Dominating the maximum likelihood estimator in predicting reliability

The reliability of a series system with n independent components are estimated. The failure distributions of these components are assumed to be Weibull with parameters α_i's and β_i's. It is assumed that the shape parameters β_i's are known. This reduces the reliability of the system to be a function of α = (α₁, ..., α_n), ξ(α), say. The MLE of ξ(α) is derived and an estimator which dominates the MLE in terms of risk, under squared error loss, is also derived. The predicted reliability using these two estimators are computed and compared.     

A recursive algorithm evaluating the exact reliability of a circular consecutive k-within-m-out-of-n:F system

A circular consecutive k-within-m-out-of-n:F system consists of n cyclically ordered components ε₁…ε_n, ie. ε_i+1 succeedes e_i, iε {1,…,n-1}, e₁ succeedes e_n. The system fails if any m consecutive components include k or more failed ones. A recursive algorithm is presented evaluating the reliability of a system with independent components whose failure probabilities may be unequal. This algorithm is computer implementable for “not too large m” (e.g. m 8 for PC machines).     

A note on the confidence interval for the availability ratio

If one is interested in determining the worth of equipment to perform a given task, often the primary concerns are reliability, maintainability, and availability. Availability consider both reliability and maintainability since it is a measure of the ratio of the operating time of the system to the operating time plus the downtime or time to repair. One method commonly used to establish a confidence interval for availability is to establish a confidence interval for the meantime between failure of the equipment and then determine a confidence interval for the meantime to repair the equipment. These two confidence intervals may then be utilized to obtain an “at least” type confidence interval for availability. This approach is not entirely satisfactory since points outside of both of the above mentioned intervals can give points inside the confidence interval for availability. This paper proposes a method for establishing an exact confidence interval for availability under the assumption that the time between failures and the time to repair are independent gamma and lognormal random variables respectively. The method requires computations of the distribution of , where Ut X²(q) and . In order to make the results usuable, tables of the cumulative distribution of W are included.     

Reliability of a two-way circular consecutively connected system with multistate components

A two-way circular consecutively connected system with multistate components (two-way circular CCSMC) consists of n cyclically ordered components e₁,…,e_n, i.e. e_i+1, succeeds e₁, iε«ng l, …, n − 1ång;, e, succeeds e_n- Each component is capable of sending a signal in left-hand and right-hand directions. All components operate independently. The whole system is functioning if both left-hand and right-hand signals are received by each component. A recursive procedure evaluating the reliability of a two-way circular CCSMC is presented.     

Exact fiducial bounds for the reliability of series systems with components having exponential time to failure distributions

Series systems with dissimilar components which have exponential times to failure, with parameters α_j, are considered. The fiducial distribution of the system parameter, φ = Σ α_j, is derived. This distribution is used to construct fiducial bounds for φ and for the reliability of the system R. Illustrative examples are given and the results are compared with three approximate fiducial bounds.     

Bayes shrinkage estimation of reliability and the parameters of a finite range failure time model

A finite range failure time distribution has been proposed and studied. For estimating the two parameters of this distribution, this paper considers a prior assumption that (1 − b) is the probability that the scale parameter θ and shape parameter p have the values θ₀ and p₀, respectively, and that the rest of the probability mass b(0 ≤ b ≤ 1) is distributed as h(p,θ) = h₁(p)h₂(θ). The value h₁(p) is a uniform density for p and h₂(θ) is an inverted gamma density for θ. With this prior density, Bayes estimators are first obtained and then Bayesian shrinkage estimators are defined. Bayesian shrinkage estimators are compared with maximum likelihood estimators (m.l.e.) and it was found that the proposed estimators are better than m.l.e. for quite a wide range of parameters.     

Weibull characteristics of n-MOSFET's with ultrathin gate oxides under FN stress and lifetime prediction

Subthreshold gate voltage shift ΔV_gw of n-MOSFET's with different oxide thicknesses aging at various stress conditions was statisticalized using Weibull distribution. Based on the statistical results, an empirical expression for the relationship between average lifetime and acceleration field was developed, and lifetime predictions were made. Results show that the shape factors (β) of intrinsic failure of the devices with 5.0, 7.0, and 9.0 nm gate oxides under 27 and 105 °C are the same, namely, the mechanisms of the intrinsic failure are the same under low and high temperatures. The proportion of the extrinsic failure increases with temperature increasing. A lifetime prediction method was developed based on the exponential relationship between lifetime and acceleration field. This method can be applied to predict the lifetime of n-MOSFET's with ultrathin gate oxides under FN stress.     

Some characteristics on a finite queue with normal partial and total failures

In this paper we consider that units arrive at a service station in a Poisson fashion with rate λ and are served exponentially by a single server with rate μ in normal working condition, and with slower rate v (v < μ) in the case of partial failure of the service channel. The total failure of a unit is repaired with repair rate β₁ and that of partial failure with repair rate β₂. The partial and total failure rates for the service channel are a₁ and a₂, respectively. The system will function even if a partial failure occurs. The waiting room is finite and the service discipline is a first come first served basis (FCFS). The purpose of this paper is to obtain a steady-state probability generating function for the number of customers present in the system for different states. The probability of various states, along with corresponding results for the particular cases of the system, has been derived.     

Study of micro-BGA solder joint reliability

A new reflow parameter, heating factor (Q_η), which is defined as the integral of the measured temperature over the dwell time above liquidus, has been proposed in this report. It can suitably represent the combined effect of both temperature and time in usual reflow process. Relationship between reliability of the micro-ball grid array (micro-BGA) package and heating factor has been discussed . The fatigue failure of micro-BGA solder joints reflowed with different heating factor in nitrogen ambient has been investigated using the bending cycle test. The fatigue lifetime of the micro-BGA assemblies firstly increases and then decreases with increasing heating factor. The greatest lifetime happens at Q_η near 500 s °C. The optimal Q_η range is between 300 and 750 s °C. In this range, the lifetime of the micro-BGA assemblies is greater than 4500 cycles. SEM micrographs reveal that cracks always initiate at the point of the acute angle where the solder joint joins the PCB pad.     

Sequential interval estimation procedures for the mean exponential survival time and reliability function

The problems of confidence interval estimation are considered (a) for estimating the mean of a one-parameter exponential distribution and (b) for estimating the reliability function associated with the one-parameter exponential distribution. For the estimation problem (a), the confidence interval of ‘preassigned width and coverage probability’ is considered. For the estimation problem (b), the confidence interval of ‘fixed-ratio width and preassigned coverage probability’ is proposed. The failure of the fixed sample size procedures to deal with these estimation problems is established and sequential procedures are proposed to deal with them. The proposed sequential procedures are proved to be ‘asymptotically efficient and consistent’ in the Chow-Robbins [Chow and Robbins, Ann. Math. Statist. 36,457–462 (1965)] sense. Asymptotic distributions of the stopping times are derived and second-order approximations are obtained for the average sample numbers associated with them.     

A recursive algorithm evaluating the exact reliability of a consecutive κ-within-m-out-of-n:F system

A consecutive κ-within-m-out-of-n:F system consists of n linearly ordered components e₁,3.e_n. The system fails if among any m subsequent components there are k or more failed ones. A recursive algorithm is presented evaluating the reliability of the system whose components are independent and have unequal failure probabilities. This algorithm is computer implementable for “not too large m” (e.g. m ≤ 15 for PC machines).     

Efficient implementation of 3X for radix-8 encoding

Several commercial processors have selected the radix-8 multiplier architecture to increase their speed, thereby reducing the number of partial products. Radix-8 encoding reduces the digit number length in a signed digit representation. Its performance bottleneck is the generation of the term 3X, also referred to as hard multiple. This term is usually computed by an adding and shifting operation, 3X=2X+X, in a high-speed adder. In a 2X+X addition, close full adders share the same input signal. This property permits simplified algebraic expressions associated to a 3X operation other than in a conventional addition. This paper shows that the 3X operation can be expressed in terms of two signals, H_i and K_i, functionally equivalent to two carries. H_i and K_i are computed in parallel using architectures which lead to an area- and speed-efficient implementation. For the purposes of comparison, implementation based on standard cells of conventional adders has been compared with the proposed circuits based on these H_i and K_i signals. As a result, the delay of the proposed serial scheme is reduced by roughly 67% without additional cost in area, the delay and area of the carry look-ahead scheme is reduced by 20% and 17%, and that of the parallel prefix scheme is reduced by 26% and 46%, respectively.     

General measures of a two-unit system

In this paper, we consider a two-unit standby system with single repair facility. The bivariate random variables (X₁, Y₁); i = 1, 2, …, n are defined as the busy and idle periods respectively of the repair facility. General measures like joint point availability and joint interval reliability are obtained using the regeneration point technique. Numerical example illustrates the results.     

Reliability characteristics of high-k dielectrics

In this paper, recent results of Weibull slopes, area scaling factors, and breakdown behaviors observed for both soft breakdown and hard breakdown are discussed. These results would help to shed light on the breakdown mechanism of HfO₂ gate dielectrics. The Weibull slope β of the hard breakdown for both the area dependence and the time-to-dielectric-breakdown distribution was found to be β=2, whereas that of the soft breakdown was about 1.4 (EOT=14 Å). We also integrated the time-to-breakdown characteristics of HfO₂ under unipolar AC voltage stress on MOS capacitors. The results show that longer lifetime of HfO₂ has been observed when compared to constant voltage stress. Higher frequency and lower duty cycle in the AC stress resulted in longer lifetime. As thickness decreases, the amount of lifetime enhancement decreases. The enhancement of unipolar t_BD is attributed to less charge trapping during the “on time”, t_on and charge detrapping during the off time, t_off. It is proposed that time (τ_in) for charge to be trapped in HfO₂ is longer than t_on of unipolar stress under high frequency. In addition to experimental results, possible solutions are discussed.     

A new mathematical approach to multi-area power system reliability evaluation

In the original interconnected system reliability model the system state space consists of a set of all n-tuples X = (x₁, x₂,…, x_l, x_l+1,…, x_n) where x₁, x₂, x₃,…, x_l are generation capacities and x_l+1, …, x_n are intertie capacities. Corresponding to each load state i we have state space S_i. For each S_i there exists a map F_i:S_i → R. For each i, S_i is then decomposed into sets of acceptable states A_ks and sets of unacceptable states B_ls. Each B_l is then classified accordingly as it is a loss of load in a particular area or not. Then the appropriate reliability indices are calculated. In this approach the maximal flow function is viewed as a map F: _iS_i → R. It is shown that F is a piecewise linear function. It is also shown that there is a one-one correspondence between B sets with the same area loss of load to each of the linear functions. A useful result which aids in the reduction of computational time of frequency calculation of loss of load is then derived.     

Evaluation of SiGe:C HBT intrinsic reliability using conventional and step stress methodologies

The reliability of SiGe:C HBT devices fabricated using the Freescale’s 0.35-μm RF-BICMOS process was evaluated using both conventional and step stress methodologies. This device technology was assessed to determine its capability for various power amplifier applications (e.g., WLAN, Bluetooth, and cellular phone), which are more demanding than conventional circuit designs. The step stress method was developed to allow a rapid evaluation of product reliability, as well as, a quick method to monitor product reliability. For all tests the collector current I_C and collector voltage V_C were kept constant throughout the test, and the current gain β (I_C/I_B) was continuously monitored. The nominal bias condition was V_C = 3.5-V and J_C = 50-kA/cm² (or 0.5-mA/μm²). The “failure criterion” for all reliability evaluations was −10% degradation in β from the initial value at the start of each stress test or interval. The median time to failure (MTTF) at a junction temperature (T_JCN) of 150 °C for the conventional stress test was 1.86E6-h, and the thermal activation energy was 1.33-eV. In contrast for the temperature step stress tests the combined results gave an MTTF at T_JCN = 150 °C of 5.2E6-h and a thermal activation energy of 1.44-eV. Considering the differences in the two test methods, these results are quite close to one another. The intrinsic reliability of this device at the nominal bias condition and T_JCN = 150 °C is more than adequate for a 5-year system life.     

Individual Overflow and Freed Carried Traffics for a Link with Trunk Reservation

Two traffic streams Φ₁, Φ₂ are offered a link. The calls of Φ_i require exponential holding times with parameter μ and are accepted if less than C_i trunks are occupied. Approximating the Φ_i by appropriate renewal processes meeting their first two moments, defined as the moments of the numbers of calls in virtual links of infinite capacity to which the traffic streams as freed traffics are virtually directed and where the calls get fresh exponential i.i.d. holding times with parameter μ, stable recursive algorithms of complexity O(max (C₁,C₂)) are derived for the first two defined as above moments of the individual overflow and freed carried traffics. The results offer a unified handling of both overflow and carried traffics in circuit switched networks with trunk reservation, providing a basis for new two-moment network dimensioning algorithms.This work was supported by a grant from the Siemens AG.     

Advanced modeling of silicon oxidation

The reaction-diffusion (i.e. the linear-parabolic) mechanism, widely in use for modeling the thermal oxidation of silicon, can be rebutted on the following issues:• There is a poor fit of the linear-parabolic law or of its derivative X² +

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X =

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(t + τ) or

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dt//dx = 2X +

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with dry oxidation data.• The experimental P_H2O dependence of

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/

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and

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contradicts the basic assumptions of the reaction-diffusion model.• The model fails to explain all technically important phenomena, such as: the nature of the fixed oxide charge, Q_f, the orientation dependence, the cross-over effect, the cleaning effects, the weak spots, the bird's beak and other 2D and 3D effects.Thermal oxidation of silicon is much better described by the extended Jorgensen model, i.e. the classical ionic-transport model modified to include non-linear conduction.• This concept leads to an excellently fitting power-parabolic growth law: X² + AX^2-α = Bt or B dt/dX = 2X + (2 - α)AX^1-α which holds for the growth data for dry and wet oxidation, including the initial part of the dry growth curves.• It gives the correct P_H2O dependence of A and B. It fully accounts for the generation of fixed oxide charge while it easily explains the Deal annealing triangle.• The model has remarkable potential to explain the orientation and cross-over effects, the cleaning effect, the weak spots, bird's beak and other 2D and 3D effects.     

Confidence limits for steady state availability of a parallel system

100 (1 − α)% confidence limits for the steady state availability of a parallel system, when the failure rate of an operable unit is a constant and the repair time of a failed unit is an Erlangian distribution are obtained.     

Shrinkage testimators for the scale parameter of an exponential distribution at single and two stage

This paper proposes some shrunken testimators at single and two stage for the scale parameter of an exponential distribution when an initial estimate of the scale parameter θ is available in the form of a point estimate θ₀. The proposed estimators are obtained by choosing different shrinkage factors useful in different situations. Expressions for bias, mean squared error and relative efficiency are derived. Comparisons of different testimators in terms of mean squared error are made and it is shown that when θ₀ is in the vicinity of the true value θ, by taking the shrinkage factor as the square of the shrinkage factor considered in Adke et al. [Commun. Statist. Theory Meth. 16, 1821 (1987)] one can get a better estimator.