Estimation of Weibull Parameters With Competing-Mode Censoring

Existing results are reviewed for the maximum likelihood (ML) estimation of the parameters of a 2-parameter Weibull life distribution for the case where the data are censored by failures due to an arbitrary number of independent 2-parameter Weibull failure modes. For the case where all distributions have a common but unknown shape parameter the joint ML estimators are derived for i) a general percentile of the j-th distribution, ii) the common shape parameter, and iii) the proportion of failures due to failure mode j. Exact interval estimates of the common shape parameter are constructable in terms of the ML estimates obtained by using i) the data without regard to failure mode, and ii) existing tables of the percentage points of a certain pivotal function. Exact interval estimates for a general percentile of failure-mode-j distribution are calculable when the failure proportion due to failure-mode-j is known; otherwise a joint s-confidence region for the percentile and failure proportion is calculable. It is shown that sudden death endurance test results can be analyzed as a special case of competing-mode censoring. Tabular values for the construction of interval estimates for the 10-th percentile of the failure-mode-j distribution are given for 17 combinations of sample size (from 5 to 30) and number of failures.     

An alternative degradation reliability modeling approach using maximum likelihood estimation

An alternative degradation reliability modeling approach is presented in this paper. This approach extends the graphical approach used by several authors by considering the natural ordering of performance degradation data using a truncated Weibull distribution. Maximum Likelihood Estimation is used to provide a one-step method to estimate the model's parameters. A closed form expression of the likelihood function is derived for a two-parameter truncated Weibull distribution with time-independent shape parameter. A semi-numerical method is presented for the truncated Weibull distribution with a time-dependent shape parameter. Numerical studies of generated data suggest that the proposed approach provides reasonable estimates even for small sample sizes. The analysis of fatigue data shows that the proposed approach yields a good match of the crack length mean value curve obtained using the path curve approach and better results than those obtained using the graphical approach.     

Sequential tests for integrated-circuit failures

We propose a sequential probability ratio test (SPRT) based on a 2-parameter Weibull distribution for integrated-circuit (IC) failure analysis. The shape parameter of the Weibull distribution characterizes the decreasing, constant, or increasing failure rate regions in the bath tub model for IC. The algorithm (SD) detects the operating region of the IC based on the observed failure times. Unlike the fixed-length tests, the SD, due to its sequential nature, uses the minimum average number of devices for the test for fixed error tolerances in the detection procedure. We find that SD is, on average, 96% more statistically efficient than the fixed-length test. SD is highly robust to the variations in the model parameters, unlike other existing sequential tests. Since the accuracy of the tests and the test length are conflicting requirements, we also propose a truncated SD which allows a better control of this tradeoff. It has both the sequential nature of examining measurements and the fixed-length property of guaranteeing that the tolerances be met approximately with a specified number of available measurements     

Relation of a Physical Process to the Reliability of Electronic Components

An ensemble of electronic components having a random variation of some parameter, such as surface contamination, is considered. A physical process is postulated which leads to a change in one of the operating characteristics of the device. When this operating characteristic attains a value outside an acceptable range, the device is considered to have failed. The failure rate is calculated directly from the time behavior of the physical process and compared, for illustration, to the Weibull failure law. The parameters of the Weibull law are then related to the parameters of the physical process and the distribution of starting parameters.     

Experimental study of Gummel-Poon model parameter correlations for bipolar junction transistors

For the purpose of statistical modeling it is necessary to choose a few independent device parameters. A number of devices are characterized in terms of their Gummel-Poon model parameters. Correlations between these parameters are studied. The parameters are divided into two groups. The first group consists of parameters depending on collector doping. Collector resistance R/SUB C/ can be chosen as the controlling parameter for this group. The second group comprises the parameters influenced by emitter and base dopings. For this group, using the statistical method of factor analysis to analyze these correlations, it is shown that the variation in the measured data can be effectively expressed in terms of only a few controlling parameters. It is shown that the saturation current I/SUB S/ shows good correlations with most of the model parameters and can be used as an independent parameter from which other parameters can then be computed.     

A class of subset selection procedures for Weibull populations

Among distributions that represent component and system life, the Weibull family is presented. Goodness of a Weibull population is defined in terms of the shape parameter. A class of subset selection-procedures based on the U-statistic (derived via the concept of convex ordering) is proposed for two cases: (1) selecting a subset from k Weibull populations which contains the population with the largest shape parameter; and (2) selecting a subset containing all the populations whose shape parameters are at least that of the control population; unknown and known control populations are considered. The approximate implementation of the selection procedures with the help of the existing tables is discussed, and a numerical example is given. Statistical simulation is used to investigate the optimal members of the proposed class in case 1. The simulation shows that a common sample-size of 11 or more can be used to implement the asymptotic results     

Approaches for reliability modeling of continuous-state devices

Three approaches for reliability modelling of continuous state devices are presented in this paper. One uses the random process to fit model parameters of a statistical distribution as functions of time. This approach allows the data set to be from any general distribution. The second approach uses the general path model to fit parameters of the model as functions of time. The relationship between the random process model and the general path model is illustrated. The third approach uses multiple linear regression to fit the distribution of lifetime directly. This approach has less restriction on the degradation data to be analyzed. All three approaches are illustrated with examples. Finally a mixture model is proposed which can be used to model both catastrophic failures and degradation failures. This mixture model also shows engineers how to design experiments to collect both hard failure data and soft failure data. Topics for further investigation in continuous device reliability modelling include further investigation of the mixture model, application of these models to practical situations, and using complex statistical distributions to fit degradation data     

Inference on Weibull Percentiles and Shape Parameter from Maximum Likelihood Estimates

Four functions of the maximum likelihood estimates of the Weibull shape parameter and any Weibull percentile are found. The sampling distributions are independent of the population parameters and depend only upon sample size and the degree of (Type II) censoring. These distributions, once determined by Monte Carlo methods, permit the testing of the following hypotheses: 1) that the Weibull shape parameter is equal to a specified value; 2) that a Weibull percentile is equal to a specified value; 3) that the shape parameters of two Weibull populations are equal; and 4) that a specified percentile of two Weibull populations are equal given that the shape parameters are. The OC curves of the various tests are shown to be readily computed. A by-product of the determination of the distribution of the four functions are the factors required for median unbiased estimation of 1) the Weibull shape parameter, 2) a Weibull percentile, 3) the ratio of shape parameters of two Weibull distributions, and 4) the ratio of a specified percentile of two Weibull distributions.     

Analysis of step-stress accelerated-life-test data: a new approach

A linear cumulative exposure model (LCEM) is used to analyze data from a step-stress accelerated-life-test, in particular, those with failure-free life (FFL). FFL is characterized by a location parameter in the distribution. For the 2-parameter Weibull distribution, the Nelson cumulative exposure model is a special case of LCEM. Under LCEM a general expression is derived for computing the maximum likelihood estimator (MLE) of stress-dependent distribution parameters under multiple censoring. The estimation procedure is simple and is illustrated by a set of experimental data using the 3-parameter Weibull distribution     

Variables sampling plans for Weibull distributed lifetimes under sudden death testing

Sudden death testing can be utilized for deciding upon the lot acceptance of manufactured parts. Variables single, and double sampling plans are proposed for the lot acceptance of parts whose life follows a Weibull distribution with known shape parameter. The proposed plans are different from the existing ones in that the lot acceptance criteria do not depend on the estimated scale parameter. Design parameters of both sampling plans are determined by using the usual two-point approach. The number of groups is determined independently of the group size, and even independently of the shape parameter. Also, the double sampling plan can reduce the average number of groups required. The effects of mis-specification of the shape parameter on the probability of accepting the lots under the single sampling plan are analyzed & discussed.     

A new small-signal modeling approach applied to GaN devices

A new small-signal modeling approach applied to GaN-based devices is presented. In this approach, a new method for extracting the parasitic elements of the GaN device is developed. This method is based on two steps, which are: 1) using cold S-parameter measurements, high-quality starting values for the extrinsic parameters that would place the extraction close to the global minimum of the objective function for the distributed equivalent circuit model are generated and 2) the optimal model parameter values are searched through optimization using the starting values already obtained. The bias-dependent intrinsic parameter extraction procedure is improved for optimal extraction. The validity of the developed modeling approach and the proposed small-signal model is verified by comparing the simulated wide-band small-signal S-parameter, over a wide bias range, with measured data of a 0.5-/spl mu/m GaN high electron-mobility transistor with a 2/spl times/50 /spl mu/m gatewidth.     

Bayesian 1-Sample Prediction for the 2-Parameter Weibull Distribution

Given the first few failures times of a sample of components whose lifetimes follow a Weibull distribution with both scale and shape parameters unknown, a Bayesian approach is used to derive prediction bounds for the failure times of the remaining components in the sample. Iterative procedures involving considerable computation are necessary for calculating prediction bounds, but some simplification is possible on restricting the shape parameter to a finite number of values. A numerical example illustrates the procedures.     

Reliability Testing of 3D-Printed Electromechanical Scanning Devices

Recent advances in the field of stereolithography based manufacturing, have led to a number of 3D-printed sensor and actuator devices, as a cost-effective and low fabrication complexity alternative to micro-electro-mechanical counterparts. Yet the reliability of such 3D-printed dynamic structures have yet to be explored. Here we perform reliability tests and analysis of a selected 3D-printed actuator, namely an electromechanical scanner. The scanner is targeted towards scanning incoming light onto the target, which is particularly useful for barcoding, display, and opto-medical tissue imaging applications. We monitor the deviations in the fundamental mechanical resonance, scan-line, and the quality factor on a number of scanners having different device thicknesses, for a total duration of 5 days (corresponding to 20–80 million cycles, depending on the device operating frequency). A total of 9 scanning devices, having 10 mm?×?10 mm die size were tested, with a highlight on device-device variability, as well as the effect of device thickness itself. An average standard deviation of < ~%10 (with respect to the mean) was observed for all tested parameters among scanners of the same type (an indicator device to device variability), while an average standard deviation of less than about 10 percent (with respect to the mean) was observed for all parameters for the duration of the entire test (as an indicator of device reliability), for a total optical scan angle of 5 degrees.     

A manufacturing sensitivity analysis of 0.35 μm LDD MOSFET's

Fundamental to successful manufacturing of integrated circuits is the achievement of sufficient control in all process steps to realize, with very high yield, fully functional circuits whose performance and reliability conform to pre-determined standards. Towards this end, it is increasingly necessary to relate in a quantitative manner the sensitivity of the electrical performance of the final devices and circuits to variations in structural parameters and doping profiles, which in turn can be related to process and tool performance variations. In this paper, we describe the results of an analysis performed to quantify the sensitivity of the electrical parameters of a 0.35 μm LDD MOSFET to variations in the doping and structural parameters of the device that are anticipated in manufacturing. A central-composite design was used to develop second-order models for six key device electrical parameters. The resulting models are manifested as second-order equations relating the device electrical parameter variations to random variations in seven key device structure and doping parameters. This set of equations thus allows one to understand quantitatively the source and nature of the device electrical parameter variations. A simple Monte Carlo approach is applied to predict the statistical distributions of the key device electrical parameters which result from the random manufacturing variations in the structure and doping parameters by using the quantitative relationships developed in this paper     

遗传算法在半导体器件参数提取软件中的实现

随着半导体器件参数的增加,目标函数的自变量空间维数变得越来越大。传统的优化算法已经不能很好地处理此类问题,主要表现在无法有效地达到目标函数的极小点。文章首先在参数提取软件中实现了遗传算法,并与传统的优化算法进行了比较,探讨了该算法在参数提取软件中的实用性,提出了将遗传算法与传统优化算法结合在一起的方法。     

Energy-conserving access protocols for identification networks

A myriad of applications are emerging, in which energy conservation is a critical system parameter for communications. Radio frequency identification device (RFID) networks, smart cards, and even mobile computing devices, in general, need to conserve energy. In RFID systems, nodes are small battery-operated inexpensive devices with radio receiving/transmitting and processing capabilities, integrated into the size of an ID card or smaller. These identification devices are designed for extremely low-cost large-scale applications, such that the replacement of batteries is not feasible. This imposes a critical energy constraint on the communications (access) protocols used in these systems, so that the total time a node needs to be active for transmitting or receiving information should be minimized. Among existing protocols, classical random access protocols are not energy conserving, while deterministic protocols lead to unacceptable delays. This paper deals with designing communications protocols with energy constraint, in which the number of time slots in which tags need to be in the active state is minimized, while the access delay meets the applications constraints. We propose three classes of protocols which combine the fairness of random access protocols with low energy requirements     

Characterization of Standard Cells for Intra-Cell Mismatch Variations

With the adoption of statistical timing across industry, there is a need to characterize all gates/cells in a digital library for delay variation (referred to as statistical characterization). Statistical characterization needs to be performed efficiently with acceptable accuracy as a function of several process and environmental parameter variations. In this paper, we propose an approach to consider intra-cell process mismatch variations to characterize a cell's delay and output transition time (output slew) variations. A straightforward approach to address this problem is to model these mismatch variations by characterizing for each device fluctuation separately. However, the runtime complexity for such characterization becomes of the order of number of devices in the cell and the number of simulations required can easily become infeasible. We analyze the fluctuations in switching and nonswitching devices and their impact on delay variations. Using these properties of the devices, we propose a clustering approach to characterize for cell's delay variations due to intra-cell mismatch variations. The proposed approach results in as much as 12X runtime improvement with acceptable accuracy, compared with Monte Carlo simulation. We show that this approach ensures an upper bound on the results while keeping the number of simulations for each cell independent of the number of devices.      

General Order Selection Combining for Nakagami and Weibull Fading Channels

In this paper, some analytical results for general order selection (GOS) over independent but not necessarily identically distributed (i.n.d.) Weibull and Nakagami fading channels are presented. The Weibull fading parameters are assumed to be equal whereas the Nakagami fading parameters are assumed to be integer-valued. It is shown that the pdf of the q-th order statistic can be expressed as a linear combination of Weibull and Nakagami pdf's respectively. Closed-form expressions for the moment generating functions and general moments are derived. In addition, exact closed-form expressions for the symbol error rate are obtained for a number of modulation schemes. Numerical results show that for the same average channel gains, the performance on i.n.d. channels may be better or worse than on i.i.d. channels.     

A cautionary tale about Weibull analysis [reliability estimation]

This paper makes three points about possible perils of unguarded fitting of Weibull distributions to data: (1) bias is introduced by incomplete data, which may have counter-intuitive effects; (2) bias is introduced into percentile estimates by using regression on log-transformed variables to fit the Weibull parameters, particularly if the percentile to be predicted lies outside the range of the data; and (3) the amount of variation associated with such estimates can be very substantial. A partial solution to the incomplete data problem using simulation is presented, and the maximum likelihood approach to parameter estimation and its advantages relative to regression estimation are explained. The problem arose in predicting life expectancy of long-lived components subject to natural aging which cannot be investigated using accelerated testing and for which the collection of data provides an incomplete life record