Bayesian Reliability——Growth Analysis for Statistical of Diverse Population Based on Non-homogeneous Poisson Process

New armament systems are subjected to the method for dealing with multi-stage system reliability-growth statistical problems of diverse population in order to improve reliability before starting mass production. Aiming at the test process which is high expense and small sample-size in the development of complex system, the specific methods are studied on how to process the statistical information of Bayesian reliability growth regarding diverse populations. Firstly, according to the characteristics of reliability growth during product development, the Bayesian method is used to integrate the testing information of multi-stage and the order relations of distribution parameters. And then a Gamma-Beta prior distribution is proposed based on non-homogeneous Poisson process（NHPP） corresponding to the reliability growth process. The posterior distribution of reliability parameters is obtained regarding different stages of product, and the reliability parameters are evaluated based on the posterior distribution. Finally, Bayesian approach proposed in this paper for multi-stage reliability growth test is applied to the test process which is small sample-size in the astronautics filed. The results of a numerical example show that the presented model can make use of the diverse information synthetically, and pave the way for the application of the Bayesian model for multi-stage reliability growth test evaluation with small sample-size. The method is useful for evaluating multi-stage system reliability and making reliability growth plan rationally.

Modeling of Atmospheric Phenomena Using Non-homogeneous Poisson Process

The PPNH （non-homogenous Poisson processes） are frequently used as models for events that come about randomly in a given time period, for example, failure times, time of accidents occurrences, etc. In this work, PPNH is used to model monthly maximum observations of urban ozone corresponding to a period of five years from the meteorological stations of Merced, Pedregal and Plateros, located in the metropolitan area of Mexico City. The interest data are the times in which the observations surpassed the permissible level of ozone of 0.11 ppm, settled by the Mexican Official Norm （NOM-020-SSA 1-1993） to preserve public health.

Immune Clone Maximum Likelihood Estimation of Improved Non-homogeneous Poisson Process Model Parameters

Aiming at the solving problem of improved nonhomogeneous Poisson process( NHPP) model in engineering application,the immune clone maximum likelihood estimation( MLE)method for solving model parameters was proposed. The minimum negative log-likelihood function was used as the objective function to optimize instead of using iterative method to solve complex system of equations,and the problem of parameter estimation of improved NHPP model was solved by immune clone algorithm. And the interval estimation of reliability indices was given by using fisher information matrix method and delta method. An example of failure truncated data from multiple numerical control( NC) machine tools was taken to prove the method. and the results show that the algorithm has a higher convergence rate and computational accuracy, which demonstrates the feasibility of the method.

Analysis of Ozone Behaviour in the City of Puebla-Mexico Using Non-Homogeneous Poisson Models with Multiple Change-Points

In this work, some non-homogeneous Poisson models are considered to study the behaviour of ozone in the city of Puebla, Mexico. Several functions are used as the rate function for the non-homogeneous Poisson process. In addition to their dependence on time, these rate functions also depend on some parameters that need to be estimated. In order to estimate them, a Bayesian approach will be taken. The expressions for the distributions of the parameters involved in the models are very complex. Therefore, Markov chain Monte Carlo algorithms are used to estimate them. The methodology is applied to the ozone data from the city of Puebla, Mexico.

Comparing the Adequacy of Some Non-Homogeneous Poisson Models to Estimate Ozone Exceedances in Mexico City

We consider some non-homogeneous Poisson models to estimate the mean number of times that a given environmental threshold of interest is surpassed by a given pollutant. Seven different rate functions for the Poisson processes describing the models are taken into account. The rate functions considered are the Weibull, exponentiated-Weibull, and their generalisation the Beta-Weibull rate function. We also use the Musa-Okumoto, the Goel-Okumoto, a generalised Goel- Okumoto and the Weibull-geometric rate functions. Whenever thought justifiable, the model allowing the presence of change-points is also going to be considered. The different models are applied to the daily maximum ozone measurements data provided by the monitoring network of the Metropolitan Area of Mexico City. The aim is to compare the adjustment of different rate functions to the data. Even though, some of the rate functions have been considered before, now we are applying them to the same data set. In previous works they were used in different data sets and therefore a comparison of the adequacy of those models were not possible. The measurements considered here were obtained after a series of environmental measures were implemented in Mexico City. Hence, the data present a different behaviour from that of earlier studies.

Non-Homogeneous Poisson Processes Applied to Count Data:A Bayesian Approach Considering Different Prior Distributions

This article discusses the Bayesian approach for count data using non-homogeneous Poisson processes, considering different prior distributions for the model parameters. A Bayesian approach using Markov Chain Monte Carlo (MCMC) simulation methods for this model was first introduced by [1], taking into account software reliability data and considering non-informative prior distributions for the parameters of the model. With the non-informative prior distributions presented by these authors, computational difficulties may occur when using MCMC methods. This article considers different prior distributions for the parameters of the proposed model, and studies the effect of such prior distributions on the convergence and accuracy of the results. In order to illustrate the proposed methodology, two examples are considered: the first one has simulated data, and the second has a set of data for pollution issues at a region in Mexico City.

One-Sample Bayesian Predictive Analyses for a Nonhomogeneous Poisson Process with Delayed S-Shaped Intensity Function Using Non-Informative Priors

The delayed S-shaped software reliability growth model (SRGM) is one of the non-homogeneous Poisson process (NHPP) models which have been proposed for software reliability assessment. The model is distinctive because it has a mean value function that reflects the delay in failure reporting: there is a delay between failure detection and reporting time. The model captures error detection, isolation, and removal processes, thus is appropriate for software reliability analysis. Predictive analysis in software testing is useful in modifying, debugging, and determining when to terminate software development testing processes. However, Bayesian predictive analyses on the delayed S-shaped model have not been extensively explored. This paper uses the delayed S-shaped SRGM to address four issues in one-sample prediction associated with the software development testing process. Bayesian approach based on non-informative priors was used to derive explicit solutions for the four issues, and the developed methodologies were illustrated using real data.

考虑不完美排错情况的NHPP类软件可靠性增长模型

针对现有NHPP类软件可靠性增长模型对故障排错过程中不完美排错情况考虑不完全的现状,提出了一种新的软件可靠性增长模型.该模型全面考虑了不完美排错的两种情况:既考虑了排错过程中引入新错误的可能性,又考虑了不完全排错的情况,并且引入了一种故障排除率随时间变化的故障排除率函数,使模型更符合实际情况.利用公开发表的两组不同的软件失效数据对该模型进行验证的结果表明,与现有的对不完美排错情况考虑不完全的模型相比,该模型能够取得更好的拟合结果和预测效果.

基于NHPP类模型的Web软件可靠性分析

结合Web软件运行的实际情况,提出一种基于BurrX测试工作量函数(TEF)的非齐次泊松过程类软件可靠性增长模型(SRGM),并将其应用到Web软件可靠性分析中。TEF能直接体现Web软件工作量与日历时间之间的关系,解决Web软件因工作量高度不均衡导致的问题。实验结果表明,与G-O模型相比,SRGM具有较好的Web软件可靠性评估效果,能较准确地描述Web软件运行的失效过程。

软件可靠性中NHPP模型趋势检验的U-图方法

对软件可靠性中一类重要的模型——非齐次泊松过程（ＮＨＰＰ）模型的趋势检验，构造了一列检验的统计量，独立同分布于均匀分布Ｕ（０，１）．并由柯尔莫哥洛夫检验判定所选择的可靠性增长趋势模型的合适性．本文方法可适用于完全数据和不完全数据．这里报告的方法，可以作为选择软件可靠性模型时的一种判断依据，以定量地评判选择的软件可靠性模型的优劣．

一种基于离散时间的NHPP软件可靠性增长模型

基于非齐次泊松过程的软件可靠性增长模型按时间域可分为连续时间模型和离散时间模型两类。现有的软件可靠性增长模型大多都是针对连续时间构造的,在一定程度上忽视了对离散时间模型的研究。利用概率生成函数构建两种基于离散时间的软件可靠性增长模型——基本模型和扩展模型,具有很大的实用性和必要性。构建的扩展模型以不完美排错情形作为基础,考虑到了由于故障排除而有可能引入新故障的问题,同时还考虑到了在软件排错过程中由于测试团队的熟练程度而引起的软件故障排除率的相对变化情况,这使得提出的模型更加符合实际。最后利用两组公开发表的失效数据集对两种模型与指数模型和S型模型进行对比验证,验证结果表明提出的两种模型具有更好的拟合能力和预测效果。

Incorporating S-shaped testing-effort functions into NHPP software reliability model with imperfect debugging

Testing-effort（TE） and imperfect debugging（ID） in the reliability modeling process may further improve the fitting and prediction results of software reliability growth models（SRGMs）. For describing the S-shaped varying trend of TE increasing rate more accurately, first, two S-shaped testing-effort functions（TEFs）, i.e.,delayed S-shaped TEF（DS-TEF） and inflected S-shaped TEF（IS-TEF）, are proposed. Then these two TEFs are incorporated into various types（exponential-type, delayed S-shaped and inflected S-shaped） of non-homogeneous Poisson process（NHPP）SRGMs with two forms of ID respectively for obtaining a series of new NHPP SRGMs which consider S-shaped TEFs as well as ID. Finally these new SRGMs and several comparison NHPP SRGMs are applied into four real failure data-sets respectively for investigating the fitting and prediction power of these new SRGMs.The experimental results show that：（i） the proposed IS-TEF is more suitable and flexible for describing the consumption of TE than the previous TEFs;（ii） incorporating TEFs into the inflected S-shaped NHPP SRGM may be more effective and appropriate compared with the exponential-type and the delayed S-shaped NHPP SRGMs;（iii） the inflected S-shaped NHPP SRGM considering both IS-TEF and ID yields the most accurate fitting and prediction results than the other comparison NHPP SRGMs.

Grey-based approach for estimating software reliability under nonhomogeneous Poisson process

Due to the randomness and time dependence of the factors affecting software reliability, most software reliability models are treated as stochastic processes, and the non-homogeneous Poisson process(NHPP) is the most used one.However, the failure behavior of software does not follow the NHPP in a statistically rigorous manner, and the pure random method might be not enough to describe the software failure behavior. To solve these problems, this paper proposes a new integrated approach that combines stochastic process and grey system theory to describe the failure behavior of software. A grey NHPP software reliability model is put forward in a discrete form, and a grey-based approach for estimating software reliability under the NHPP is proposed as a nonlinear multi-objective programming problem. Finally, four grey NHPP software reliability models are applied to four real datasets, the dynamic R-square and predictive relative error are calculated. Comparing with the original single NHPP software reliability model, it is found that the modeling using the integrated approach has a higher prediction accuracy of software reliability. Therefore, there is the characteristics of grey uncertain information in the NHPP software reliability models, and exploiting the latent grey uncertain information might lead to more accurate software reliability estimation.

NHPP软件可靠性模型的推广

在一般的意义上对一类非齐次泊松过程模型作了推广,用多项式函数来拟合纠错经验函数,在一定程度上简化了模型的选择.给出了上述推广的可行性的证明,并对模型参数的性质进行了讨论.

Nonlinear Modeling for a Two-Stage Degradation System Based on Nonhomogeneous Poisson Process

The degradation process modeling is one of research hotspots of prognostic and health management(PHM),which can be used to estimate system reliability and remaining useful life(RUL).In order to study system degradation process,cumulative damage model is used for degradation modeling.Assuming that damage increment is Gamma distribution,shock counting subjects to a homogeneous Poisson process(HPP)when degradation process is linear,and shock counting is a non-homogeneous Poisson process(NHPP)when degradation process is nonlinear.A two-stage degradation system is considered in this paper,for which the degradation process is linear in the first stage and the degradation process is nonlinear in the second stage.A nonlinear modeling method for considered system is put forward,and reliability model and remaining useful life model are established.A case study is given to validate the veracities of established models.

An Improved NHPP Model with Time-Varying Fault Removal Delay

In this paper, an improved NHPP model is proposed by replacing constant fault removal time with time-varying fault removal delay in NHPP model, proposed by Daniel R Jeske. In our model, a time-dependent delay function is established to fit the fault removal process. By using two sets of practical data, the descriptive and predictive abilities of the improved NHPP model are compared with those of the NHPP model, G-O model, and delayed S-shape model. The results show that the improved model can fit and predict the data better.

A transient process observation method based on the non-homogeneous Poisson process model

The current-mode-counting method is a new approach to observing transient processes,especially in transient nuclear fusion,based on the non-homogeneous Poisson process(NHPP)model.In this paper,a new measurement process model of the pulsed radiation field produced by transient nuclear fusion is built based on the NHPP.A simulated measurement is performed using the model,and the current signal from the detector is obtained by simulation based on Poisson process thinning.The neutron time spectrum is reconstructed and is in good agreement with the theoretical value,with its maximum error of a characteristic parameter less than 2.3%.Verification experiments were carried out on a CPNG-6 device at the China Institute of Atomic Energy,with a detection system with a nanosecond response time.The experimental charge amplitude spectra are in good agreement with those obtained by the traditional counting mode,and the characteristic parameters of the time spectrum are in good agreement with the theoretical values.This shows that the current-mode-counting method is effective for the observation of transient nuclear fusion processes.

多台数控机床最小维修的可靠性评估

为了解决数控(NC)机床最小维修的可靠性评估问题,提出了基于随机点过程的多台NC机床时间截尾的可靠性评估的可修系统方法,建立了故障时间的非齐次泊松过程(NHPP)模型.用Fisher信息矩阵(FIM)法给出了模型参数的点估计和区间估计,同时也给出了NC机床的累积平均无故障间隔时间(MTBF)和累积故障强度在截尾时间的点估计和区间估计.AIC信息准则计算结果表明,对于故障时间具有某种单调趋势的NC机床最小维修可靠性评估,可修系统方法优于统计分布方法.模型通过了趋势检验、更新过程检验和拟合优度检验.

劣化系统周期预防性维修策略的优化

以长期运行费用率为评估指标,研究了劣化系统的周期预防性维修策略。假定开展的周期预防性维修介于最小维修和完全维修之间,维修间隔期间的故障过程用非齐次泊松过程进行描述,在此基础上采用比例寿命回退模型对预防性维修效果进行建模,给出了最佳预防性维修周期T和预防性维修次数N的计算方法以及参数应当满足的条件,同时结合实际情况给出了(N,T)策略的优化求解算法,最后通过数值算例,验证了所提方法的可行性。

考虑软件不同失效过程偏差的软件可靠性模型

软件可靠性分析是根据软件失效数据等信息,通过合理建模来对软件可靠性进行预计和评价.现有的基于随机过程的可靠性模型一般采用均值过程来描述软件失效数据,然而,软件失效数据的模型化实质上应该是使其成为某个随机过程的一个样本轨迹.文中建立了考虑软件不同失效过程偏差的软件可靠性模型,用NHPP过程表示失效过程均值函数的变化趋势,ARMA过程表示实际失效过程对均值过程的偏差序列.在两组公开发表的真实数据集上对模型的实验表明,新模型较之一些广泛使用的NHPP软件可靠性模型在拟合能力及适用性上有明显的提高,并且保持了较好的预测能力.