Objectives
To provide an introduction to the statistical analysis of
failure time data
To discuss the impact of data censoring on data analysis
To demonstrate software tools for reliability data analysis
Organization
Reliability definition
Characteristics of reliability data
Statistical analysis of censored reliability data
Weibull Analysis is an important tool for Reliability Engineering. It can be used verifying the design life at component level, comparing two designs and warranty analysis.
This is a three parts lecture series. The parts will cover the basics and fundamentals of reliability engineering. Part 1 begins with introduction of reliability definition and other reliability characteristics and measurements. It will be followed by reliability calculation, estimation of failure rates and understanding of the implications of failure rates on system maintenance and replacements in Part 2. Then Part 3 will cover the most important and practical failure time distributions and how to obtain the parameters of the distributions and interpretations of these parameters. Hands-on computations of the failure rates and the estimation of the failure time distribution parameters will be conducted using standard Microsoft Excel.
Part 2. Reliability Calculations
1.Use of failure data
2.Density functions
3.Reliability function
4.Hazard and failure rates
You’ve heard about Weibull Analysis, and want to know what it can be used for, OR you’ve used Weibull Analysis in the past, but have forgotten some of the background and uses….
This webinar looks at giving you the background of Weibull Analysis, and its use in analyzing failure modes. Starting from basics and giving examples of its uses in answering the questions:
• How many do I test, for how long?
• Is our design system wrong?
• How many more failures will I have in the next month, year, 5 years?
Sit in and listen and ask your questions … not detailed “How to” but “When & Why to”!
This is a presentation to the top management as to why reliability is important and what is the difference between a maintenance engineer and a reliability engineer.
Objectives
To understand Weibull distribution
To be able to use Weibull plot for failure time analysis and
diagnosis
To be able to use software to do data analysis
Organization
Distribution model
Parameter estimation
Regression analysis
Weibull Analysis is an important tool for Reliability Engineering. It can be used verifying the design life at component level, comparing two designs and warranty analysis.
This is a three parts lecture series. The parts will cover the basics and fundamentals of reliability engineering. Part 1 begins with introduction of reliability definition and other reliability characteristics and measurements. It will be followed by reliability calculation, estimation of failure rates and understanding of the implications of failure rates on system maintenance and replacements in Part 2. Then Part 3 will cover the most important and practical failure time distributions and how to obtain the parameters of the distributions and interpretations of these parameters. Hands-on computations of the failure rates and the estimation of the failure time distribution parameters will be conducted using standard Microsoft Excel.
Part 2. Reliability Calculations
1.Use of failure data
2.Density functions
3.Reliability function
4.Hazard and failure rates
You’ve heard about Weibull Analysis, and want to know what it can be used for, OR you’ve used Weibull Analysis in the past, but have forgotten some of the background and uses….
This webinar looks at giving you the background of Weibull Analysis, and its use in analyzing failure modes. Starting from basics and giving examples of its uses in answering the questions:
• How many do I test, for how long?
• Is our design system wrong?
• How many more failures will I have in the next month, year, 5 years?
Sit in and listen and ask your questions … not detailed “How to” but “When & Why to”!
This is a presentation to the top management as to why reliability is important and what is the difference between a maintenance engineer and a reliability engineer.
Objectives
To understand Weibull distribution
To be able to use Weibull plot for failure time analysis and
diagnosis
To be able to use software to do data analysis
Organization
Distribution model
Parameter estimation
Regression analysis
This is a three parts lecture series. The parts will cover the basics and fundamentals of reliability engineering. Part 1 begins with introduction of reliability definition and other reliability characteristics and measurements. It will be followed by reliability calculation, estimation of failure rates and understanding of the implications of failure rates on system maintenance and replacements in Part 2. Then Part 3 will cover the most important and practical failure time distributions and how to obtain the parameters of the distributions and interpretations of these parameters. Hands-on computations of the failure rates and the estimation of the failure time distribution parameters will be conducted using standard Microsoft Excel.
Part 3. Failure Time Distributions
1.Constant failure rate distributions
2.Increasing failure rate distributions
3.Decreasing failure rate distributions
4.Weibull Analysis – Why use Weibull?
Design for reliability (DFR) is an industry-wide practice and a philosophy of considering reliability in an early stage of product design and development, to achieve a highly-reliable product while with sustainable cost. Physical of Failure (PoF) is recognized as a key approach of implementing DFR in a product design and development process. The author will present a case study to illustrate predicting and identifying product failure early in the design phase with the help of a quantitative PoF model based analysis tool.
Achieving high product reliability has become increasingly vital for manufacturers in order to meet customer expectations amid the threat of strong global competition. Poor reliability can doom a product and jeopardize the reputation of a brand or company. Inadequate reliability also presents financial risks from warranty, product recalls, and potential litigation. When developing new products, it is imperative that manufacturers develop reliability specifications and utilize methods to predict and verify that those reliability specifications will be met. This 4-Hour course provides an overview of quantitative methods for predicting product reliability from data gathered from physical testing or from field data
Accelerated life testing plans are designed under multiple objective consideration, with the resulting Pareto optimal solutions classified and reduced using neural network and data envelopement analysis, respectively.
This seminar session provides an overview of major aspects of reliability engineering, including general introduction of reliability engineering (definition of reliability, function of reliability engineering, a brief history of reliability, etc.), reliability basics (metrics used in reliability, commonly-used probability distributions in reliability, bathtub curve, reliability demonstration test planning, confidence intervals, Bayesian statistics application in reliability, strength-stress interference theory, etc.), accelerated life testing (ALT) (types of ALT, Arrhenius model, inverse power law model, Eyring model, temperature-humidity model, etc.), reliability growth (reliability-based growth models, MTBF-based growth model, etc.), systems reliability & availability (reliability block diagram, non-repairable or repairable systems, reliability modeling of series systems, parallel systems, standby systems, and complex systems, load sharing reliability, reliability allocation, system availability, Monte Carlo simulation, etc.), and degradation-based reliability (introduction of degradation-based reliability, difference between traditional reliability and degradation-based reliability, etc.).
Authors: (i) Prashanth Lakshmi Narasimhan,
(ii) Mukesh Ravichandran
Industry: Automobile -Auto Ancillary Equipment ( Turbocharger)
This was presented after the completion of our 2 months internship at Turbo Energy Limited during our 3rd Year Summer holidays (2013)
This is a three parts lecture series. The parts will cover the basics and fundamentals of reliability engineering. Part 1 begins with introduction of reliability definition and other reliability characteristics and measurements. It will be followed by reliability calculation, estimation of failure rates and understanding of the implications of failure rates on system maintenance and replacements in Part 2. Then Part 3 will cover the most important and practical failure time distributions and how to obtain the parameters of the distributions and interpretations of these parameters. Hands-on computations of the failure rates and the estimation of the failure time distribution parameters will be conducted using standard Microsoft Excel.
Part 1. Reliability Definitions
1.Reliability---Time dependent characteristic
2.Failure rate
3.Mean Time to Failure
4.Availability
5.Mean residual life
How do you use the Weibull Distribution? It’s just one of many useful statistical distribution we have to master as reliability engineers. Let’s explore an array of distribution and the problems they can help solve in our day to day work.
Detailed Information: When confronted with a set of time to failure data, what is your goto analysis approach. For me it’s a Weibull plot. It’s quick, often provides some insight to ask better questions, and easy to explain to others. A histogram is another great starting point. If we know a little about the source of the data, we may favor the normal or lognormal distributions. If discreet data, then binomial is the first choice, yet Poisson or hypergeometric have uses, too. A basic understanding of statistical distributions provides you a way to summarize data providing insights to identify or solve problems. In this webinar we’ll explore a few distributions useful for reliability engineering work and talk about how to select a distribution, basics on interpreting distributions and just touch on judging if you have selected the right distribution.
This Accendo Reliability webinar originally broadcast on 14 April 2015.
This is a three parts lecture series. The parts will cover the basics and fundamentals of reliability engineering. Part 1 begins with introduction of reliability definition and other reliability characteristics and measurements. It will be followed by reliability calculation, estimation of failure rates and understanding of the implications of failure rates on system maintenance and replacements in Part 2. Then Part 3 will cover the most important and practical failure time distributions and how to obtain the parameters of the distributions and interpretations of these parameters. Hands-on computations of the failure rates and the estimation of the failure time distribution parameters will be conducted using standard Microsoft Excel.
Part 3. Failure Time Distributions
1.Constant failure rate distributions
2.Increasing failure rate distributions
3.Decreasing failure rate distributions
4.Weibull Analysis – Why use Weibull?
Design for reliability (DFR) is an industry-wide practice and a philosophy of considering reliability in an early stage of product design and development, to achieve a highly-reliable product while with sustainable cost. Physical of Failure (PoF) is recognized as a key approach of implementing DFR in a product design and development process. The author will present a case study to illustrate predicting and identifying product failure early in the design phase with the help of a quantitative PoF model based analysis tool.
Achieving high product reliability has become increasingly vital for manufacturers in order to meet customer expectations amid the threat of strong global competition. Poor reliability can doom a product and jeopardize the reputation of a brand or company. Inadequate reliability also presents financial risks from warranty, product recalls, and potential litigation. When developing new products, it is imperative that manufacturers develop reliability specifications and utilize methods to predict and verify that those reliability specifications will be met. This 4-Hour course provides an overview of quantitative methods for predicting product reliability from data gathered from physical testing or from field data
Accelerated life testing plans are designed under multiple objective consideration, with the resulting Pareto optimal solutions classified and reduced using neural network and data envelopement analysis, respectively.
This seminar session provides an overview of major aspects of reliability engineering, including general introduction of reliability engineering (definition of reliability, function of reliability engineering, a brief history of reliability, etc.), reliability basics (metrics used in reliability, commonly-used probability distributions in reliability, bathtub curve, reliability demonstration test planning, confidence intervals, Bayesian statistics application in reliability, strength-stress interference theory, etc.), accelerated life testing (ALT) (types of ALT, Arrhenius model, inverse power law model, Eyring model, temperature-humidity model, etc.), reliability growth (reliability-based growth models, MTBF-based growth model, etc.), systems reliability & availability (reliability block diagram, non-repairable or repairable systems, reliability modeling of series systems, parallel systems, standby systems, and complex systems, load sharing reliability, reliability allocation, system availability, Monte Carlo simulation, etc.), and degradation-based reliability (introduction of degradation-based reliability, difference between traditional reliability and degradation-based reliability, etc.).
Authors: (i) Prashanth Lakshmi Narasimhan,
(ii) Mukesh Ravichandran
Industry: Automobile -Auto Ancillary Equipment ( Turbocharger)
This was presented after the completion of our 2 months internship at Turbo Energy Limited during our 3rd Year Summer holidays (2013)
This is a three parts lecture series. The parts will cover the basics and fundamentals of reliability engineering. Part 1 begins with introduction of reliability definition and other reliability characteristics and measurements. It will be followed by reliability calculation, estimation of failure rates and understanding of the implications of failure rates on system maintenance and replacements in Part 2. Then Part 3 will cover the most important and practical failure time distributions and how to obtain the parameters of the distributions and interpretations of these parameters. Hands-on computations of the failure rates and the estimation of the failure time distribution parameters will be conducted using standard Microsoft Excel.
Part 1. Reliability Definitions
1.Reliability---Time dependent characteristic
2.Failure rate
3.Mean Time to Failure
4.Availability
5.Mean residual life
How do you use the Weibull Distribution? It’s just one of many useful statistical distribution we have to master as reliability engineers. Let’s explore an array of distribution and the problems they can help solve in our day to day work.
Detailed Information: When confronted with a set of time to failure data, what is your goto analysis approach. For me it’s a Weibull plot. It’s quick, often provides some insight to ask better questions, and easy to explain to others. A histogram is another great starting point. If we know a little about the source of the data, we may favor the normal or lognormal distributions. If discreet data, then binomial is the first choice, yet Poisson or hypergeometric have uses, too. A basic understanding of statistical distributions provides you a way to summarize data providing insights to identify or solve problems. In this webinar we’ll explore a few distributions useful for reliability engineering work and talk about how to select a distribution, basics on interpreting distributions and just touch on judging if you have selected the right distribution.
This Accendo Reliability webinar originally broadcast on 14 April 2015.
Application of Survival Data Analysis- Introduction and Discussion (存活数据分析及应用- 简介和讨论), will give an overview of survival data analysis, including parametric and non-parametric approaches and proportional hazard model, providing a real life example of survival data-based field return analysis. Several common issues in survival data analysis will also be discussed.
Determining the right sample size for a reliability test is always challenging. If the sample size is too small, not enough failure information can be generated. If the sample is too large, cost and time probably will be wasted. In this presentation, we will discuss several commonly used methods on determining the right sample size for 1) reliability demonstration tests, 2) operational life tests under use condition, 3) accelerated life tests under elevated stresses. The theory behind these methods will be discussed first, and then examples of applying these methods will be provided using commercial software tools.
Estimation of Reliability Indices of Two Component Identical System in the Pr...IJLT EMAS
Progress in science & technology has made
engineering systems more powerful than ever. The intensity of
sophistication in high-tech industrial producers emerged with
reliability problems. Therefore the problem of reliability
continue to exist and more likely to require complex solutions.
Consequently, the field of reliability analysis and statistical
probability modeling of the systems and components were
growing. Ever since the theory of reliability was formally
recognized statistical and modeling of the components/ systems
analysis was used to develop various reliability measures that are
important to assess the system performance. In this research
paper, an attempt is made to find an approach of estimation
method, which could establish a formal estimation procedure to
estimate the reliability measures and also developed estimates of
the system reliability indices practically under the influence of
common cause shock failures as well as intrinsic failures. From
the results, it is seen that maximum likelihood approach used
was found useful in the estimation process to find estimate for
the reliability measures of the system, where small sample is
essential point of interest in the case of reliability analysis. The
estimates so derived using empirical procedure do possess the
property that MSE in each case is well within the prescribed
error, i.e. coincides even to the three decimal places are more.
Application of Lifetime Models in Maintenance (Case Study: Thermal Electricit...iosrjce
IOSR Journal of Mathematics(IOSR-JM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Application of Reliability Analysis for Predicting Failures in Cement Industrytheijes
This research entails the use of reliability analysis for predicting failures of machines used in the cement industries and was done by evaluating machine down times data. This research work was necessitated by the need to accurately predict failures of the machines used in the cement industries and come up with an effective planning, for preventive maintenance schedule and reducing down times through developed mathematical model for the machines. The failure frequency variation with time was determined and a regression analysis using least squares methods. Correlation was done to ascertain the suitability of linear regression of the data and also to determine that, the independent variable is a good predictor of the dependent variable. The reliability model of the machines was achieved by applying the down times and the regression analysis result of the machines studied for a period of six years to the Weibull model. Two critical components of the machines were identified; contributing a total of 55 % of the down time. It was concluded that the critical components indicate the trend of failure of the machines. Therefore, reducing the failure rate of these components will increase the useful life of the machines and the obtained failure ratemodel, could be used as an important tool for predicting future failures and hence, effectively planning against such failures.
A Comparative Reliability Analysis of Bulldozers arriving at workshop in East...IOSR Journals
Study of reliabilities of machinery used in any kind of production is of utmost necessity for optimum use of man power and resources to make the process cost effective and with minimum downtime. This is applicable for all large and small industries alike. But in small industries data is not accurately stored and it becomes difficult to estimate product reliabilities. This paper focuses on a case study to estimate the reliabilities of two competing machines, when the only available data is Time To Failure. The Weibull Parameters are calculated using Microsoft Excel 2010. The results show that after knowing the reliabilities of both the Bulldozers at different lengths of time, we can ascertain which of them is preferable to use at which time period.
Environmental Stress Screening (ESS) is performed on most of the Electrical/Electronic products. However Failure Rate/Time distribution analysis is not conducted always to evaluate the effectiveness of the Screening Process
Draft comparison of electronic reliability prediction methodologiesAccendo Reliability
A draft version of the paper that was eventually published as “J.A.Jones & J.A.Hayes, ”A comparison of electronic-reliability prediction models”, IEEE Transactions on reliability, June 1999, Volume 48, Number 2, pp 127-134”
Provide with the kind permission of the author, J.A.Jones
On Duty Cycle Concept in Reliability - Definitions, Pitfalls, and Clarifications
By Frank Sun, Ph.D.
Product Reliability Engineering
HGST, a Western Digital company
For ASQ Reliability Division Webinar
August 14, 2014
With the increase in global competition, more and more costumers consider reliability as one of their primary deciding factors, when purchasing new products. Several companies have invested in developing their own Design for Reliability (DFR) processes and roadmaps in order to be able to meet those requirements and compete in today’s market. This presentation will describe the DFR roadmap and how to effectively use it to ensure the success of the reliability program by focusing on the following DFR elements.
Improved QFN Reliability Process by John Ganjei. John will talk about the improvements in the reliability process in this webinar.
It is free to attend - see www.reliabilitycalendar.org/webinars/ to register for upcoming events.
GridMate - End to end testing is a critical piece to ensure quality and avoid...ThomasParaiso2
End to end testing is a critical piece to ensure quality and avoid regressions. In this session, we share our journey building an E2E testing pipeline for GridMate components (LWC and Aura) using Cypress, JSForce, FakerJS…
Epistemic Interaction - tuning interfaces to provide information for AI supportAlan Dix
Paper presented at SYNERGY workshop at AVI 2024, Genoa, Italy. 3rd June 2024
https://alandix.com/academic/papers/synergy2024-epistemic/
As machine learning integrates deeper into human-computer interactions, the concept of epistemic interaction emerges, aiming to refine these interactions to enhance system adaptability. This approach encourages minor, intentional adjustments in user behaviour to enrich the data available for system learning. This paper introduces epistemic interaction within the context of human-system communication, illustrating how deliberate interaction design can improve system understanding and adaptation. Through concrete examples, we demonstrate the potential of epistemic interaction to significantly advance human-computer interaction by leveraging intuitive human communication strategies to inform system design and functionality, offering a novel pathway for enriching user-system engagements.
Generative AI Deep Dive: Advancing from Proof of Concept to ProductionAggregage
Join Maher Hanafi, VP of Engineering at Betterworks, in this new session where he'll share a practical framework to transform Gen AI prototypes into impactful products! He'll delve into the complexities of data collection and management, model selection and optimization, and ensuring security, scalability, and responsible use.
Observability Concepts EVERY Developer Should Know -- DeveloperWeek Europe.pdfPaige Cruz
Monitoring and observability aren’t traditionally found in software curriculums and many of us cobble this knowledge together from whatever vendor or ecosystem we were first introduced to and whatever is a part of your current company’s observability stack.
While the dev and ops silo continues to crumble….many organizations still relegate monitoring & observability as the purview of ops, infra and SRE teams. This is a mistake - achieving a highly observable system requires collaboration up and down the stack.
I, a former op, would like to extend an invitation to all application developers to join the observability party will share these foundational concepts to build on:
zkStudyClub - Reef: Fast Succinct Non-Interactive Zero-Knowledge Regex ProofsAlex Pruden
This paper presents Reef, a system for generating publicly verifiable succinct non-interactive zero-knowledge proofs that a committed document matches or does not match a regular expression. We describe applications such as proving the strength of passwords, the provenance of email despite redactions, the validity of oblivious DNS queries, and the existence of mutations in DNA. Reef supports the Perl Compatible Regular Expression syntax, including wildcards, alternation, ranges, capture groups, Kleene star, negations, and lookarounds. Reef introduces a new type of automata, Skipping Alternating Finite Automata (SAFA), that skips irrelevant parts of a document when producing proofs without undermining soundness, and instantiates SAFA with a lookup argument. Our experimental evaluation confirms that Reef can generate proofs for documents with 32M characters; the proofs are small and cheap to verify (under a second).
Paper: https://eprint.iacr.org/2023/1886
GraphSummit Singapore | The Future of Agility: Supercharging Digital Transfor...Neo4j
Leonard Jayamohan, Partner & Generative AI Lead, Deloitte
This keynote will reveal how Deloitte leverages Neo4j’s graph power for groundbreaking digital twin solutions, achieving a staggering 100x performance boost. Discover the essential role knowledge graphs play in successful generative AI implementations. Plus, get an exclusive look at an innovative Neo4j + Generative AI solution Deloitte is developing in-house.
Communications Mining Series - Zero to Hero - Session 1DianaGray10
This session provides introduction to UiPath Communication Mining, importance and platform overview. You will acquire a good understand of the phases in Communication Mining as we go over the platform with you. Topics covered:
• Communication Mining Overview
• Why is it important?
• How can it help today’s business and the benefits
• Phases in Communication Mining
• Demo on Platform overview
• Q/A
Securing your Kubernetes cluster_ a step-by-step guide to success !KatiaHIMEUR1
Today, after several years of existence, an extremely active community and an ultra-dynamic ecosystem, Kubernetes has established itself as the de facto standard in container orchestration. Thanks to a wide range of managed services, it has never been so easy to set up a ready-to-use Kubernetes cluster.
However, this ease of use means that the subject of security in Kubernetes is often left for later, or even neglected. This exposes companies to significant risks.
In this talk, I'll show you step-by-step how to secure your Kubernetes cluster for greater peace of mind and reliability.
GraphRAG is All You need? LLM & Knowledge GraphGuy Korland
Guy Korland, CEO and Co-founder of FalkorDB, will review two articles on the integration of language models with knowledge graphs.
1. Unifying Large Language Models and Knowledge Graphs: A Roadmap.
https://arxiv.org/abs/2306.08302
2. Microsoft Research's GraphRAG paper and a review paper on various uses of knowledge graphs:
https://www.microsoft.com/en-us/research/blog/graphrag-unlocking-llm-discovery-on-narrative-private-data/
Enchancing adoption of Open Source Libraries. A case study on Albumentations.AIVladimir Iglovikov, Ph.D.
Presented by Vladimir Iglovikov:
- https://www.linkedin.com/in/iglovikov/
- https://x.com/viglovikov
- https://www.instagram.com/ternaus/
This presentation delves into the journey of Albumentations.ai, a highly successful open-source library for data augmentation.
Created out of a necessity for superior performance in Kaggle competitions, Albumentations has grown to become a widely used tool among data scientists and machine learning practitioners.
This case study covers various aspects, including:
People: The contributors and community that have supported Albumentations.
Metrics: The success indicators such as downloads, daily active users, GitHub stars, and financial contributions.
Challenges: The hurdles in monetizing open-source projects and measuring user engagement.
Development Practices: Best practices for creating, maintaining, and scaling open-source libraries, including code hygiene, CI/CD, and fast iteration.
Community Building: Strategies for making adoption easy, iterating quickly, and fostering a vibrant, engaged community.
Marketing: Both online and offline marketing tactics, focusing on real, impactful interactions and collaborations.
Mental Health: Maintaining balance and not feeling pressured by user demands.
Key insights include the importance of automation, making the adoption process seamless, and leveraging offline interactions for marketing. The presentation also emphasizes the need for continuous small improvements and building a friendly, inclusive community that contributes to the project's growth.
Vladimir Iglovikov brings his extensive experience as a Kaggle Grandmaster, ex-Staff ML Engineer at Lyft, sharing valuable lessons and practical advice for anyone looking to enhance the adoption of their open-source projects.
Explore more about Albumentations and join the community at:
GitHub: https://github.com/albumentations-team/albumentations
Website: https://albumentations.ai/
LinkedIn: https://www.linkedin.com/company/100504475
Twitter: https://x.com/albumentations
Encryption in Microsoft 365 - ExpertsLive Netherlands 2024Albert Hoitingh
In this session I delve into the encryption technology used in Microsoft 365 and Microsoft Purview. Including the concepts of Customer Key and Double Key Encryption.
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
3. Outlines
1/11/2014Webinar for ASQ Reliability Division3
Objectives
To provide an introduction to the statistical analysis of
failure time data
To discuss the impact of data censoring on data analysis
To demonstrate software tools for reliability data analysis
Organization
Reliability definition
Characteristics of reliability data
Statistical analysis of censored reliability data
4. Reliability
1/11/2014Webinar for ASQ Reliability Division4
Meeker and Escobar (1998) ‒ “Reliability is often
defined as the probability that a system, vehicle,
machine, device, and so on will perform its intended
function under operating conditions, for a specified
period of time.”
Condra (2001) ‒ “Reliability is quality over time.”
Leemis (1995) ‒ “The reliability of an item is the
probability that it will adequately perform its specified
purpose for a specified period of time under specified
environmental conditions.
5. Reliability Function
1/11/2014Webinar for ASQ Reliability Division5
The reliability function is the probability that an item
performs its function for a fixed period of time:
The time at which an item fails to perform its intended
function is called its failure time.
The failure time of an item is a continuous
nonnegative random variable, often denoted T
𝑹 𝒕 = 𝑷𝒓𝒐𝒃(𝐢𝐭𝐞𝐦 𝐩𝐞𝐫𝐟𝐨𝐫𝐦𝐬 𝐢𝐧𝐭𝐞𝐧𝐝𝐞𝐝 𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧 𝐮𝐧𝐝𝐞𝐫
𝐢𝐧𝐭𝐞𝐧𝐝𝐭𝐞𝐝 𝐜𝐨𝐧𝐝𝐢𝐭𝐢𝐨𝐧 𝐟𝐨𝐫 𝐚𝐭 𝐥𝐞𝐚𝐬𝐭 𝐭 𝐭𝐢𝐦𝐞 𝐮𝐧𝐢𝐭𝐬)
)Pr()( tTtR )Pr()(1)( tTtRtF
6. Understanding Hazard Function
1/11/2014Webinar for ASQ Reliability Division6
Reliability function
Define a hazard function
Instantaneous failure
Is a function of time
Only exponential distribution has constant hazard (failure rate)
Relationships between reliability function and hazard function
t
tTttTt
th t
)|(Pr
lim)( 0
)(
)(
)(
tR
tf
th
t
dxxhtH
0
)()( )(
)( tH
etR
)(1)( tRtF
dt
tdF
tf
)(
)(
)Pr()( tTtR
7. Characteristics of Reliability Data
1/11/2014Webinar for ASQ Reliability Division7
Failure time censoring
Right censoring
Left censoring
Interval censoring
Data from reliability tests
Type-I censoring (time censoring)
Type-II censoring (failure censoring)
Read-outs (multiple censoring)
8. Right Censoring
1/11/2014Webinar for ASQ Reliability Division8
Actual failure time exceeds
observation
In life tests
Type-I censoring (time
censoring)
Type-II censoring (failure
censoring)
9. Example
1/11/2014Webinar for ASQ Reliability Division9
Low-cycle fatigue test of nickel super alloy (Meeker &
Escobar (1998), p. 638, attr. Nelson (1990), p. 272)
kCycles Censor
211.626 0
200.027 0
57.923 1
155 0
13.949 0
112.968 1
152.68 0
156.725 0
138.114 1
56.723 0
121.075 0
122.372 1
112.002 0
43.331 0
12.076 0
13.181 0
18.067 0
21.3 0
15.616 0
13.03 0
8.489 0
12.434 0
9.75 0
11.865 0
6.705 0
5.733 0
10. Left Censoring
1/11/2014Webinar for ASQ Reliability Division10
Actual life time less than observation
May occur when the item is inspected at a fixed
time point
Less often than
right censoring
11. Example
1/11/2014Webinar for ASQ Reliability Division11
In the nickel super alloy example, suppose that the
observation starts only after 10,000 cycles.
kCycles start kCycles end
211.626 211.626
200.027 200.027
57.923 *
155 155
13.949 13.949
112.968 *
152.68 152.68
156.725 156.725
138.114 *
56.723 56.723
121.075 121.075
122.372 *
112.002 112.002
43.331 43.331
12.076 12.076
13.181 13.181
18.067 18.067
21.3 21.3
15.616 15.616
13.03 13.03
8.489 8.489
12.434 12.434
* 10
11.865 11.865
* 10
* 10
12. Interval Censoring
1/11/2014Webinar for ASQ Reliability Division12
Observation gives upper and lower bound on failure
time
Occurs often with scheduled inspections
Right censoring and left censoring are special cases
Read-outs
Grouped data
13. Example
1/11/2014Webinar for ASQ Reliability Division13
In the nickel super alloy example, suppose that the test
units are inspected at 25, 50, 100, 200 kCycles.
kCycles start kCycles end read-outs
0 25 12
25 50 2
50 100 2
100 200 8
200 * 2
14. Multiple Censored Data
1/11/2014Webinar for ASQ Reliability Division14
More than one censoring mechanisms are
employed
Exact failure times, right censoring times and
interval censoring times are very common in
practice
May not be easily recognized
Calendar time vs. lifetime
15. Example
1/11/2014Webinar for ASQ Reliability Division15
Adapted from an example in Meeker & Escobar (1998), p. 8.
Nuclear power plant use heat exchangers to transfer energy from the
reactor to stream turbines. A typical heat exchanger contains
thousands of tubes. With age, heat exchanger tubes develop cracks.
Suppose there are three plants. Plant 1 had been in operation for 3
years, Plant 2 for 2 years, and Plant 3 for only 1 year. All heat
exchangers are of the same design and operated under similar
conditions. At the beginning, each plant has 100 new tubes. Failed
tubes will be removed from the heat exchanger during operation.
Plant 1: 1 failure in the first year, 2 failures in the second year, and 2
failures in the third year.
Plant 2: 2 failures in the first year, 3 failures in the second year.
Plant 3: 1 failure in the first year.
17. Example (cont.)
1/11/2014Webinar for ASQ Reliability Division17
initial year 1 year 2 year 3
Plant 1 100 1 2 2
Plant 2 100 2 3
Plant 3 100 1
start year end year at risk removed failed
0 1 300 99 4
1 2 197 95 5
2 3 97 2
start year end year readouts
* 1 4
1 * 99
1 2 5
2 * 95
2 3 2
3 * 95
Data by group Data by age
Data by censoring type
18. Simulation
1/11/2014Webinar for ASQ Reliability Division18
Monte Carlo method
Assume a probabilistic model
Generate random numbers
Compute the statistics of interest
Repeat it many times
Estimate confidence intervals
Useful for evaluating and comparing data
analysis methods
Useful for evaluating and comparing life test
plans
19. Simulate Censored Failure
Time
1/11/2014Webinar for ASQ Reliability Division19
In MS Excel®
Many build-in functions for generated random numbers from
specific distribution, such as NORMINV(p, mu, sigma)
Utilize GAMMAINV(p, alpha, beta) to generate exponentially
distributed failure times
Set p=rand(), alpha=1, beta=mean failure time
Utilize NORMAINV(p, mu, sigma) to generate the failure times with
lognormal distribution
Set p=rand(), mu and sigma are the parameters of the lognormal
distribution
Compute exp(normal random number)
No build-in inverse function for Weibull distribution
Use function [(-ln(1-rand()))/a]^(1/b)
Where a is the intrinsic failure rate of Weibull distribution, b is a shape
parameter
Use If() function to create censored failure times
20. Features of Lifetime
Distribution
1/11/2014Webinar for ASQ Reliability Division20
Failure data from electrical appliance test (Lawless, p.7.
Attr. Nelson (1970))
Variable: cycles to failure (exact failure time)
Nonnegative
Right (positively) skewed
Some long life observations
Normal distribution may not be a good idea!
21. Exponential Distribution
1/11/2014Webinar for ASQ Reliability Division21
The simplest lifetime distribution
One parameter
or
Constant failure rate (constant mean-time-to-
failure, MTTF)
Memoryless property
Regardless of past experience, the chance of failure
in future is the same.
Closure property
System’s failure time is still exponential, if its
components’ failure times are exponential and they
are in a series configuration.
)exp()|( ttf )/exp(/1)|( ttf
22. Weibull Distribution
1/11/2014Webinar for ASQ Reliability Division22
When the hazard function is a power function of time
Two common forms
Two parameters
Either characteristic life or intrinsic failure rate and shape
parameter
Relationship with exponential distribution
When the shape parameter is known
1
)(
t
th
tt
tf exp)(
1
t
tR exp)(
1
)(
tth
ttR exp)(
tttf
exp)( 1
/1
23. Rectification
Plot failure probability on a complementary log-log scale
Plot time on a log scale
Some important features on the plot
Slope is the shape parameter
Characteristic life can be found at where the failure
probability is (1-1/e)=0.632
Reliability at a given lifetime depends on distribution
parameters, except at the characteristic life
1/11/2014Webinar for ASQ Reliability Division23
Weibull Plot
)log(log))](1log(log[ ttF
25. Lognormal Distribution
1/11/2014Webinar for ASQ Reliability Division25
From normal to lognormal and vice versa
If T has a lognormal distribution, then log(T) has a
normal distribution
If X has a normal distribution, then exp(X) has a
lognormal distribution
Median failure time
Log(t50) is a robust estimate of the scale parameter
of lognormal distribution
26. Parametric Distribution Models
1/11/2014Webinar for ASQ Reliability Division26
Maximum likelihood estimation (MLE)
Likelihood function
Find the parameter estimate such that the chance of having such failure
time data is maximized
Contribution from each observation to likelihood function
Exact failure time
Failure density function
Right censored observation
Reliability function
Left censored observation
Failure function
Interval censored observation
Difference of failure functions
)(tR
)(tF
)()(
tFtF
)(tf
27. Exponential Distribution
1/11/2014Webinar for ASQ Reliability Division27
Exact failure times
Failure density function
Likelihood function
Failure rate estimate
Type-I censoring
Reliability function
Likelihood function
Failure rate estimate
it
i etf
)(
n
i itn
n etttL 1
),...,,;( 21
n
i it
n
1
ˆ
ct
c etR
)(
c
r
i i trntr
ccr etttttL
)(
21
1
),...,,...,,;(
r
i ci trnt
r
1
)(
ˆ
28. Exponential Distribution (cont.)
1/11/2014Webinar for ASQ Reliability Division28
Type-II censoring
Likelihood function
Failure rate estimate
General formula for Exponential failure times
r
r
i i trntr
rrr etttttL
)(
121
1
),...,,...,,;(
r
i ri trnt
r
1
)(
ˆ
timetestingtotal
failuresofnumber
RateFailure
failuresofnumber
timetestingtotal
MTTF
29. Precision of Failure Rate
Estimate
1/11/2014Webinar for ASQ Reliability Division29
Sum of exponential distributions becomes gamma
distribution
Independently identical distributed (i.i.d.)
Exact confidence intervals for the cases of exact failure
time and type-II censoring
An approximated confidence interval for the case of type-
I censoring
n
i
i ngammaT
1
),(~
TTTTTT
rr
2
,
2
2
2/1,2
2
2/,2
)ˆ.(.ˆ),ˆ.(.ˆ
2/2/ eszesz
r
es
2ˆ
)ˆ.(.
31. Final Remarks
1/11/2014Webinar for ASQ Reliability Division31
Be aware of censoring when analyzing reliability data
Ignoring censored data will bias failure(reliability) estimates
Often underestimate reliability
The amount of information of censored data depends on the
censoring type
Nonparametric methods are based on ranks
Often utilize the ratio of number of failures and number of items at risk
Parametric methods are based on likelihood functions
Maximum likelihood estimation
Computation becomes complicated
Use software
Simulation is a very useful tool for studying the effect of
sample size or censoring