Weibull Failure Research Articles

Cost benefit analysis of a two dissimilar unit cold standby system with Weibull failure and repair laws

This paper deals with cost benefit analysis of a single server two non-identical unit cold standby system with two modes of each unit: Normal and Total failure. One unit gets priority in operation as well as in repair over the other unit. The switching device to put the standby unit into operation is always perfect and instantaneous. Failure and repair time distributions of each unit are taken as Weibull with common shape parameter and different scale parameters. The system is observed at suitable regenerative epochs to obtain various measures of system effectiveness of interest to system designers and operation managers.

RELIABILITY AND AVAILABILITY ANALYSIS OF MECHANICAL SYSTEMS USING STOCHASTIC PETRI NET MODELING BASED ON DECOMPOSITION APPROACH

Reliability and availability assessment of a complex system in a single model, considering binary states of individual components, using Markov technique is difficult due to state space explosion problem. Inclusion of various types of dependencies in the model further aggravates it. To overcome this, Stochastic Petri net (SPN) modeling based on decomposition is proposed for reliability and availability assessment of mechanical systems. The decomposition is based on three aspects of the system: Hierarchical level, basic structure and dependency. The model is demonstrated at three hierarchical levels. Individual component (level '3') SPN model is developed assuming Weibull failure distribution, while the individual subsystem (level '2') SPN model is developed considering the arrangement of components within the subsystem. The individual model at level '2' is reduced to an equivalent single net model and its equivalent transition rate is derived from its basic structure assuming the independence of components. This along with the dependencies (e.g. repair, standby redundancy) are included in the system model (level '1'). The repair distribution in this model is assumed exponential. Reachable markings are generated for the system model to obtain the reduced state space of Semi-Markov model for assessment of reliability and availability of the system. The steps of the proposed methodology are suggested and these are illustrated for a pumping system, with two pumps; one in standby.

Bimodal Weibull Distribution of Metal/High-$\kappa$ Gate Stack TDDB—Insights by Scanning Tunneling Microscopy

We provide new insights, via nanoscale TDDB testing, into the bimodal Weibull failure distribution obtained from area scaling of high-κ (HK) gate stack. Time-to-breakdown (BD) statistics for grain boundary (GB) and grain in a polycrystalline HK gate stack are obtained individually from localized constant voltage stressing via a scanning tunneling microscope. In spite of an initial difference in the preexisting defect density, no apparent difference in the Weibull slope is observed for the two sets of BD statistics. The bimodal Weibull distribution is shown to be a combined effect: 1) The steep Weibull slope of the lower percentile, arising from large-area devices, is related to BD at GBs, and 2) the upper percentile, arising from small-area devices, is mostly related to grain BDs. In this case, the Weibull slope is reduced by a small fraction of these devices exhibiting early failures due to GB BDs. We show directly that structural defects in an HK dielectric, particularly GBs, play an important role on its BD distribution.

Bayesian Estimation of Two‐Parameter Weibull Distribution Using Extension of Jeffreys′ Prior Information with Three Loss Functions

The Weibull distribution has been observed as one of the most useful distribution, for modelling and analysing lifetime data in engineering, biology, and others. Studies have been done vigorously in the literature to determine the best method in estimating its parameters. Recently, much attention has been given to the Bayesian estimation approach for parameters estimation which is in contention with other estimation methods. In this paper, we examine the performance of maximum likelihood estimator and Bayesian estimator using extension of Jeffreys prior information with three loss functions, namely, the linear exponential loss, general entropy loss, and the square error loss function for estimating the two‐parameter Weibull failure time distribution. These methods are compared using mean square error through simulation study with varying sample sizes. The results show that Bayesian estimator using extension of Jeffreys′ prior under linear exponential loss function in most cases gives the smallest mean square error and absolute bias for both the scale parameter α and the shape parameter β for the given values of extension of Jeffreys′ prior.

A GLM approach to step-stress accelerated life testing with interval censoring

In this paper, we present a statistical inference procedure for the step-stress accelerated life testing (SSALT) model with Weibull failure time distribution and interval censoring via the formulation of generalized linear model (GLM). The likelihood function of an interval censored SSALT is in general too complicated to obtain analytical results. However, by transforming the failure time to an exponential distribution and using a binomial random variable for failure counts occurred in inspection intervals, a GLM formulation with a complementary log–log link function can be constructed. The estimations of the regression coefficients used for the Weibull scale parameter are obtained through the iterative weighted least square (IWLS) method, and the shape parameter is updated by a direct maximum likelihood (ML) estimation. The confidence intervals for these parameters are estimated through bootstrapping. The application of the proposed GLM approach is demonstrated by an industrial example.

Evaluations of intrinsic time dependent dielectric breakdown of dielectric copper diffusion barriers

Study of the intrinsic time dependent dielectric breakdown (TDDB) of dielectric copper diffusion barriers was realized using a unique planar capacitor (Pcap) test structure. The test vehicle has several advantages over the metal dot method. The most important one is the elimination of field enhancement effect which could be more than 50% in the case of metal dots. In order to test the effectiveness of this test vehicle, two different dielectric barriers, one conventional SiCN film and the other a BN film still under development, were selected for TDDB evaluations. Significant differences were observed between the two films in TDDB lifetime, the shape parameter β of the Weibull failure time distributions, and the characteristics of the current versus time curves. The BN film had a degradation of more than four orders of magnitude in TDDB lifetime than SiCN. The Weibull shape parameter β of SiCN was 8.4, among the highest values that have been reported for any dielectric barrier films, versus 1.4 for the boron nitride film. Furthermore, the SiCN film showed a decrease in current versus time curves during the initial stage of the stressing, a trend which is typically related to charge trapping in defect-free films. This was not the case for the BN film where the leakage currents were much higher. The results demonstrate that the Pcap test structure is an effective vehicle to evaluate the intrinsic reliability of dielectric barrier films.

A diagnostic approach to Weibull‐Weibull stress‐strength model and its generalization

PurposeThe objective of the paper is to consider the problem of the strength of a manufactured item against stress, when the component follows Weibull failure law. Different cases of stress and strength with varying parameters are discussed for the Weibull‐Weibull stress‐strength model considered in this paper. The application of the proposed technique will help in understanding the design methodology of the system and addressing the risks involved in perceived quality and reliability levels by eliminating or at least reducing the risk impact at the design phase.Design/methodology/approachGeneralised Weibull‐Weibull stress‐strength models have been analysed for different cases of shape parameters for stress and strength to estimate the reliability of the system. The model is generalized using semi‐regenerative stochastic processes with the help of a state space approach to include a repair facility.FindingsDifferent cases of stress and strength with varying parameters have been discussed for the Weibull‐Weibull stress‐strength models considered in this paper. The results show how the stress‐strength reliability model is affected by changes in the parameters of stress and strength. The application of the proposed technique will help in understanding the design methodology of the system, and also lead to the problem of addressing the risks involved in perceived quality and reliability levels by eliminating or at least reducing the risk impact in the design phase.Research limitations/implicationsThe present study is limited to a few special cases of Weibull‐Weibull stress‐strength models. The authors propose to continue to study the behaviour of general Weibull strength against exponential stress in particular and to identify the shape parameter that maximises the strength reliability.Practical implicationsThe application of the proposed technique will help in understanding the design methodology of the system, and also lead to the problem of addressing the risks involved in perceived quality and reliability levels by eliminating or at least reducing the risk impact at the design phase. The model has been extended and generalized to include a repair facility under the assumption that all the random variables involved in the analysis are arbitrarily distributed (i.e. general).Originality/valueIn the Weibull‐Weibull stress‐strength model of reliability, different cases have been considered. In the first case, both parameters of stress‐strength have the same values and are independent of the distribution. In the second case, if the shape parameter of the strength is twice that of the stress, the probability will have a normal distribution with different parameter values. In the third case, if the shape parameter of the stress is twice that of the strength, then probability distribution is a parabolic cylindrical function. The study shows how to proceed in all cases. The model is generalized to include a repair facility, with all the random variables involved in the analysis being arbitrarily distributed using semi‐regenerative stochastic processes.

Economic statistical design for x-bar control charts under non-normal distributed data with Weibull in-control time

This paper proposes an approach which simultaneously considers the properties of cost and quality based on the Burr distribution to determine three parameters (including sample size, sampling interval between successive samples, and the control limits) when an x-bar chart monitors a manufacturing process with Weibull failure characteristic and non-normal data. Also, the cost model of Banerjee and Rahim (1988) is used as the objective function, and the probability density function of the Burr distribution is applied to derive the statistical constraints of economic statistical design of the x-bar control charts for non-normal data. The example of Banerjee and Rahim (1988) is adopted to indicate the solution procedure and sensitivity analyses. Meanwhile, the design parameters of the x-bar control charts can be obtained through the grid search method. The results show that an increase of skewness coefficient (α3) results in a slight decrease for sample size (n), but is robust to the control limit width (L). Also, an increase of kurtosis coefficient (α4) leads to a wider control limit width.

Gate oxide reliability at the nanoscale evaluated by combining conductive atomic force microscopy and constant voltage stress

The gate oxide integrity of different thin films (silicon dioxide, silicon nitride, and hafnium oxide) was analyzed by constant voltage stress (CVS) at the nanoscale using conductive atomic force microscopy (cAFM) with the probe tip directly in contact to the dielectric layer. The results were evaluated assuming a Weibull failure distribution for the dielectrics under voltage stress, and a good fit was obtained for the measurement data. This indicates that CVS measurements at the nanoscale can be applied for inline characterization of as-deposited dielectrics without the need for gate electrodes. In addition, time-to-breakdown extracted from the CVS using cAFM was compared to data retrieved from CVS measurements using conventional current-voltage measurements on samples with gate electrodes. In particular, area scaling of CVS data over eight orders of magnitude using cAFM was performed for the first time. The evaluation indicates a decent match between macroscopic and nanoscale CVS measurements for the hafnium oxide and silicon dioxide samples. In contrast, significant discrepancies are evident for the silicon nitride samples, which may be related to charge trapping effects.

Strong Dependence of Mechanical Properties on Fiber Diameter for Polymer−Nanotube Composite Fibers: Differentiating Defect from Orientation Effects

We have prepared polyvinylalcohol-SWNT fibers with diameters from ∼1 to 15 μm by coagulation spinning. When normalized to nanotube volume fraction, V(f), both fiber modulus, Y, and strength, σ(B), scale strongly with fiber diameter, D: Y/V(f) ∝ D(-1.55) and σ(B)/V(f) ∝ D(-1.75). We show that much of this dependence is attributable to correlation between V(f) and D due to details of the spinning process: V(f) ∝ D(0.93). However, by carrying out Weibull failure analysis and measuring the orientation distribution of the nanotubes, we show that the rest of the diameter dependence is due to a combination of defect and orientation effects. For a given nanotube volume fraction, the fiber strength scales as σ(B) ∝ D(-0.29)D(-0.64), with the first and second terms representing the defect and orientation contributions, respectively. The orientation term is present and dominates for fibers of diameter between 4 and 50 μm. By preparing fibers with low diameter (1-2 μm), we have obtained mean mechanical properties as high as Y = 244 GPa and σ(B) = 2.9 GPa.

Guaranteeing Service Availability in SLAs; a Study of the Risk Associated with Contract Period and Failure Process

Service Level Agreements (SLAs) are a common means to define the obligations of network/service providers and users in business relationships. The terms that define the guaranteed availability for a given period are an important element of these contracts. The appropriate selection values is difficult due to the large number of variables involved, the complexities of the network and service provision and the computational challenge posed by the transient solution, as opposed to a steady state, that is needed. A common policy taken to solve it, is using the steady state availability as a reference. Nevertheless this simplification may put on risk the contract fulfillment as stochastic variation of the measured availability is significant over a typical contract period. This paper analyzes the relevance that the interval availability analysis has on SLAs, and provides suggestions to the network providers on the selection of adequate availability guarantees. The interval availability of unprotected and shared protected connections is studied under exponential and Weibull failure and repair distributions. It is observed that for a single path scenario, a small reduction of the guaranteed availability below the steady state value improve the probability to meet the requirements considerably. The same is the case for connections with shared backup protection. However performing this analysis in the transient domain is quite demanding. Hence, to simplify it, it is proposed to obtain the steady state results and introduce a safeguard factor to control that the availability guarantee is meet. For the Weibull distributed times between failures, where the shape factor is less than one, as observed in operational networks, the the probability of meeting a guaranteed availability over a finite contract period, decrease more radically than for the commonly assumed Poisson failure process. This increases the importance of making a transient analysis.

Thermal cycling of flip chips on FR‐4 and PI substrates with parylene C coating

PurposeThe purpose of this paper is to study the effect of conformal coating on the thermal cycling reliability of anisotropically conductive adhesive film (ACF) joined flip chip components on FR‐4 and polyimide (PI) substrates.Design/methodology/approachTest chips were joined using flip chip technology and an anisotropically conductive adhesive. The conformal coating used was parylene C and it was applied using the vapour deposition polymerisation method. The reliability of ACF joined flip chip components on FR‐4 and PI substrates was evaluated using −40/+85°C thermal cycling testing. Test lots with and without parylene C coating were studied. Additionally, one test lot with initial moisture inside the coating layer and a PI substrate was subjected to the test. The reliability results were analyzed using Weibull analysis and failure analysis was performed to study the failure mechanisms using cross sectioning and optical and scanning electron microscopy.FindingsThe results show a clear difference between the FR‐4 and PI substrate materials. PI substrate material proved to be reliable enough to withstand the thermal cycling testing. Two different occurrences of the first failures are seen and analyzed with FR‐4 substrates. The conformal coating layer did not seem to impair the reliability. Parylene C coating proved to be a reliable choice to protect, and even improve, the thermal cycling reliability of flip chip devices.Originality/valueUsually, conformal coatings are studied in humidity tests. However, it is also vital to know whether the conformal coatings affect the reliability in thermal cycling and there is a lack of reliability studies in this area. This paper gives reliability data for conformal coating users about the influence of thermal cycling.

Some testimators for the scale parameter and reliability function of Weibull lifetime data

The Weibull failure model is considered in this paper. Some testimators are proposed of the scale parameter (β) and of the reliability function ( R( t)) using life-testing data when a prior value about the unknown parameter (β) is available. The expressions for the bias, mean squared error and relative efficiency are obtained and compared with existing estimators.

The finite-time-period preventive maintenance policies with failure rate reduction under a warranty consideration

In practice, a system or equipment has a finite useful life. When an aged system is replaced by a new system, the new system seldom has exactly the same characteristics, functions, investment cost, maintenance expenses, etc., as those of the aged system. However, the literature has shown that most researchers are devoted to studying maintenance problems in an infinite time span, which assumes that the replaced system in each replacement cycle (or period) has the same conditions and costs. Apparently, the assumption might not be practical, since the useful operating life of most systems is finite in the real world. Therefore, the purpose of this research is to develop an optimal preventive maintenance (PM) policy having failure rate reduction within a finite time span under a warranty consideration by minimizing the expected total maintenance cost. The PM cost is assumed to be a linear function of each PM effect. Two cases are studied and compared; one is with no warranty provided and the other is with a given warranty period. For the second case, we examine two policies: (1) no PM within the warranty period is assumed and (2) the PM will be performed within the warranty period. In this article, the algorithm for searching the optimal solution is presented. Examples with Weibull failure cases are given and the sensitivity analysis of the optimal solution is also provided.

Glass Strengthening Via High-Intensity Plasma-Arc Heating

A high-intensity plasma-arc lamp was used to irradiate the surface of soda–lime silicate glass tiles to determine whether an increase in strength could be achieved. The lamp has a maximum power density of 3500 W/cm2, a processing area of 2 cm × 10 cm, a broad-spectrum energy distribution between 0.2 and 1.4 μm, and was controlled to unidirectionally sweep across 50 mm2 tiles at a constant speed of 8 mm/s. Ring-on-ring (RoR) equibiaxial flexure and four-point unidirectional flexure testings of entire tiles were used to measure and compare uncensored and unimodal Weibull failure stress distributions of treated and untreated glass. Even under nonoptimized processing conditions, the RoR Weibull characteristic failure stress increased by approximately 25% and the four-point-bend Weibull characteristic failure stress increased by approximately 65%. Additionally, the failure stresses of the weakest samples of the heat-treated glass were significantly higher than those of the untreated glass. Strengthening was due to a fire-polishing-like mechanism. The arc-lamp heat treatment caused the location of the strength-limiting flaws in the four-point-bend tiles to change; namely, failure initiation occurred on the gauge section surface for the treated glass, whereas it occurred at a gauge section edge for the untreated glass. Arc-lamp heat treatment is attractive not only because it provides strengthening, but because it can (noncontact) process large amounts of glass quickly and inexpensively, and is a process that either a glass manufacturer or an end-user can readily use.

Microstructure and Reliability Comparison of Different Pb-Free Alloys Used for Wave Soldering and Rework

This paper describes the results of an intensive microstructural and reliability study of pin-through-hole (PTH) and surface mount technology (SMT) components which were wave solder assembled using three groups of alloys: (1) near-eutectic Sn-Ag-Cu alloys such as SAC405 and SAC305, (2) low-Ag off-eutectic Pb-free alloys with an Ag content of about 1% and lower, and (3) eutectic Sn-Cu alloys with Ni and other additives. Both primary attach and reworked solder connections using solder fountain and hand rework were studied. The PTH connector types and SMT components were wave solder assembled on a test vehicle. Accelerated thermal cycling (ATC) was conducted at 0°C to 100°C for 6000 cycles. The difference in microstructures, intermetallic formation, Cu dissolution, grain coarsening, and crack formation is shown. The influence of the microstructure after assembly and rework on Weibull plot parameters and failure modes is described for 2512 resistors. Interconnect defects such as nonuniform phase distribution and void formation are discussed. The Sn-Cu-Ni- and Sn-Cu-Ag-Bi-based alloys tested in this study are recommended as potential suitable replacements for SAC305/405 in the wave solder process; no failure was detected up to 6000 cycles at 0°C to 100°C. Although SAC405 demonstrated better barrel fill and lower rate of crack propagation during ATC, after PTH rework, both of the alternative Pb-free alloys have a much lower Cu dissolution rate and definitely outperform SAC405 in ATC. SAC405 glue and wave resistors after primary attachment and rework demonstrate higher reliability than alternative alloys. Early failures relate to alternative alloy characteristics and should be considered for some applications.

Reliability analysis for wind turbines with incomplete failure data collected from after the date of initial installation

Reliability has an impact on wind energy project costs and benefits. Both life test data and field failure data can be used for reliability analysis. In wind energy industry, wind farm operators have greater interest in recording wind turbine operating data. However, field failure data may be tainted or incomplete, and therefore it needs a more general mathematical model and algorithms to solve the model. The aim of this paper is to provide a solution to this problem. A three-parameter Weibull failure rate function is discussed for wind turbines and the parameters are estimated by maximum likelihood and least squares. Two populations of German and Danish wind turbines are analyzed. The traditional Weibull failure rate function is also employed for comparison. Analysis shows that the three-parameter Weibull function can obtain more accuracy on reliability growth of wind turbines. This work will be helpful in the understanding of the reliability growth of wind energy systems as wind energy technologies evolving. The proposed three-parameter Weibull function is also applicable to the life test of the components that have been used for a period of time, not only in wind energy but also in other industries.

Numerical computation algorithms for sequential checkpoint placement

This paper concerns sequential checkpoint placement problems under two dependability measures: steady-state system availability and expected reward per unit time in the steady state. We develop numerical computation algorithms to determine the optimal checkpoint sequence, based on the classical Brender’s fixed point algorithm and further give three simple approximation methods. Numerical examples with the Weibull failure time distribution are devoted to illustrate quantitatively the overestimation and underestimation of the sub-optimal checkpoint sequences based on the approximation methods.

Planning of progressive group-censoring life tests with cost considerations

This paper considers a life test under progressive type I group censoring with a Weibull failure time distribution. The maximum likelihood method is used to derive the estimators of the parameters of the failure time distribution. In practice, several variables, such as the number of test units, the number of inspections, and the length of inspection interval are related to the precision of estimation and the cost of experiment. An inappropriate setting of these decision variables not only wastes the resources of the experiment but also reduces the precision of estimation. One problem arising from designing a life test is the restricted budget of experiment. Therefore, under the constraint that the total cost of experiment does not exceed a pre-determined budget, this paper provides an algorithm to solve the optimal decision variables by considering three different criteria. An example is discussed to illustrate the proposed method. The sensitivity analysis is also studied.

Weibull methodologies-accurate reliability predictors

Weibull Statistical Methods and Failure Modes and Effects Analysis (FMEA) are technologies that greatly contributed to continually improving reliability, product integrity and many organizations and their processes. Weibull Statistical Methods are founded on the work of Swedish Professor Weibull.