if and only if within m consecutive components, there are at least k failures. Let Tn be the system's lifetime. Then, under quite general conditions we prove that there is a positive constant a such that the random variable n1/kaTn converges to a Weibull distribution as n →∞.

A direct algorithm for computing reliability of a consecutive-k cycle

A consecutive-k cycle is a circular system such that the system fails if and only if any i consecutive components all fail. Reliabilities for consecutive-k cycles are usually computed by recursive equations. However, most recursive equations proposed so far for the cycle involve reliabilities for consecutive-k lines, requiring two passes where the first pass computes only the line reliabilities. A recursive equation involving cycles only is proposed that is simpler in form but much harder to understand on intuitive grounds. Another advantage is that the proposed cycle recursion has the same form as a line recursion previously proposed. Thus a uniform treatment of lines and cycles is possible. This uniform approach is used to obtain some explicit solutions of both line and cycle reliabilities for 2<or=k<or=4.<<ETX>>

Relayed consecutive-k-out-of-n:F lines

A consecutive-k-out-of-n:F line is a system of components in a sequence such that the system fails if and only if k consecutive components all fail. Relayed systems often quoted as examples of such systems differ from the definition by the fact that the first component must work to initiate the relay (in some cases the last component also must work). Such systems are differentiated from ordinary consecutive-k-out-of-n:F lines by adding the word 'relayed'. It is shown that the main properties of the reliabilities of consecutive-k-out-of-n:F lines are preserved under this modification.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

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.

A limit theorem for the reliability of a consecutive-k-out-of-n system

A consecutive-k-out-of-n system consists of n identical and linearly ordered components. The system will fail if and only if at least k consecutive components fail. Let Tn be the system&amp;amp;s lifetime. Then, under very general conditions we prove that there is a positive constant a, so that the distribution of the random variable n (1/ka) Tn converges to a Weibull distribution, as n→∞

A limit theorem for the reliability of a consecutive-k-out-of-n system

A consecutive-k-out-of-n system consists of n identical and linearly ordered components. The system will fail if and only if at least k consecutive components fail. Let Tn be the system&amp;s lifetime. Then, under very general conditions we prove that there is a positive constant a, so that the distribution of the random variable n(1/ka)Tn converges to a Weibull distribution, as n→∞

Subcortical correlates of the P300 potential complex in man to auditory stimuli

Bipolar and referential responses correlated to the surface P300 auditory potential complex (components N2, P3 and N4) were studied in different subcortical structures of parkinsonian and epileptic patients with implanted electrodes used as an electrophysiological procedure for surgical treatment. 1. (1) Bipolar and referential subcortical responses were recorded from a wide subcortical region which included subthalamo-thalamic (Sth-Th), dorsothalamic (DTh), orbifronto-hippocampal (OFH) and striatal (STR) structures. They were formed by 3 consecutive components C, D and E correlated to components N2,P and N4 of the surface responses. Component D correlated to surface P3 was either absent or very small in all subcortical bipolar responses and in referential responses recorded from OFH structures. 2. (2) Polarity of bipolar component C was inverted between DTh and OFH while that of bipolar component E was inverted between Sth-Th and DTh, between OFH and STR and between DTh-Th and STR. Polarity of the referential component C was inverted between STh-Th and DTh and between OFH and STR while that of referential components D and E remained unchanged along all subcortical structures. 3. (3) Referential components C-E and D attained maximal amplitudes and minimal latencies at the subthalamic and at the medial geniculate thalamic nucleus regions respectively. From here, their amplitudes decreased and latencies increased with distance in other structures rostrally and caudally located, with the exception of those at the orbitofrontal (components C-E) and dorsal thalamic and striatal regions (component D) where no amplitude-latency gradients were observed.

Geometric invariants for liaison of space curves

Let k be an algebraically closed field and let S= k[X,, X,, X2, X3]. For a curve Cc Pi, the Hartshorne-Rao module M(C) = BneiE H’(P3, Xc(n)) is a graded S-module of finite length and, up to duals and shifts, is a complete invariant of liaison (cf. [ 12 1). Given a graded S-module M = @,, z M, of finite length, the action of S, = P’(P’, 6( 1)) between two consecutive components (i.e., Ql,,: S, + Hom(M,, M, + ,)) gives rise to a degeneracy locus, which can be thought of as lying in PSI = (P’)*. Namely, let I’,,,= P’(LES, Irk4,,(L)<r) and let V, = V,,s where s = max(r I V,., 5 ( P3)* ) (or else V, = 0). These loci are isomo~hism invariants and are preserved under duals and shifts. For M = N(C) these loci are related to the geometry of C itself. The main philosophy that emerges (Section 2) is that the degeneracy locus generally corresponds to those planes H in P3 which meet C nongenerically, either containing a component of C or having Cn N impose an unusually small number of conditions on some plane curves on H. We conclude Section 2 by applying these ideas to derive some necessary conditions for M(C) to have components in negative degrees. The remaining sections give various apphcations of the techniques introduced in Section 2. First, in Section 3 we classify sets of skew lines up to liaison. That is, if C and C’ consist of t 2 3 and t’ skew lines, respectively, then we ask when C can be linked to C’. The answer in general is “never,” but the situation changes signi~cantly if C lies on a quadric surface (cf. Theorem 3.1). A large part of the answer can be read from the degeneracy locus V,.

Subcortical correlates of the somatic, auditory and visual vertex activities. II. Referential EEG responses

A systematic analysis of polarity, amplitude and latency of the referential EEG responses correlated to surface somatic (SVA), auditory (AVA) and visual (VVA) vertex activities (VA) was done in various subcortical structures of non-specific system of the same group of patients with electrodes implanted and reported in a previous work. These structures included the caudal and rostral mesencephalic reticular formations (cttc and rttc), the subthalamic region (Sth), the centro-median (Ce), parafascicular (Pf), ventrolateral (VL), dorsolateral (DL) and reticularis (Rt) thalamic nuclei; the orbito-frontal cortex (GO), the anterior commissure (Acm), the pallidum medialis and lateralis (Pm and P1), the putamen (Pt) and the amygdala (Am). Subcortical SVA, AVA and VVA were formed by 4 consecutive components O, A, B and C correlated to components P1, N1, P2 and N2 of the surface VA. Components A and B of the subcortical SVA reversed polarity 4 times when recorded in structures arranged in a caudo-rostral order: between rttc-Ce and VL, between Rt and GO, between GO and Acm and between Acm and Am. Components A and b of subcortical AVA and VVA reversed polarity two times each: AVA between cttc and rttc and between GO and Acm; while VVA between GO and Acm and between Acm and Pm. Components A and B of SVA and VVA attained maximal amplitudes and minimal latencies at GO. From here, their amplitude decreased and latency increased with distance along other structures rostrally and caudally located. Components A and B of AVA attained maximal amplitudes and minimal latencies at cttc. From here, their amplitude decreased and latency increased with distance along other structures rostrally located; except those at the Acm, where amplitudes increased and latencies decreased irrespective to distance. Components A and B of responses recorded at Sth, Pf, DL and Pl-Pt were similar to those recorded at rttc, Ce, Rt and Pm, respectively, and showed similar polarity shifts and amplitude-latency gradients.

Subcortical correlates of the somatic, auditory and visual vertex activities in man. I. Bipolar EEG responses and electrical stimulation

Bipolar responses correlated to somatic, auditory and visual vertex activities were studied in different subcortical structures of patients with implanted electrodes used as an electrophysiological procedure for surgical treatment. In addition, we studied the subjective responses of patients to electrical stimulation of subcortical structures where vertex-like activities (VA) were recorded, in order to determine their possible role in somatic, auditory and visual sensations. 1. (1) Typical subcortical VA evoked by somatic, auditory and visual stimuli, were recorded together from a single structure of a non-specific system including: reticulo-, medial dorsal-thalamic, orbito-frontal, limbic and striatal structures. Typical subcortical VA were formed by 4 consecutive components: O, A, B and C correlated to P1, N1, P2 and N2 of the surface VA. Polarity of component A was inverted between reticulo- and medial-thalamic structures while that of component B was inverted between medial thalamic and dorsal thalamic and between fronto-limbic and striatal structures. 2. (2) Typical subcortical VA were absent wiithin specific thalamic nuclei and pathways in presence or absence of short latency responses evoked by specific sensory stimuli. 3. (3) Atypical subcortical VA of 1, 2 and 3 sensory modalities with inconsistent amplitudes, latencies and polarities were recorded from an intermediate zone surrounding and between specific and non-specific structures. 4. (4) Electrical stimulation of the non-specific structures at the reticulo- and medial thalamic levels elicited sensation of falling down and increase tension and tremor of the contralateral hand while no subjective responses were elicited by the stimulation at the dorsal thalamic, fronto-limbic and striatal levels. Stimulation of the specific structures and the intermediate zone elicited somatic, auditory and visual sensations with and without and apparent topographical organization, respectively.

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.

Consecutive K-Out-of-N Failure Systems: Reliability, Availability, Component Importance, and Multistate Extensions

SYNOPTIC ABSTRACTCoherent systems which fail if and only if at least k consecutive components fail are studied, when the components are independent and identically distributed. An expression for the reliability of such systems is derived by examining the waiting time for a run of a certain length using Markov chain techniques. Approximations for systems with large numbers of components are obtained, and measures of component importance are studied. Extensions are made to availability and multistate models.