Parallel System Reliability Research Articles

Impact of copulas for modeling bivariate distributions on system reliability

A copula-based method is presented to investigate the impact of copulas for modeling bivariate distributions on system reliability under incomplete probability information. First, the copula theory for modeling bivariate distributions as well as the tail dependence of copulas are briefly introduced. Then, a general parallel system reliability problem is formulated. Thereafter, the system reliability bounds of the parallel systems are generalized in the copula framework. Finally, an illustrative example is presented to demonstrate the proposed method. The results indicate that the system probability of failure of a parallel system under incomplete probability information cannot be determined uniquely. The system probabilities of failure produced by different copulas differ considerably. Such a relative difference in the system probabilities of failure associated with different copulas increases greatly with decreasing component probability of failure. The maximum ratio of the system probabilities of failure for the other copulas to those for the Gaussian copula can happen at an intermediate correlation. The tail dependence of copulas has a significant influence on parallel system reliability. The copula approach provides new insight into the system reliability bounds in a general way. The Gaussian copula, commonly used to describe the dependence structure among variables in practice, produces only one of the many possible solutions of the system reliability and the calculated probability of failure may be severely biased.

On multinormal integrals by Importance Sampling for parallel system reliability

The ability to compute multinormal integrals to any required accuracy is a key issue for an efficient computation of failure probabilities, particular important in context with system reliability analysis. Hence in this paper, an accurate Importance Sampling procedure to compute multinormal integrals in high dimensions is presented. The novel method allows to sample exclusively in the failure domain which substantially increases the efficiency of the Importance Sampling procedure. The proposed approach is extended for slightly non-linear limit state functions which typically result from non-Gaussian distributed input variables and linear limit state functions of the response in linear structural analysis. The suggested procedure is easy to implement, accurate and convenient for practical applications.

Reliability Analysis of a Suspension Bridge Affected by Hydrogen Induced Cracking Based upon Response Surface Method

This paper deals with the Hydrogen Induced Cracking (HIC) of wires in a cable of a suspension bridge. The main objective is to compare the safety of a bridge system locally and globally. In a deterministic analysis stage, HIC in cable wires is calculated in order to evaluate the local safety of the main cable of suspension bridge. In the local analysis, a decoupling technique has been developed for the evaluation of crack propagations in a wire section driven by the hydrogen diffusion to a wire section, in terms of two 2-dimensional finite element models. One is for the fracture analysis in a longitudinal section, and the other is for a hydrogen diffusion model in a horizontal section. In a stochastic analysis stage, an ultimate limit state function for the cracking in cable wires is considered for the local safety of main cable. Using the ultimate limit state, the reliability of time-dependent and crack depth-dependent HIC of a cable wire has been calculated in component and parallel system reliability analysis. Globally, a serviceability limit state based on the global responses of a stiffening girder is also evaluated. Based on the observed difference of safeties between the global and local behaviors of the suspension bridge system, the proposed solutions are discussed.

Analytical derivation of the outcrossing rate in time-variant reliability problems

The usual approach to time-variant reliability problems is based on the computation of the outcrossing rate through the limit state surface and its time integration. The so-called PHI2 method allows one to compute the outcrossing rate by solving a two-component parallel system reliability problem using the First Order Reliability Method (FORM). Following this approach, the present paper provides new analytical expressions of the outcrossing rate and their implementation. The corresponding improvement of the PHI2 method in terms of accuracy is shown. The method is first validated using a simple time-variant reliability problem, for which an analytical expression of the associated outcrossing rate exists. Then, it is applied to evaluate the reliability of a corroded steel beam submitted to a midspan random load.

Damage Equivalent Method of Fatigue Reliability Analysis of Load-Sharing Parallel System

As for load-sharing parallel system like multi-engine system and wire cable, dependence-failure must occur due to load redistributing, so the component life distributions changed. After the analysis of the disadvantage of failure probability equivalent principle and the transformation of equivalent working time of different life distribution based on damage equivalent principle, the parallel system reliability model applying full probability formula is established. The established reliability model provides a new method for reliability analysis of load-sharing parallel system whose component life follows any distribution.

Time-Variant Reliability Analysis of Ship Structure Subjected to Fatigue and Corrosion

Based on the characteristics of loading condition of ship structure in service and the degeneration of the strength under fatigue and corrosion, the random time-variant model of ship hull section modulus was set up using the theory of random process. Then the time-variant reliability was analyzed with the method of up-crossing analysis and parallel system reliability. This method could not only present successive decrease of reliability of the ship structure under fatigue and corrosion, but also avoid the complicated numerical integration. It could be seen that this method was easy to achieve. Furthermore, the time-variant reliability was compared with the annual instantaneous reliability through an example, which indicated that this method was more precise, and could provide some references to the decision-making of the reliability, maintenance and safeguard.

On Bayesian Estimation in a Parallel System

This paper presents a Bayesian approach to the estimation of the reliability of a parallel system with k components. It is assumed that the strengths of the k components are subjected to a common stress which is independent of the strengths of the k components. The reliability of the parallel system or system reliability is estimated assuming that the strengths and the stress are exponentially distributed random variables. The performed Bayesian analysis is based on conjugated prior and non-informative prior distributions, respectively. A simulation data set is considered in order to study the performance of the Bayesian solutions. Some observations are discussed concerning alternative methods and possible extensions.

Analysis of approximations for multinormal integration in system reliability computation

In the context of first-order reliability analysis, the computation of multivariate normal integrals is a key step in the analysis of the system probability of failure. Several approximate methods for multinormal integration have been developed, since the direct numerical integration in large dimensions (30–50) is not feasible. The product of conditional marginal (PCM) method was proposed as a simple and effective method for system reliability computation. Although PCM is fairly accurate in computing parallel system reliability, it can result in a large overestimation of the failure probability of series systems with highly reliable elements. The paper presents an improved version, referred to as I-PCM, to eliminate this shortcoming of the original method. The I-PCM method employs a simple modification of bivariate integrals based on the additive law of probability. Detailed error analyses and numerical examples are presented in the paper to illustrate the improved accuracy and efficiency of the proposed I-PCM method.

Linear Stochastic Dynamical System under Uncertain Load: Inverse Reliability Analysis

The reliability of a linear dynamical system driven by a partially known Gaussian load process, specified only through its total average energy, is studied. A simple dynamic parallel and series system reliability network is investigated for the failure analysis using the crossing theory of stochastic processes. The critical input power spectral density of the load process which maximizes the mean crossing rate of a parallel or series system network emerges to be fairly narrow banded and hence fails to represent the erratic nature of the random input realistically. Consequently, a tradeoff curve between the maximum mean crossing rate of the reliability network and the disorder in the input, quantitatively measured through its entropy rate, is generated. Mathematically, the generation of the tradeoff curve of nondominated solutions, known as the Pareto optimal set, leads to a nonlinear, nonconvex, and multicriteria optimization problem with conflicting objectives. A recently developed Pareto optimization te...

Assessment of Reliability of Aging Reinforced Concrete Structures

Aggressive environment and in-service loading can cause the resistance degradation of existing reinforced concrete structures. They must be considered for the safety evaluation and service-life prediction of deteriorating structures. This paper presents an integrated approach for time-dependent system reliability analysis considering the stochastic nature of load processes, uncertainties in strength, degradation initiation time, and strength degradation mode. The approach takes into account the partial correlation among the failures of structural elements, which arises from the fact that elements of a structural system are subjected to the same load processes and/or depend on some common basic random variables. The approach is efficient, since it does not require simulation. Analysis results obtained by using the proposed approach indicate that the reliability of series systems is relatively insensitive to the correlation between the failure of structural elements. However, the reliability of parallel system is highly sensitive to the correlation. The use of the time-dependent reliability obtained by neglecting possible correlation as a safety measure, therefore, should be avoided.

Failure Time Prediction of Deteriorating Fail-Safe Structures

Structural evaluation and design have been dominated by component safety checks. In fact, current reliability-based codes for buildings and bridges are largely limited to individual structural component checks and do not generally include requirements for system reliability. Also, these codes do not consider the potential damaged states of a structural system and the ability of the damaged system to continue to carry loads. Furthermore, current codes do not entail the prediction of the failure time of deteriorating structural systems under time-varying loads. In this study, the reliability of deteriorating fail-safe structures is investigated using a time-variant parallel system reliability approach in which both load and resistance are time dependent. Monte Carlo simulation is used to find the cumulative-time system failure probability. Numerical examples are presented to illustrate the effects of various parameters such as variabilities in live loads, live load occurrence rate, live to dead load ratio, strength loss rate, resistance correlation, and number of degrading members on the cumulative-time failure probability of a deteriorating fail-safe system. Postfailure behavior of components and load redistribution effects are also considered. The results can be used to better predict the failure time of deteriorating fail-safe structures, such as redundant reinforced concrete bridges under corrosion attack, and to develop optimal lifetime reliability-based maintenance strategies for these structures.

ON THE RELIABILITY OF GAVER'S PARALLEL SYSTEM SUSTAINED BY A COLD STANDBY UNIT AND ATTENDED BY TWO REPAIRMEN

We analyse the reliability of Gaver's parallel system sustained by a cold standby unit and attended by two identical repairmen. The system satisfies the usual conditions (i.i.d. random variables, perfect repair, instantaneous and perfect switch, queueing). Each operative unit has a constant failure rate but a general repair time distribution. Our reliability analysis is based on a time dependent version of the supplementary variable method. We transform the basic equation into an integro-differential equation of the (mixed) Fredholm type. The equation generalizes Takacs' integro-differential equation. In order to present computational results, we outline the solution procedure for a repair time distribution with an arbitrary rational Laplace-Stieltjes transform. A particular numerical example displays the survivor function with the security interval that ensures a reliability level of at least 95%.

Estimation of system reliability from stress-strength relationship

In this paper, we estimate the reliability of parallel system with two components. We assume that strengths of these components follow a bi-variate exponential(BVE) distribution. These two componen...

Estimation of reliability in freund model for two component system

The simultaneous failure of a pair of components of a system is so rare or to say, impossible and the failure of one of the two components of the system may result in more (less) workload on the other component, Freund(1961) bivariate model is applicable to such cases. The uniformly minimum variance unbiased estimator (UMVUE) of the parallel system reliability for the Freund bivariate exponential distribution (BVED) is obtained.

Reliability‐Based Optimization of Series Systems of Parallel Systems

This study investigated whether already-developed techniques for design, including optimization of system where failure is related to single-component failure, can be extended to more complex systems. It is shown how equivalent elements for the parallel system can be formulated based on a joint beta-point formulation of the parallel system reliability estimation. Sensitivity tools for the optimization are described. Computational aspects are described, including a discussion of whether it is possible to neglect derivatives in the sensitivity analysis.

Stochastic analysis of a parallel system with common-cause failures and critical human errors

This paper presents reliability and availability analysis of a n-unit parallel system with common-cause failures and human errors. Laplace transforms of state probabilities, system availability, and reliability are presented. A formula to estimate system mean time to failure is developed. Special case expressions for steady-state probabilities and system availability are derived.

A classification system for reliability models : Tony Bendell. IEEE Trans. Reliab.R-33, 160 (1984)

This paper presents reliability and availability analysis of a n-unit parallel system with common-cause failures and human errors. Laplace transforms of state probabilities, system availability, and reliability are presented. A formula to estimate system mean time to failure is developed. Special case expressions for steady-state probabilities and system availability are derived.

Reliability Modelling of Non‐Maintained Parallel Systems with Multiple Failure Modes

This article presents four newly developed mathematical models representing non‐maintained parallel systems with hardware failures, common‐cause failures and human error. The Markov method was used to develop expressions for parallel system state probabilities, system reliability and mean time to failure (MTTF). System reliability and MTTF plots are shown. These plots clearly show the effect of common‐cause failures and human error on system reliability.

Bayesian shrinkage estimation of reliability in parallel system with exponential failure of the components

The paper presents a Bayesian shrinkage method of estimating the reliability of a parallel system with identical exponential distribution of failure times. Data is available from type II censored samples. Thompson ( J. Am. Statist. Ass. 63, 113–123, 1968) proposed the ordinary shrunken estimator ϰθ + (1- ϰ)θ 0, with ϰ a constant (0 ⩽ ϰ ⩽ 1) and θ 0 the prior or guess value of θ, which is more efficient than the unbiased estimator θ if θ is close to θ 0 and is less efficient otherwise. Here we postulate a prior distribution of θ around θ 0 and use an ordinary Bayes estimator of θ instead of θ 0 in the ordinary shrunken estimator. The Bayesian shrinkage estimator of parallel system reliability has thus been obtained and Monte Carlo study of its efficiency with respect to unbiased and maximum likelihood estimators (mles) was conducted. It was found that the proposed estimator is better.

``An Algorithm to Determine Parallel System Reliability''

``An Algorithm to Determine Parallel System Reliability''