Introduction to Reliability Engineering

Lecture 4: Basic Probability and Statistics Background

Life Distributions

Review

� Random Variable

� Probability Density Function (pdf)

Review

� Cumulative Distribution Function (CDF)

� Reliability Function

Review

� Design Life

� Median of continuous distribution

� MTTF

� Variance and Standard Deviation

Review

� Hazard function

� Relationship between pdf, cdf, reliability function and hazard function for continuous rv X

Hazard Function and Reliability Function

� A particular hazard function will uniquely determine a reliability function

� �(�) = � (�)� (�) =

� �(�) � ⋅ "

� (�)

Example

� Given the hazard rate function � � = 5 × 10�)�where t is measured in operating hours, what is the design life if a 0.98 reliability is desired?

Hazard Function

� IFR: increasing failure rate– Let X be a rv with cdf F and hazard rate h. X has an

increasing failure rate (IFR) or, equivalently, F is an IFR distribution if h(x) is weakly increasing for all x such that F(x) < 1.

� CFR: constant failure rate

� DFR: decreasing failure rate

� An important form of hazard function

The Bathtub Curve

X XI II III

t1 t2 t3t0 Time (t)

I: Burn-in or Infant Mortality ZoneII: Useful Life ZoneIII: Wearout Zone

Haz

ard

Rat

e (λ) Overall Life Characteristic Curve

Stress Related FailuresQuality Related Failures

Wearout Failures

More on the Bathtub Curve

Hazard rate Cause by Reduced by

Burn-in DFR Manufacturing defects, Welding flaws, Cracks, Defective parts, Poor quality control, Contamination, Poorworkmanship

Burn-in testing, Screening,Quality control, Acceptance testing

Useful life CFR Environment, Random loads, Human error, Chance events

Redundancy, Excess strength

Wearout IFR Fatigue, Corrosion, Aging, Friction, Cyclical loading

Derating, Preventive maintenance,Parts replacement

Conditional Reliability

� Useful in describing the reliability of a component or system following a burn-in period 23 or after a warranty period 23

� Probability that the system survives for another t periods after 23 is reached

Example

� Let � � = 3.4"333

�"333

�3.4, where t is in year. Is the

hazard IFR, CFR or DFR?

� Find the design life if the reliability of 0.9 is desired.

� Next, consider that the system is functional after 6 months. What is the new design life (starting from month 6) for this system if the reliability of 0.9 is desired?

Example

� � � = 3.4"333

�"333

�3.4; The design life if the

reliability of 0.9 is desired?

Example

� The system is functional after 6 months. The design life to achieve reliability 0.9 starting month 6?

� Compare ___ yrs with ___ yrs

Life Data Analysis

� The study and modeling of observed product lives

� Life Data Analysis is commonly referred to as “Weibull Analysis”

� The objective is to make predictions about the life of all products in the population by fitting a statistical distribution (model) to life data from a representative sample of units

Life Data Analysis

� Reliability prediction relies on life data (time-to-failure)

� Prediction accuracy is directly affected by the quality and completeness of the data used

� Population: – EVERY data point that has ever been or ever will be

generated from a given characteristic

� Sample: – A portion (or subset) of the population, either at one time

or over time

Why do we sample?

Life Data Analysis

� Steps to perform Life Data Analysis:

1. Collect life data for the unit under consideration

2. Select the lifetime distribution that best fits the data

3. Estimate the distribution parameters

4. Generate plots and estimate life charecteristcis (e.g. MTTF, R(t))

Types of Data: Alternative Terminology

� Attribute Data– Discrete data

– Finite or countable

– How many/how often/what kind?

� Variable Data– Continuous data

– Infinite

– Data value could be any real number (e.g., 5.013 inches)

– How long/what volume/how much time?

Types of Data: Alternative Terminology

� What data type is:

– # of units with defects

– Current reading (amps) for an electric motor during final test

– Time-to-failure

Life Data Classification

� Complete Data

– The time-to-failure values are known for all samples

� Example:

Life Data Classification

� Incomplete Data

– The time-to-failure values are not known for all samples

• Some of the samples didn’t fail

• Exact time-to-failure is not known

– Right Censored Data

– Interval Censored Data

– Left Censored Data

– Singly-censored

– Multiply-censored

Life Data Classification

� Right Censored Data

– Most common case of Incomplete Data

– The time-to-failure values are known for some of the samples

� Example

Life Data Classification

� Interval Censored Data

– Exact time-to-failure is unknown

– Also called inspection data

� Example

Life Data Classification

� Left Censored Data

– Time-to-failure is only known to be before a certain time

� Example

Life Data Classification

Sample Time to Failure (hrs) Failure Mode

1 100 A

2 75 B

3 122 C

4 105 A

5 118 C

6 93 A

7 78 B

8 69 B

9 124 C

10 80 B

Life Data Classification

� What is your objective?

– Analyze the time-to-failure regardless of the failure mode

Sample 1

Sample 2

Sample 3

Sample 4

Sample 5

Sample 6

Sample 7

Sample 8

Sample 9

Sample 10

20 40 60 80 100 120

Life Data Classification

� What is your objective?

– Analyze the time-to-failure for a specific failure mode (e.g. A)

Sample 1

Sample 2

Sample 3

Sample 4

Sample 5

Sample 6

Sample 7

Sample 8

Sample 9

Sample 10

20 40 60 80 100 120

Life Data Analysis

� Probability Density Function f(t)– Total area under curve is 1

� Cumulative Density Function F(t) – Integral of f(t) from –∞ to t

� Reliability Function R(t) – Integral of f(t) from t to ∞

� Hazard Function h(t)

– �(�) = � (�)� (�)

Life Distributions

� Most commonly used distributions:– Exponential

• Constant failure rate, due to purely random events

– Weibull• Model material strength or times-to-failure of electronic and

mechanical components, equipment or systems

– Normal• Used for simple electronic and mechanical components

– Lognormal• Used when failure modes are of a fatigue-stress nature

Parameter Types

� These distributions are parametric models or parametric families, i.e., they have a finite number of parameters

� Usually, life distributions are limited to the following 3 paramaters:

– Location

– Scale

– Shape

Parameter Types

� Location parameter– Used to shift the distribution in one direction or the

other

� Scale parameter– Defines where the majority of the distribution lies or

how much spread it has

� Shape parameter– Defines the shape of the distribution– Some distributions don’t have shape parameter (e.g.

normal distribution) as they have a predefined shape

Weibull Distribution

� Weibull distribution is one of the most widely used distributions in Reliability

� Used to model:– Material strength

– Times-to-failure of electronic and mechanical components, equipment or systems

� Two common versions of Weibull distribution– 2-Parameter Weibull (β = shape parameter; η = scale

parameter)

– 3-Parameter Weibull (minimum life �0 = location parameter)

Weibull Distribution

� Density function:

5 � = 67

�7

6�" ⋅ e� 89

:, ; > 0, = > 0, � ≥ 0

� Reliability function: R � = e� 89

:

� Hazard function: � � =– Behavior of � � ?

Example

� Time to failure for a capacitor follows a weibull distribution with β = 2 and η = 300 months.

� What is the reliability at 200 months?

� What is the probability of failure in 200 months:

� What is design life if the reliability of 0.9 is desired?

Weibull Distribution

� Gamma function (you may use tables)

Γ A = B CD�"E�FGCH

3� What’s Γ 1 ?

� Note that Γ A = A − 1 Γ x − 1 . What’s Γ n , K ∈ ℤN?

� MTTF: E T = ; ⋅ Γ(1 + "R)

� Variance: ST = ;T ⋅ Γ 1 + TR − Γ 1 + "

RT

QUESTIONS?