Bounding the stochastic performance of continuum structure functions. I

1986 ◽  
Vol 23 (3) ◽  
pp. 660-669 ◽  
Author(s):  
Laurence A. Baxter ◽  
Chul Kim
Keyword(s):  
Structure Function ◽  
Random Variables ◽  
Structure Functions ◽  
Unit Interval ◽  
Minimal Path ◽  
Modular Decomposition ◽  
Associated Random Variables ◽  
Unit Hypercube ◽  
Continuum Structure

A continuum structure function γ is a non-decreasing mapping from the unit hypercube to the unit interval. Minimal path (cut) sets of upper (lower) simple continuum structure functions are introduced and are used to determine bounds on the distribution of γ (Χ) when X is a vector of associated random variables and when γ is right (left)-continuous. It is shown that, if γ admits of a modular decomposition, improved bounds may be obtained.

Bounding the stochastic performance of continuum structure functions. I

1986 ◽  
Vol 23 (03) ◽  
pp. 660-669 ◽  
Author(s):  
Laurence A. Baxter ◽  
Chul Kim
Keyword(s):  
Structure Function ◽  
Random Variables ◽  
Structure Functions ◽  
Unit Interval ◽  
Minimal Path ◽  
Modular Decomposition ◽  
Associated Random Variables ◽  
Unit Hypercube ◽  
Continuum Structure

A continuum structure function γ is a non-decreasing mapping from the unit hypercube to the unit interval. Minimal path (cut) sets of upper (lower) simple continuum structure functions are introduced and are used to determine bounds on the distribution of γ (Χ) when X is a vector of associated random variables and when γ is right (left)-continuous. It is shown that, if γ admits of a modular decomposition, improved bounds may be obtained.

Bounding the stochastic performance of continuum structure functions. II

1987 ◽  
Vol 24 (03) ◽  
pp. 609-618 ◽  
Author(s):  
Laurence A. Baxter ◽  
Chul Kim
Keyword(s):  
Structure Function ◽  
Random Variables ◽  
Structure Functions ◽  
Unit Interval ◽  
Modular Decomposition ◽  
Associated Random Variables ◽  
Unit Hypercube ◽  
Continuum Structure

A continuum structure function γ is a non-decreasing mapping from the unit hypercube to the unit interval. Block and Savits (1984) use the sets and to determine bounds on the distribution of γ (X) when X is a vector of associated random variables. It is shown that, if γ admits of a modular decomposition, improved bounds may be obtained.

Bounding the stochastic performance of continuum structure functions. II

1987 ◽  
Vol 24 (3) ◽  
pp. 609-618 ◽  
Author(s):  
Laurence A. Baxter ◽  
Chul Kim
Keyword(s):  
Structure Function ◽  
Random Variables ◽  
Structure Functions ◽  
Unit Interval ◽  
Modular Decomposition ◽  
Associated Random Variables ◽  
Unit Hypercube ◽  
Continuum Structure

A continuum structure function γ is a non-decreasing mapping from the unit hypercube to the unit interval. Block and Savits (1984) use the sets and to determine bounds on the distribution of γ (X) when X is a vector of associated random variables. It is shown that, if γ admits of a modular decomposition, improved bounds may be obtained.

Structure functions with finite minimal vector sets

1989 ◽  
Vol 26 (1) ◽  
pp. 196-201 ◽  
Author(s):  
Laurence A. Baxter ◽  
Seung Min Lee
Keyword(s):  
Structure Function ◽  
Random Variables ◽  
Structure Functions ◽  
Unit Interval ◽  
Independent Random Variables ◽  
Unit Hypercube ◽  
Continuum Structure ◽  
Minimal Vectors

A continuum structure function (CSF) y is a non-decreasing mapping from the unit hypercube to the unit interval. Define whereas γ (γ) < α for all y < x}, the set of minimal vectors to level α. This paper examines CSFs for which each Pα is finite. It is shown that if γ is such a CSF and X is a vector of independent random variables, the distribution of γ (X) is readily calculated. Further, if γ is an arbitrary right-continuous CSF, the distribution of γ (X) may be approximated arbitrarily closely by that of γ′(X) where γ′ is a right-continuous CSF for which each minimal vector set is finite.

Structure functions with finite minimal vector sets

1989 ◽  
Vol 26 (01) ◽  
pp. 196-201
Author(s):  
Laurence A. Baxter ◽  
Seung Min Lee
Keyword(s):  
Structure Function ◽  
Random Variables ◽  
Structure Functions ◽  
Unit Interval ◽  
Independent Random Variables ◽  
Unit Hypercube ◽  
Continuum Structure ◽  
Minimal Vectors

A continuum structure function (CSF) y is a non-decreasing mapping from the unit hypercube to the unit interval. Define whereas γ (γ) &lt; α for all y &lt; x}, the set of minimal vectors to level α. This paper examines CSFs for which each Pα is finite. It is shown that if γ is such a CSF and X is a vector of independent random variables, the distribution of γ (X) is readily calculated. Further, if γ is an arbitrary right-continuous CSF, the distribution of γ (X) may be approximated arbitrarily closely by that of γ′(X) where γ′ is a right-continuous CSF for which each minimal vector set is finite.

Reliability importance for continuum structure functions

1987 ◽  
Vol 24 (3) ◽  
pp. 779-785 ◽  
Author(s):  
Chul Kim ◽  
Laurence A. Baxter
Keyword(s):  
Structure Function ◽  
Structure Functions ◽  
Unit Interval ◽  
System Level ◽  
Unit Hypercube ◽  
Continuum Structure ◽  
Definition Of

A continuum structure function is a non-decreasing mapping from the unit hypercube to the unit interval. A definition of the reliability importance, ℛi(α) say, of component i at system level α(0 < α ≦ 1) is proposed. Some properties of this function are deduced, in particular conditions under which and conditions under which ℛi(α) is positive (0 < α < 1).

Reliability importance for continuum structure functions

1987 ◽  
Vol 24 (03) ◽  
pp. 779-785
Author(s):  
Chul Kim ◽  
Laurence A. Baxter
Keyword(s):  
Structure Function ◽  
Structure Functions ◽  
Unit Interval ◽  
System Level ◽  
Unit Hypercube ◽  
Continuum Structure ◽  
Definition Of

A continuum structure function is a non-decreasing mapping from the unit hypercube to the unit interval. A definition of the reliability importance, ℛi(α) say, of component i at system level α(0 &lt; α ≦ 1) is proposed. Some properties of this function are deduced, in particular conditions under which and conditions under which ℛi(α) is positive (0 &lt; α &lt; 1).

Further Properties of Reliability Importance for Continuum Structure Functions

1989 ◽  
Vol 3 (2) ◽  
pp. 237-246 ◽  
Author(s):  
Laurence A. Baxter ◽  
Seung Min Lee
Keyword(s):  
Continuous Function ◽  
Structure Function ◽  
Simple Algorithm ◽  
Structure Functions ◽  
Unit Interval ◽  
System Level ◽  
Unit Hypercube ◽  
Continuum Structure ◽  
Definition Of ◽  
Nondecreasing Mapping

A continuum structure function (CSF) is a nondecreasing mapping from the unit hypercube to the unit interval. The Kim-Baxter definition of the reliability importance of component i in a CSF at system level α, Ri(α), say, is reviewed. Conditions under which Ri(α) is positive, under which Ri(α) is a continuous function of α, and under which Ri(α) ≥ Rj(α) uniformly in α are presented. A simple algorithm for evaluating Ri(α) is described.

Continuum structures. II

1986 ◽  
Vol 99 (2) ◽  
pp. 331-338 ◽  
Author(s):  
Laurence A. Baxter
Keyword(s):  
Structure Function ◽  
Unit Interval ◽  
Continuum Structures ◽  
Unit Hypercube ◽  
Continuum Structure ◽  
The Subject ◽  
Nondecreasing Mapping

AbstractA continuum structure function is a nondecreasing mapping from the unit hypercube to the unit interval. This paper continues the author's work on the subject, extending Griffith's definitions of coherency to such functions and studying the analytic properties of a continuum structure function based on Natvig's ‘second suggestion’.

Continuum structures I

1984 ◽  
Vol 21 (04) ◽  
pp. 802-815 ◽  
Author(s):  
Laurence A. Baxter
Keyword(s):  
Structure Function ◽  
Functional Form ◽  
First Passage Time ◽  
Passage Time ◽  
Unit Interval ◽  
The State ◽  
Continuum Structures ◽  
Critical Elements ◽  
Closure Theorems ◽  
Continuum Structure

A generalisation of multistate coherent structures is proposed where the state of each component in a binary coherent structure can take any value in the unit interval, as can the structure function. The notions of duality, critical elements and strong coherency for such a structure are discussed and the functional form of the structure function is analysed. An expression is derived for the distribution function of the state of the system, given the distributions of the states of the components, and generalisations of the Moore–Shannon and IFRA and NBU closure theorems are proved. The states of the components are then permitted to vary with time and a first-passage-time distribution is discussed. A simple model for such a process, based on the concept of partial availability, is then proposed. Lastly, an alternative continuum structure function is introduced and discussed.

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