Why the mixture failure rate decreases

It is well known that mixtures of decreasing failure rate (DFR) distributions are always decreasing. It turns out that very often mixtures of increasing failure rate (IFR) distributions can decrease as well. In this paper several types of continuous mixtures of IFR distributions are considered. The corresponding limiting behavior of the mixture failure rate function is analyzed for the specific case of mixing which can be interpreted in terms of the proportional hazards model. It is shown that the conditional characteristics (expectation and variance) of the mixing parameter are crucial for the limiting behavior. Several examples are presented and possible generalizations are discussed.     

A flexible Weibull extension

We propose a new two-parameter ageing distribution which is a generalization of the Weibull and study its properties. It has a simple failure rate (hazard rate) function. With appropriate choice of parameter values, it is able to model various ageing classes of life distributions including IFR, IFRA and modified bathtub (MBT). The ranges of the two parameters are clearly demarcated to separate these classes. It thus provides an alternative to many existing life distributions. Details of parameter estimation are provided through a Weibull-type probability plot and maximum likelihood. We also derive explicit formulas for the turning points of the failure rate function in terms of its parameters. This, combined with the parameter estimation procedures, will allow empirical estimation of the turning points for real data sets, which provides useful information for reliability policies.     

The Weibull log-logistic mixture distributions: Model,theory and application to lifetime data

Lifetime data collected from reliability tests are among data that often exhibit significant heterogeneity caused by variations in manufacturing, which makes standard lifetime models inadequate. Finite mixture models provide more flexibility for modeling such data. In this paper, the Weibull-log-logistic mixture distributions model is introduced as a new class of flexible models for heterogeneous lifetime data. Some statistical properties of the model are presented including the failure rate function, moments generating function, and characteristic function. The identifiability property of the class of all finite mixtures of Weibull-log-logistic distributions is proved. The maximum likelihood estimation (MLE) of model parameters under the Type I and Type II censoring schemes is derived. Some numerical illustrations are performed to study the behavior of the obtained estimators. The model is applied to the hard drive failure data made by the Backblaze data center, where it is found that the proposed model provides more flexibility than the univariate life distributions (Weibull, Exponential, logistic, log-logistic, Frechet). The failure rate of hard disk drives (HDDs) is obtained based on MLE estimates. The analysis of the failure rate function on the basis of SMART attributes shows that the failure of HDDs can have different causes and mechanisms.     

An approach for the Bayesian estimation in the case of ordered piecewise constant failure rates

In this paper we present an approach for the Bayesian estimation of piecewise constant failure rates under the constraint that the constant value of the failure rate in an interval of time is greater than a function of its values in the prior intervals. We apply this approach to the estimation of piecewise constant failure rates for conditional IFR, IFRA and NBU distributions. The prior distribution for the failure rate in each interval is specified through gamma distributions with functions of the failure rate values corresponding to the rest of the intervals as location parameters. Using this approach the prior distribution parameters have interpretations through prior means and variances of the values of the piecewise constant failure rate. The posterior distributions and expected values can be found in terms of gamma functions, without the necessity of numerical integrations. We apply this approach to a model for reliability estimation when two operational modes exists and the number of failures in each operational mode is unknown. Finally a numerical example is presented in which simulations of posterior densities are carried out.     

淀粉对聚乙烯膨胀阻燃体系降解和阻燃的影响

以淀粉作为膨胀阻燃体系中的成炭剂,聚代或部分代取了膨胀型阻燃剂中的季戊四醇,研究了淀粉对膨胀阻燃剂及其与线性低密度聚乙烯膨胀体系的热降解行为和阻燃性。研究表明,聚磷酸铵可明显地改变淀粉的热降解行为促进成炭;尽管淀粉可提高ＩＦＲ的成炭量和膨胀体系的膨胀倍数,但它却在一定程度上降低了ＬＬＤＰＥ的膨胀体系的阻燃性,也即是降低了极限氧指数和提高了热释放速率峰值,而用淀粉部分取代ＰＥＲ,对其阻燃性很小,可用     

三聚氰胺磷酸盐和季戊四醇在EVA中的阻燃研究

研究了三聚氰胺磷酸盐(M P)和季戊四醇(PER)作为膨胀型阻燃剂(IFR)在乙烯-醋酸乙烯酯共聚物(EVA)中的阻燃作用。采用氧指数法和垂直燃烧法研究了M P和PER不同配比对EVA阻燃效果的影响。实验结果表明,M P和PER的配比不同对体系的阻燃有很大影响。在M P和PER总添加量为50%时,M P/PER质量比为2∶1时显示出最好的阻燃效果,阻燃EVA体系氧指数最高,垂直燃烧达到V-0级。采用热分析研究了膨胀型阻燃EVA体系的热分解特性,以及采用激光拉曼光谱等手段对材料燃烧后形成的膨胀炭层进行了表征。     

Otimal Run Quantities for an Assembly System with Random Yields

An assembly system consists of m production lines, each of which produces a certain type of component. The components are then fed into an assembly stage, where each set of m components, one unit from each different type, are assembled. Due to uncertainties involved in the production lines, the component yields are random. The optimization problem is to find the run quantities of the components, in order to minimize the total setup cost while guarantee at a certain level the probability of achieving a given target yield from the assembly stage. Solution procedures are developed that efficiently solve the problem under convex setup costs and increasing failure rate (IFR) component yield distributions. Also studied are the computation of the expected yield and the effect of variability.     

Modeling lifetimes with unknown initial age

The failure rate of a distribution, which describes a lifetime random variable with unknown initial age, is studied. A specific model of mixing formalizes this setting. In accordance with this model, it turns out that under certain assumptions the operation of mixing distributions with increasing failure rates can result in the mixture failure rate with a bathtub shape: decreasing in the initial interval of time and then increasing, asymptotically approaching the failure rate of the baseline distribution. Two specific cases of the baseline distribution are considered: the Weibull and the truncated extreme value distributions. A possibility of mixing via the mean residual lifetime function is discussed.     

Enhancement on physical properties of flame retarded ethylene-vinyl acetate copolymer/ferric pyrophosphate composites through electron beam irradiation

Ferric pyrophosphate (FePP) with silica-gel microencapsulated ammonium polyphosphate (MCAPP) and char forming agent (CFA) in the flame retardation of ethylene-vinyl acetate copolymer has been studied through LOI and UL-94 test. The UL-94 data show that EVA with 20 wt.% IFR only has V-2 rating. However, with the addition of 1 wt.% FePP and 19 wt.% IFR, it can reach V-0 rating. The properties of the composites before and after the electron beam irradiation are compared. The volume resistivity, mechanical and thermal properties of the irradiated EVA composites are also evidently improved at appropriate irradiation dose as compared with those of un-irradiated ones. The tensile strength of EVA/IFR/FePP composites with 160 kGy irradiation is 21.8 Mpa, which is much higher than that of un-irradiated one (12.7 MPa). Furthermore, the thermal aged test and water resistance test results demonstrate that EVA/IFR/FePP composites have good thermal aging property and water durability.     

Aging property of unimodal failure rate models

There is a family of statistical models, whose failure rate is unimodal or reverse bathtub shaped. For the situations where the failure is mainly caused by fatigue or corrosion, the time to failure is often represented by such models. This paper studies the aging property of such a class of models. It shows that a unimodal failure rate can be effectively viewed as increasing, decreasing, or constant, depending on the model parameters. Relevant quantitative measures are developed to identify to which case a given unimodal failure rate model belongs. Such two models, lognormal and inverse Weibull distributions, are analyzed.     

磷钼酸盐中金属离子对聚丙烯阻燃效率的提升

多金属氧酸盐是一类新型催化剂, 在很多领域表现出良好的应用价值。改变多金属氧酸盐的金属离子类型可以调节其性能。本研究将磷钼酸(PMA)阴离子与三种不同的金属(镍(Ni)、钠(Na)、锌(Zn))离子反应形成磷钼酸盐(PMos), 将其作为催化剂提升膨胀阻燃聚丙烯(PP)的阻燃效率。结果表明, 单独添加膨胀阻燃剂(IFR)时, 添加量达到25wt%才能使PP复合材料的阻燃等级达到UL-94 V0级别, 然而在PP/IFR中添加0.5wt%的磷钼酸钠(NaPMo)或磷钼酸锌(ZnPMo)后, 仅需添加14.5wt%IFR即可使PP复合材料达到UL-94 V0级别, 而同样的配方下, 磷钼酸镍(NiPMo)只能使PP复合材料达到UL-94 V1级别。不同的金属离子在PP/IFR中具有不同的催化活性, 其中NaPMo和ZnPMo与IFR的匹配性较好, NiPMo较差。PMos通过促进IFR反应, 缓和燃烧过程的热释放速率, 并且形成阻隔作用更优良的炭层, 提高PP与IFR的匹配性, 进而提高了其在UL-94测试中的阻燃效率。     

煤基聚乙烯/蒙脱土复合材料的阻燃特性

研究了以聚磷酸铵(APP)、季戊四醇(PER)和三聚氰胺(MEL)复配成的膨胀型阻燃荆、煤及蒙脱土阻燃协同作用及其对煤基聚乙烯／蒙脱土复合材料阻燃性能的影响。少量蒙脱土即可显著提高聚乙烯的极限氧指数。超细煤粉的含量在一定范围内可明显提高聚乙烯／蒙脱土复合材料的极限氧指数．煤、蒙脱土及膨胀型阻燃剂之间存在良好的协同阻燃效应．可使聚乙烯的极限氧指数得到一定的提高。热氧化分解及红外光谱分析表明，煤、蒙脱土及膨胀型阻燃荆之间的协同阻燃效应与煤基聚乙烯／蒙脱土阻燃材料具有良好的热稳定性和较高的残炭率有关。     

金属氧化物对膨胀阻燃涂层耐火及成炭性能的影响

金属氧化物(MO)可显著影响膨胀阻燃体系的热解成炭过程, 进而改善膨胀阻燃涂层的耐火性能。将Fe₂O₃、ZnO、TiO₂分别添加到双环笼状磷酸酯膨胀阻燃环氧涂层中, 研究了MO对涂层耐火及成炭性能的影响规律。燃烧背温测试结果表明, MO可产生显著的协效耐火作用, 三种MO对耐火性能的增效能力为Fe₂O₃>ZnO>TiO₂。热失重(TGA), 激光拉曼光谱(LRS)和X射线光电子能谱(XPS)分析表明, MO促进了残炭的耐高温氧化性能及类石墨化程度的提高, 增加了涂层的高温残炭量, 三种MO提升涂层成炭性能的能力为Fe₂O₃>ZnO>TiO₂。     

用CONE/TG研究含淀粉膨胀阻燃聚丙烯体系的阻燃和烟释放

利用锥形量热仪（ＣＯＮＥ）在５０ｋＷ／ｍ＾２热辐照条件下,并配合ＴＧ和极限氧指数（ＬＯＩ）对含淀粉膨胀阻燃聚丙烯（ＰＰ）体系的阻燃和烟释放进行了研究,通过对获得的最大热释放速率（ｐｋ－ＨＲＲ）,总烟释放量（ＴＳＰ）、平均比消光面积（ａｖ－ＳＥＡ）及质量损失速度燃烧热（ａｖ－ＥＨＣ）、最大烟产生速率（ｐｋ－ＳＰＲ）、总烟释放量（ＴＳＰ）、平均比消光面积（ａｖ－ＳＥＡ）及质量损失速度（ＭＬＲ）等参数和     

无卤阻燃PES热熔胶的制备与阻燃性能

以聚磷酸铵(APP)、季戊四醇(PER)和三聚氰胺(MEL)组成的膨胀阻燃体系(IFR)作为阻燃剂,以4A分子筛作为协效剂,制备了无卤阻燃共聚酯(PES)热熔胶。研究了IFR对PES阻燃性能的影响及4A分子筛的协效作用。结果表明,少量4A分子筛可促进IFR的阻燃作用。IFR添加量为30%时,阻燃PES氧指数达30.7%,垂直燃烧达V-0级,最大热释放速率大幅降低;加入3%的4A分子筛,氧指数达35.1%。热重分析(TG)、扫描电镜(SEM)及X射线光电子能谱(XPS)结果表明,少量4A分子筛可催化IFR酯化反应,促进体系形成致密炭层,高温时4A分子筛分解并参与成炭反应,稳定炭层。     

聚磷酸铵型膨胀阻燃剂对聚丙烯的阻燃作用

在改性聚磷酸铵中加入聚己内酰胺（ＰＡ６）,可显著提高由它们组成的膨胀型阻燃剂（ＩＦＲ）对聚丙烯（ＰＰ）的阻燃作用,ＰＡ６在其中主要起成炭剂的作用。热重分析表明,当ＩＦＲ－ＰＰ受热燃烧时,ＩＦＲ参与了ＰＰ的成热分解反应并促使部分碳化,元素分析和红外光谱结果表明,ＩＦＲ－ＰＰ受热燃烧时磷主要积聚在燃烧端面并以磷酸及其相应的铵盐存在,它们的形成与ＩＦＲ受热燃烧时的一系列变化有关,并有助于焦化物的进一步炭     

Effective use of fly ash slurry as fill material

A lot of effort has been put into increasing coal ash utilization; however, 50% of total amount is disposed of on land and in the sea. Several attempts have been reported recently concerning slurried coal fly ash use for civil engineering materials, such as for structural fill and backfill. The authors have studied this issue for more than 15 years and reported its potential for (1) underwater fills, (2) light weight backfills, and (3) light weight structural fills, through both laboratory tests and construction works. This paper is an overview of the results obtained for slurry, focusing on the following. (1) Coal fly ash reclaimed by slurry placement shows lower compressibility, higher ground density, and higher strength than by the other methods. This higher strength increases stability against liquefaction during earthquake. (2) Higher stability of the fly ash ground formed by slurry placement is caused by higher density and its self-hardening property. (3) Stability of fly ash reclaimed ground can be increased by increasing density and also by strength enhancement by cement addition. (4) Technical data obtained through a man-made island construction project shows the advantages of fly ash slurry in terms of mechanical properties such as higher stability against sliding failure, sufficient ground strength, and also in terms of cost saving. (5) Concentration in leachates from the placed slurry is lower than the Japanese environmental law. (6) In order to enlarge the fly ash slurry application toward a lightweight fill, mixtures of air foam, cement and fly ash were examined. Test results shows sufficient durability of this material against creep failure. This material was then used as lightweight structural fill around a high-rise building, and showed sufficient quality. From the above data, it can be concluded that coal fly ash slurry can be effectively utilized in civil engineering projects.     

A reliability analysis for electronic devices under an extension of exponentiated perks distribution

This paper presents a reliability analysis for electronic devices (ED) with bathtub curve-shaped failure times. An extension of the exponentiated perks distribution (EPD) is proposed for the analysis. The extension of this new distribution is based on the Alpha Power Transformation, so the Alpha Exponentiated Perks Distribution (AEXP) is introduced. The AEXP has three shape parameters and one scale parameter, allowing greater flexibility to represent failure rates in an increasing, decreasing, or bathtub curve form. Some useful properties in the reliability engineering context are presented. AEXP parameters were estimated via the Maximum Likelihood Method. Finally, two case studies focused on ED are used to compare the proposed distribution and other distributions with similar failure rate representation properties. The obtained results show that the AEXP better describes the behavior of ED than the distributions considered in the analysis.     

Simple bounds for counting processes with monotone rate of occurrence of failures

The article discusses some aspects of analogy between certain classes of distributions used as models for time to failure of nonrepairable objects, and the counting processes used as models for failure process for repairable objects. The notion of quantiles for the counting processes with strictly increasing cumulative intensity function is introduced. The classes of counting processes with increasing (decreasing) rate of occurrence of failures are considered. For these classes, the useful nonparametric bounds for cumulative intensity function based on one known quantile are obtained. These bounds, which can be used for repairable objects, are similar to the bounds introduced by Barlow and Marshall [Barlow, R. Marshall, A. Bounds for distributions with monotone hazard rate, I and II. Ann Math Stat 1964; 35: 1234–74] for IFRA (DFRA) time to failure distributions applicable to nonrepairable objects.     

Relationship Between System Failure Rate and Component Failure Rates

A simple sufficient condition is given for a system to have an increasing failure rate when the identical components comprising it have an increasing failure rate. Systems which function if and only if at least k of the n components function (“k out of n” systems) satisfy this condition. For systems of non-identical components, upper and lower bounds on failure rate are obtained in terms of component failure rates. These bounds are increasing functions of time for “k out of n” structures having components with increasing failure rates.