Extreme Value Metamodeling for System Reliability With Time-Dependent Functions

2015 ◽  
Author(s):  
Zhifu Zhu ◽  
Xiaoping Du
Keyword(s):  
Computational Efficiency ◽  
System Reliability ◽  
Extreme Values ◽  
Random Variables ◽  
Probability Of Failure ◽  
High Accuracy ◽  
Extreme Value ◽  
Time Dependent ◽  
System Failure ◽  
Kriging Method

The reliability of a system is usually measured by the probability that the system performs its intended function in a given period of time. Estimating such reliability is a challenging task when the probability of failure is rare and the responses are nonlinear and time variant. The evaluation of the system reliability defined in a period of time requires the extreme values of the responses in the predefined period of time during which the system is supposed to function. This work builds surrogate models for the extreme values of responses with the Kriging method. For the sake of computational efficiency, the method creates Kriging models with high accuracy only in the region that has high contributions to the system failure; training points of random variables and time are sampled simultaneously so that their interactions could be considered automatically. The example of a mechanism system shows the effectiveness of the proposed method.

Extreme values of phase-type and mixed random variables with parallel-processing examples

1999 ◽  
Vol 36 (01) ◽  
pp. 194-210 ◽  
Author(s):  
Sungyeol Kang ◽  
Richard F. Serfozo
Keyword(s):  
Parallel Processing ◽  
Asymptotic Distribution ◽  
Extreme Values ◽  
Random Variables ◽  
Extreme Value Distribution ◽  
Rates Of Convergence ◽  
Value Theory ◽  
Extreme Value ◽  
Independent Random Variables ◽  
Phase Type

A basic issue in extreme value theory is the characterization of the asymptotic distribution of the maximum of a number of random variables as the number tends to infinity. We address this issue in several settings. For independent identically distributed random variables where the distribution is a mixture, we show that the convergence of their maxima is determined by one of the distributions in the mixture that has a dominant tail. We use this result to characterize the asymptotic distribution of maxima associated with mixtures of convolutions of Erlang distributions and of normal distributions. Normalizing constants and bounds on the rates of convergence are also established. The next result is that the distribution of the maxima of independent random variables with phase type distributions converges to the Gumbel extreme-value distribution. These results are applied to describe completion times for jobs consisting of the parallel-processing of tasks represented by Markovian PERT networks or task-graphs. In these contexts, which arise in manufacturing and computer systems, the job completion time is the maximum of the task times and the number of tasks is fairly large. We also consider maxima of dependent random variables for which distributions are selected by an ergodic random environment process that may depend on the variables. We show under certain conditions that their distributions may converge to one of the three classical extreme-value distributions. This applies to parallel-processing where the subtasks are selected by a Markov chain.

Time-Dependent System Reliability Analysis Using Random Field Discretization

2015 ◽  
Vol 137 (10) ◽  
Author(s):  
Zhen Hu ◽  
Sankaran Mahadevan
Keyword(s):  
Random Field ◽  
Reliability Analysis ◽  
System Reliability ◽  
Probability Of Failure ◽  
Two Dimensions ◽  
Time Dependent ◽  
Time Interval ◽  
System Reliability Analysis ◽  
Reliability Method ◽  
Time Dependent System

This paper proposes a novel and efficient methodology for time-dependent system reliability analysis of systems with multiple limit-state functions of random variables, stochastic processes, and time. Since there are correlations and variations between components and over time, the overall system is formulated as a random field with two dimensions: component index and time. To overcome the difficulties in modeling the two-dimensional random field, an equivalent Gaussian random field is constructed based on the probability equivalency between the two random fields. The first-order reliability method (FORM) is employed to obtain important features of the equivalent random field. By generating samples from the equivalent random field, the time-dependent system reliability is estimated from Boolean functions defined according to the system topology. Using one system reliability analysis, the proposed method can get not only the entire time-dependent system probability of failure curve up to a time interval of interest but also two other important outputs, namely, the time-dependent probability of failure of individual components and dominant failure sequences. Three examples featuring series, parallel, and combined systems are used to demonstrate the effectiveness of the proposed method.

scholarly journals Extreme value distributions for dependent jointly ln,p-symmetrically distributed random variables

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
K. Müller ◽  
W.-D. Richter
Keyword(s):  
Extreme Values ◽  
Random Variables ◽  
Geometric Approach ◽  
Extreme Value ◽  
Symmetric Distribution ◽  
Tail Behavior ◽  
Extreme Value Distributions ◽  
Symmetric Distributions ◽  
Dependent Random Variables ◽  
Elliptically Contoured

AbstractA measure-of-cone representation of skewed continuous ln,p-symmetric distributions, n ∈ N, p > 0, is proved following the geometric approach known for elliptically contoured distributions. On this basis, distributions of extreme values of n dependent random variables are derived if the latter follow a joint continuous ln,p-symmetric distribution. Light, heavy, and extremely far tails as well as tail indices are discussed, and new parameters of multivariate tail behavior are introduced.

Extreme values of phase-type and mixed random variables with parallel-processing examples

1999 ◽  
Vol 36 (1) ◽  
pp. 194-210 ◽  
Author(s):  
Sungyeol Kang ◽  
Richard F. Serfozo
Keyword(s):  
Parallel Processing ◽  
Asymptotic Distribution ◽  
Extreme Values ◽  
Random Variables ◽  
Extreme Value Distribution ◽  
Rates Of Convergence ◽  
Value Theory ◽  
Extreme Value ◽  
Independent Random Variables ◽  
Phase Type

A basic issue in extreme value theory is the characterization of the asymptotic distribution of the maximum of a number of random variables as the number tends to infinity. We address this issue in several settings. For independent identically distributed random variables where the distribution is a mixture, we show that the convergence of their maxima is determined by one of the distributions in the mixture that has a dominant tail. We use this result to characterize the asymptotic distribution of maxima associated with mixtures of convolutions of Erlang distributions and of normal distributions. Normalizing constants and bounds on the rates of convergence are also established. The next result is that the distribution of the maxima of independent random variables with phase type distributions converges to the Gumbel extreme-value distribution. These results are applied to describe completion times for jobs consisting of the parallel-processing of tasks represented by Markovian PERT networks or task-graphs. In these contexts, which arise in manufacturing and computer systems, the job completion time is the maximum of the task times and the number of tasks is fairly large. We also consider maxima of dependent random variables for which distributions are selected by an ergodic random environment process that may depend on the variables. We show under certain conditions that their distributions may converge to one of the three classical extreme-value distributions. This applies to parallel-processing where the subtasks are selected by a Markov chain.

scholarly journals Resilience Assessment Based on Time-Dependent System Reliability Analysis

2016 ◽  
Vol 138 (11) ◽  
Author(s):  
Zhen Hu ◽  
Sankaran Mahadevan
Keyword(s):  
Reliability Analysis ◽  
System Reliability ◽  
Assessment Method ◽  
Time Dependent ◽  
Design Stage ◽  
System Failure ◽  
System Reliability Analysis ◽  
System Resilience ◽  
Resilience Assessment ◽  
Time Dependent System

Significant efforts have been recently devoted to the qualitative and quantitative evaluation of resilience in engineering systems. Current resilience evaluation methods, however, have mainly focused on business supply chains and civil infrastructure, and need to be extended for application in engineering design. A new resilience metric is proposed in this paper for the design of mechanical systems to bridge this gap, by investigating the effects of recovery activity and system failure paths on system resilience. The defined resilience metric is connected to design through time-dependent system reliability analysis. This connection enables us to design a system for a specific resilience target in the design stage. Since computationally expensive computer simulations are usually used in design, a surrogate modeling method is developed to efficiently perform time-dependent system reliability analysis. Based on the time-dependent system reliability analysis, dominant system failure paths are enumerated and then the system resilience is estimated. The connection between the proposed resilience assessment method and design is explored through sensitivity analysis and component importance measure (CIM). Two numerical examples are used to illustrate the effectiveness of the proposed resilience assessment method.

scholarly journals GLOBAL FLUCTUATIONS IN PHYSICAL SYSTEMS: A SUBTLE INTERPLAY BETWEEN SUM AND EXTREME VALUE STATISTICS

2008 ◽  
Vol 22 (20) ◽  
pp. 3311-3368 ◽  
Author(s):  
MAXIME CLUSEL ◽  
ERIC BERTIN
Keyword(s):  
Extreme Values ◽  
Random Variables ◽  
Extreme Value ◽  
Physical Models ◽  
Extreme Value Statistics ◽  
Specific Class ◽  
Qualitative Properties ◽  
Rigorous Results ◽  
General Mapping ◽  
Global Fluctuations

Fluctuations of global additive quantities, like total energy or magnetization for instance, can in principle be described by statistics of sums of (possibly correlated) random variables. Yet, it turns out that extreme values (the largest value among a set of random variables) may also play a role in the statistics of global quantities, in a direct or indirect way. This review discusses different connections that may appear between problems of sums and of extreme values of random variables, and emphasizes physical situations in which such connections are relevant. Along this line of thought, standard convergence theorems for sums and extreme values of independent and identically distributed random variables are recalled, and some rigorous results as well as more heuristic reasonings are presented for correlated or non-identically distributed random variables. More specifically, the role of extreme values within sums of broadly distributed variables is addressed, and a general mapping between extreme values and sums is presented, allowing us to identify a class of correlated random variables whose sum follows (generalized) extreme value distributions. Possible applications of this specific class of random variables are illustrated on the example of two simple physical models. A few extensions to other related classes of random variables sharing similar qualitative properties are also briefly discussed, in connection with the so-called BHP distribution.

A Radial-Based Centralized Kriging Method for System Reliability Assessment

2018 ◽  
Vol 140 (7) ◽  
Author(s):  
Yao Wang ◽  
Dongpao Hong ◽  
Xiaodong Ma ◽  
Hairui Zhang
Keyword(s):  
System Reliability ◽  
High Efficiency ◽  
Reliability Assessment ◽  
System Failure ◽  
Kriging Method ◽  
Limit States ◽  
Density Region ◽  
High Probability Density ◽  
Failure Region ◽  
Computationally Intensive

System reliability assessment is a challenging task when using computationally intensive models. In this work, a radial-based centralized Kriging method (RCKM) is proposed for achieving high efficiency and accuracy. The method contains two components: Kriging-based system most probable point (MPP) search and radial-based centralized sampling. The former searches for the system MPP by progressively updating Kriging models regardless of the nonlinearity of the performance functions. The latter refines the Kriging models with the training points (TPs) collected from pregenerated samples. It concentrates the sampling in the important high-probability density region. Both components utilize a composite criterion to identify the critical Kriging models for system failure. The final Kriging models are sufficiently accurate only at those sections of the limit states that bound the system failure region. Its efficiency and accuracy are demonstrated via application to three examples.

System Optimization Design Under Time Variant Reliability Constraints

2011 ◽  
Author(s):  
Xiao-Ling Zhang ◽  
Hong-Zhong Huang ◽  
Zhong-Lai Wang ◽  
Pei-Nan Ge
Keyword(s):  
Probability Theory ◽  
Random Variables ◽  
System Optimization ◽  
Probability Of Failure ◽  
Time Dependent ◽  
Design Parameters ◽  
Total Probability ◽  
Engineering Design Problems ◽  
Reliability Method ◽  
Time Invariant

Due to the degradation, input loading and uncertainty in the design parameters usually involve random variables and random processes, reliability analysis for engineering design problems are usually time dependent. Many problems related to degradation have been treated as monotonic or statistically independent, therefore, the probability of failure only at the end of the lifetime of the structure are considered. To the issues of parameters with stochastic process, the outcrossing rate methods have been extensively developed to calculate the upper bound of time-dependent reliability. In these methods, the issue of proper choice of time interval is crucial and difficult. In this paper, a new method for time dependent reliability optimization based on the total probability theory and universal generating function is proposed. In the proposed method, firstly, Parameters with stochastic processes are discretized into some discrete random variables. Secondly, the discrete parameters are reformed into a new random process by the operation of the universal generating functions. Finally, based on the total probability theory, the probability of failure for each limit state function is analyzed using sequential optimization and time invariant reliability assessment method. Only the time invariant reliability method is needed in the proposed method, by conditioning the continuous random variables on the discrete random parameters. Numerical example is presented to demonstrate the performance of the proposed method.

Time-Dependent System Reliability Analysis for Bivariate Responses

2017 ◽  
Vol 4 (3) ◽  
Author(s):  
Zhen Hu ◽  
Zhifu Zhu ◽  
Xiaoping Du
Keyword(s):  
Stochastic Processes ◽  
Reliability Analysis ◽  
System Reliability ◽  
Random Variables ◽  
General System ◽  
Time Dependent ◽  
Time Duration ◽  
Reliability Method ◽  
Bivariate Responses ◽  
Time Dependent System

Time-dependent system reliability is computed as the probability that the responses of a system do not exceed prescribed failure thresholds over a time duration of interest. In this work, an efficient time-dependent reliability analysis method is proposed for systems with bivariate responses which are general functions of random variables and stochastic processes. Analytical expressions are derived first for the single and joint upcrossing rates based on the first-order reliability method (FORM). Time-dependent system failure probability is then estimated with the computed single and joint upcrossing rates. The method can efficiently and accurately estimate different types of upcrossing rates for the systems with bivariate responses when FORM is applicable. In addition, the developed method is applicable to general problems with random variables, stationary, and nonstationary stochastic processes. As the general system reliability can be approximated with the results from reliability analyses for individual responses and bivariate responses, the proposed method can be extended to reliability analysis of general systems with more than two responses. Three examples, including a parallel system, a series system, and a hydrokinetic turbine blade application, are used to demonstrate the effectiveness of the proposed method.

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