Computing minimal signature of coherent systems through matrix-geometric distributions

2021 ◽  
Vol 58 (3) ◽  
pp. 621-636
Author(s):  
Serkan Eryilmaz ◽  
Fatih Tank
Keyword(s):  
Geometric Distribution ◽  
Probability Generating Function ◽  
Random Variable ◽  
Tail Probability ◽  
System Structure ◽  
Coherent System ◽  
Challenging Problem ◽  
Coherent Systems ◽  
Binary Trials ◽  
Geometric Random Variable

AbstractSignatures are useful in analyzing and evaluating coherent systems. However, their computation is a challenging problem, especially for complex coherent structures. In most cases the reliability of a binary coherent system can be linked to a tail probability associated with a properly defined waiting time random variable in a sequence of binary trials. In this paper we present a method for computing the minimal signature of a binary coherent system. Our method is based on matrix-geometric distributions. First, a proper matrix-geometric random variable corresponding to the system structure is found. Second, its probability generating function is obtained. Finally, the companion representation for the distribution of matrix-geometric distribution is used to obtain a matrix-based expression for the minimal signature of the coherent system. The results are also extended to a system with two types of components.

Some Reliability and Other Properties of Beta-Transformed Random Variables

2018 ◽  
Vol 33 (2) ◽  
pp. 83-92
Author(s):  
M. Sreehari ◽  
E. Sandhya ◽  
V. K. Mohamed Akbar
Keyword(s):  
Power Series ◽  
Geometric Distribution ◽  
Random Variables ◽  
Random Variable ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Power Series Distribution ◽  
Geometric Random Variable ◽  
Necessary And Sufficient

Abstract The reliability properties of beta-transformed random variables are discussed. A necessary and sufficient condition for a beta-transformed geometric random variable to follow a power series distribution is derived. It is shown that a beta-transformed member of the Katz family does not belong to the Katz family unless it is a geometric distribution, thereby getting a characterization.

scholarly journals The number of failed components in a coherent working system when the lifetimes are discretely distributed

Metrika ◽  
2021 ◽  
Author(s):  
Krzysztof Jasiński
Keyword(s):  
Random Variables ◽  
Continuous Case ◽  
Coherent System ◽  
Coherent Systems ◽  
Discrete Random Variables ◽  
Independent And Identically Distributed

AbstractIn this paper, we study the number of failed components of a coherent system. We consider the case when the component lifetimes are discrete random variables that may be dependent and non-identically distributed. Firstly, we compute the probability that there are exactly i, $$i=0,\ldots ,n-k,$$ i = 0 , … , n - k , failures in a k-out-of-n system under the condition that it is operating at time t. Next, we extend this result to other coherent systems. In addition, we show that, in the most popular model of independent and identically distributed component lifetimes, the obtained probability corresponds to the respective one derived in the continuous case and existing in the literature.

On the equivalence of systems of different sizes, with applications to system comparisons

2016 ◽  
Vol 48 (2) ◽  
pp. 332-348 ◽  
Author(s):  
Bo H. Lindqvist ◽  
Francisco J. Samaniego ◽  
Arne B. Huseby
Keyword(s):  
Optimization Problem ◽  
Coherent System ◽  
Sufficient Condition ◽  
Equivalent System ◽  
Coherent Systems ◽  
Two Systems ◽  
Equivalent Systems ◽  
Engineered Systems ◽  
Signature Vector ◽  
Mixed Systems

Abstract The signature of a coherent system is a useful tool in the study and comparison of lifetimes of engineered systems. In order to compare two systems of different sizes with respect to their signatures, the smaller system needs to be represented by an equivalent system of the same size as the larger system. In the paper we show how to construct equivalent systems by adding irrelevant components to the smaller system. This leads to simpler proofs of some current key results, and throws new light on the interpretation of mixed systems. We also present a sufficient condition for equivalence of systems of different sizes when restricting to coherent systems. In cases where for a given system there is no equivalent system of smaller size, we characterize the class of lower-sized systems with a signature vector which stochastically dominates the signature of the larger system. This setup is applied to an optimization problem in reliability economics.

scholarly journals On relative ageing of coherent systems with dependent identically distributed components

2020 ◽  
Vol 52 (1) ◽  
pp. 348-376
Author(s):  
Nil Kamal Hazra ◽  
Neeraj Misra
Keyword(s):  
Sufficient Conditions ◽  
System Level ◽  
Hazard Rates ◽  
Coherent System ◽  
Coherent Systems ◽  
Component Level ◽  
Distributed Components ◽  
Two Systems

AbstractRelative ageing describes how one system ages with respect to another. The ageing faster orders are used to compare the relative ageing of two systems. Here, we study ageing faster orders in the hazard and reversed hazard rates. We provide some sufficient conditions for one coherent system to dominate another with respect to ageing faster orders. Further, we investigate whether the active redundancy at the component level is more effective than that at the system level with respect to ageing faster orders, for a coherent system. Furthermore, a used coherent system and a coherent system made out of used components are compared with respect to ageing faster orders.

Moments for multidimensional Mandelbrot cascades

2016 ◽  
Vol 53 (1) ◽  
pp. 1-21
Author(s):  
Chunmao Huang
Keyword(s):  
Random Walk ◽  
Random Matrices ◽  
Random Variable ◽  
Tail Probability ◽  
Decay Rates ◽  
Branching Random Walk ◽  
Sufficient Condition ◽  
Random Vectors ◽  
Products Of Random Matrices ◽  
Sums Of Products

Abstract We consider the distributional equation Z =D ∑k=1NAkZ(k), where N is a random variable taking value in N0 = {0, 1, . . .}, A1, A2, . . . are p x p nonnegative random matrices, and Z, Z(1), Z(2), . . ., are independent and identically distributed random vectors in R+p with R+ = [0, ∞), which are independent of (N, A1, A2, . . .). Let {Yn} be the multidimensional Mandelbrot martingale defined as sums of products of random matrices indexed by nodes of a Galton–Watson tree plus an appropriate vector. Its limit Y is a solution of the equation above. For α > 1, we show a sufficient condition for E|Y|α ∈ (0, ∞). Then for a nondegenerate solution Z of the distributional equation above, we show the decay rates of Ee-t∙Z as |t| → ∞ and those of the tail probability P(y ∙ Z ≤ x) as x → 0 for given y = (y1, . . ., yp) ∈ R+p, and the existence of the harmonic moments of y ∙ Z. As an application, these results concerning the moments (of positive and negative orders) of Y are applied to a special multitype branching random walk. Moreover, for the case where all the vectors and matrices of the equation above are complex, a sufficient condition for the Lα convergence and the αth-moment of the Mandelbrot martingale {Yn} are also established.

scholarly journals Three Methods to Calculate the Probability of Ruin

1989 ◽  
Vol 19 (1) ◽  
pp. 71-90 ◽  
Author(s):  
François Dufresne ◽  
Hans U. Gerber
Keyword(s):  
Stationary Distribution ◽  
Geometric Distribution ◽  
Random Variable ◽  
Claim Amount ◽  
Adjustment Coefficient ◽  
Exponential Distributions ◽  
Probability Of Ruin ◽  
The Third ◽  
Compound Geometric Distribution ◽  
Aggregate Loss

AbstractThe first method, essentially due to GOOVAERTS and DE VYLDER, uses the connection between the probability of ruin and the maximal aggregate loss random variable, and the fact that the latter has a compound geometric distribution. For the second method, the claim amount distribution is supposed to be a combination of exponential or translated exponential distributions. Then the probability of ruin can be calculated in a transparent fashion; the main problem is to determine the nontrivial roots of the equation that defines the adjustment coefficient. For the third method one observes that the probability, of ruin is related to the stationary distribution of a certain associated process. Thus it can be determined by a single simulation of the latter. For the second and third methods the assumption of only proper (positive) claims is not needed.

scholarly journals Component importance in coherent systems with exchangeable components

2015 ◽  
Vol 52 (3) ◽  
pp. 851-863 ◽  
Author(s):  
Serkan Eryilmaz
Keyword(s):  
Importance Measure ◽  
Coherent System ◽  
Coherent Systems ◽  
System A ◽  
Component Importance

This paper is concerned with the Birnbaum importance measure of a component in a binary coherent system. A representation for the Birnbaum importance of a component is obtained when the system consists of exchangeable dependent components. The results are closely related to the concept of the signature of a coherent system. Some examples are presented to illustrate the results.

ON RELATIVE AGING COMPARISONS OF COHERENT SYSTEMS WITH IDENTICALLY DISTRIBUTED COMPONENTS

2020 ◽  
pp. 1-15 ◽  
Author(s):  
Nil Kamal Hazra ◽  
Neeraj Misra
Keyword(s):  
Hazard Rate ◽  
Sufficient Conditions ◽  
Stochastic Orders ◽  
Coherent System ◽  
Coherent Systems ◽  
Cumulative Hazard ◽  
Numerical Examples ◽  
Distributed Components ◽  
Rate Functions ◽  
Important Notion

The relative aging is an important notion which is useful to measure how a system ages relative to another one. Among the existing stochastic orders, there are two important orders describing the relative aging of two systems, namely, aging faster orders in the cumulative hazard and the cumulative reversed hazard rate functions. In this paper, we give some sufficient conditions under which one coherent system ages faster than another one with respect to the aforementioned stochastic orders. Further, we show that the proposed sufficient conditions are satisfied for k-out-of-n systems. Moreover, some numerical examples are given to illustrate the applications of proposed results.

Pseudo-modular decompositions and ‘refined bounds’ for the interval reliability and the availability for binary coherent systems

1989 ◽  
Vol 21 (01) ◽  
pp. 91-122
Author(s):  
N. Mazars
Keyword(s):  
Traditional Approach ◽  
Applied Mathematics ◽  
Divide And Conquer ◽  
Second Step ◽  
Coherent System ◽  
Coherent Systems ◽  
Complete Extension ◽  
Interval Reliability

‘Divide and conquer’ is a traditional approach in various fields of applied mathematics. In reliability, only modules have been proposed to decompose complex coherent systems. However, a system may include no modules, except ‘trivial’ ones. This paper is the second step of a study concerned with module generalizations : in the first step, pseudo-modules have been retained as the most general coherent subsystems which can yield a complete extension of the fundamental results concerning modules; in addition, it has been proved that they concern any binary coherent system; in this paper, it is shown that pseudo-modular decompositions are the most general coherent decompositions which can yield a complete extension of all the ‘refined bounds’ for the interval reliability and the availability currently proposed in terms of modular decompositions. This study also yields some fundamental results to extend all the ‘refined bounds’ currently proposed for multistate coherent systems, using an easier approach.

scholarly journals NON-EMPTINESS OF MODULI SPACES OF COHERENT SYSTEMS

2008 ◽  
Vol 19 (07) ◽  
pp. 777-799 ◽  
Author(s):  
L. BRAMBILA-PAZ
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Algebraic Curve ◽  
Coherent System ◽  
Coherent Systems ◽  
Generic Element

Let X be a general smooth projective algebraic curve of genus g ≥ 2 over ℂ. We prove that the moduli space G(α:n,d,k) of α-stable coherent systems of type (n,d,k) over X is empty if k > n and the Brill–Noether number β := β(n,d,n + 1) = β(1,d,n + 1) = g - (n + 1)(n - d + g) < 0. Moreover, if 0 ≤ β < g or β = g, n ∤g and for some α > 0, G(α : n,d,k) ≠ ∅ then G(α : n,d,k) ≠ ∅ for all α > 0 and G(α : n,d,k) = G(α′ : n,d,k) for all α,α′ > 0 and the generic element is generated. In particular, G(α : n,d,n + 1) ≠ ∅ if 0 ≤ β ≤ g and α > 0. Moreover, if β > 0 G(α : n,d,n + 1) is smooth and irreducible of dimension β(1,d,n + 1). We define a dual span of a generically generated coherent system. We assume d < g + n1≤ g + n2and prove that for all α > 0, G(α : n1,d, n1+ n2) ≠ ∅ if and only if G(α : n2,d, n1+ n2) ≠ ∅. For g = 2, we describe G(α : 2,d,k) for k > n.

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