1

1

Introduction to Reliability

• Reliability is:

– An inherent feature of design

– Concerned with performance in the field, as opposed toquality of production (conformance to design specs)

• Definition

– Reliability is the probability that a system will performin a satisfactory manner for a given period of timewhen used under specified operating conditions.

2

Introduction to Reliability (cont)

• What is Satisfactory ?

– All critical functions

– Time-oriented quantitative factors--MTBFP (X>to), with X = Lifetime

– Qualitative factors, too

• Operating Conditions

– Use

– Handling, Transport, Installation, Storage

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3

Reliability in the System Life-Cycle

• Conceptual Design Phase– Define reliability requirements of a system

– Plan Reliability Program

• Preliminary Design Phase– Allocate reliability requirements

– Predict reliability of components/subsystems

– Provide reliability estimates to cost estimating and designtrade-off studies

– Participate in design reviews

– Assess subsystem/ component supplier reliability estimates

4

Reliability in the System Life-Cycle(cont)

• Detail Design Phase

– More detailed reliability prediction

– Assist in detail design decisions

– Assist in logistic support analysis

– Assist in prototype development

– Recommend changes prior to production

– Evaluate reliability of prototype

– Participate in other test and evaluation activities asrelated to reliability

3

5

Reliability in the System Life-Cycle(cont)

• Production/ Construction Phase

– Monitor production

– Perform reliability tests of selected items

• Qualification Tests -Prior to production, repetitive tests todetermine MTBF, degradation, failure modes

• Acceptance Tests- Random or 100%, testing of itemsexiting production to assure that reliability demonstratedduring qualifying-test is being achieved in production items.

– Collect and analyze data on operational test (product evaluationtests at a designated site)

– Recommend Corrective action

– Continue to update reliability models and predictions

6

Reliability in the System Life-Cycle(cont)

• System Use Phase

– Data collection and analysis

– Reliability improvement studies

– Change recommendations

– Equipment redesign projects

4

7

Measures of Reliability

• Let T = Random Variable Measuring “Lifetime” of an item(time to first - next - failure)

• Range Space of T={t:t 0}

• Tests to establish PDF & Parameters of T are called “LifeTesting”

• Cum, Distribution Function F(t)=P(T t) is called theFailure Distribution Function

≥

≥

≤

8

Measures of Reliability(cont)• The Reliability Function is:

– R(t)=P(T>t)=1-F(t)=

• Four ways to determine R(t) for a particular system– Test many systems to failure. Develop curve empirically.

– Test many subsystems, use historical field data on others,develop subsystem reliability functions, use a reliability systemmodel to combine.

– Extrapolate past experience with similar systems.

– Physical properties--Hypothesize a certain distribution.

f t dtt

( )∞

∫

0

1

Prob

t

F (t)

R (t)

Reliability Density Function

5

9

Failures and Failure Rates

• 3 Types of Failure (See Figure 12.4)

– Initial ( Failure at t=0)

– Random

– Wearout

• IF initial failures are to be disregarded in your analysis,then use, , t>0; as density for

[ ])0(1)(

)(=−

=TPtg

tf ( )0>TT

10

Failures and Failure Rates(cont)

• The Hazard Function is the instantaneous failure rate attime t, given survival up to t has formula:

• Note: H(t)= number failures in [0,t]= is calledthe failure count function

)(

)(

)(

)()(

tR

tf

tR

tRth =

′−=

h x dxt

( )0∫

6

11

Failures and Failure Rates(cont)

• How are H(t),R(t),F(t) Related?

• So, R(t)=

H tR x

R xdx R x R t Re

tt

e e( )( )( )

log ( ) log ( ) log ( )]= − ′ = − = − +∫ 0

00

)(tHe−

0

12

Mean Lifetime (Time BetweenFailures)

• Mean Life = θ ≡ E(T)=

• Example:

– Random failures often are modeled by time-to-failure isexponential with rate λ:

, or

1- F(t)0

tf t dt

dt R t dt

( )

( )

0

0

∞

∞ ∞

∫

∫ ∫[ ] =

f t e t F t e

R t e

t t

t

( ) , ( )

( )

= ≥ = −

=

− −

−

λ λ λ

λ

= 0 otherwise

0 1

7

13

Example (cont)

• Then, Constant

• Also, because , H(t)=λt Linear in t and

θ=E(T)=

• P(T< θ)=F(θ)=

• P(T≥ θ)=.3679 , Independent of λ (or θ)

λλλ

λ

=== −

−

t

t

ee

tRtf

th)()(

)(

)()( tHetR −=

λ1

6321.3679.11

111 1 =−=−=−=− −−

eee λθ

14

Examples on Pages 349

• Example 1

– 5 Components did not fail in 600 hours

– 5 Others failed at various points

• Example 2

– Operating Cycle = 168.8 hours

– Downtime = 26.8 hours

– Operating Time = 142

001196.0hours 4180

failures 5 ==λ

8

15

Examples on Pages 352-353(cont)

• Number of failures = 6

• λ = 6 / 142 = 0.042

• MTBF = 23.81 hours = 1 / λ

• Operational Availability =841.0

4666.481.2381.23 =

+=

+MDTMTBMMTBM

Only if we treatMTBF = MTBM(instant maintenance)

Other examples are on handoutsHines and Montgomery, example 15-7Halpern, examples 10-1 thru 10-6

Note: For exponential failure module is the first Term in a poisson distribution with parameter x.

R(t) = e- tλ

16

What if Failure Rate Not Constant?

• Distribution Failure Rate h(t) Behavior Normal Increasing Function

• Lognormal Various Shapes

• Weibull Decreasing β<1

Constant β =1

Increasing β>1

• Gamma Decreasing n<1 Constant n =1 Increasing n>1

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17

What if Failure Rate NotConstant(cont)

• Have different h(t) for each time interval where rate isconstant

• use average failure rate (AFR) between t1 and t2

AFR t t

h t dt

t t

H t H t

t t

R t R t

t tt

t

( , )

( )( ) ( ) ln ( ) ln ( )

1 22 1

2 1

2 1

1 2

2 1

1

2

=−

= −−

= −−

∫

Note: AFR (0, t) = H (t)

t =

- ln R(t)t

18

Concepts Our Text Skips

• Renewal Rate Function r(t) = Instantaneous failure rate at time Taccounting for replacement of failed items with new componentsfrom same population as original parts

• Censored Type I Data : A fixed test duration T is pre-set. Unitsthat do not fail before T are “censored” in that the data doesn’taccount for their survival beyond time T. If T is poorly chosen,may get no failures by time T--then what?

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19

Concepts Our Text Skips (cont)

• Censored Type II Data : A fixed number of failures is pre-specified, n items are tested until r fail. If r is poorlychosen, test make take too long.

• Readout Time Data : Record actual failure times of eachfailed component

20

Estimation of λ for Exponential Life

• λ = (number of failures) / (total unit test hours)

• Type I Censored Data

– n items, r failures

• Type II Censored Data

• If system has n components and system fails when firstcomponent fails

λ =+ −

=∑

r

t n r Tii

r

( )1

λ =+ −

=∑

r

t n r ti ri

r

( )1

λ λs ii

n

= ∑=

1

ti = time of i failureth

(ends at r failure time t )thr

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21

System Reliability Models

• Defined: Math models of the system that show functionalrelationships among subsystems, components, etc.

• Examples

– Reliability block diagram

• Shows all possible success/failure combinations

• Series and parallel; also k-out-of-n configuration

• Any closed path through system is success

• May not resemble system physically

• Standby redundancy

22

System Reliability Models(cont)

• Coherent systems models

• Fault tree analysis and other cause-consequence diagrams

– Work from top level events (failures)

– To primary events ( causes)

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23

Series Configuration

• Static Model:

• Dynamic Model:

R R R R Rs ii

n

n= ==

∏ 1

1 2* * ...

R t R t

h t h t

H t H t

s ii

n

s ii

n

s ii

n

( ) ( )

( ) ( )

( ) ( )

=

=

=

=

=

=

∏

∑

∑

1

1

1

1 2 n

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Example

• Exponential Subsystem Failure Models

R t e

h t

MTBF

st

s ii

n

ii

n

n( )

( )

...=

=

= =

− + + +( )

=

=

∑

∑

λ λ λ

λ

θλ

1 2

1

1

1

Constant

See example on page 354

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25

Active Parallel Configuration

• Static:

• Dynamic:

• Identical Components:

R Ra ii

n

= − −( )=

∏1 11

R t R ta ii

n

( ) ( )= − −( )=

∏1 11

R t R tan( ) ( )= − −[ ]1 1

1

2

n

System fails onlyif all n subsystemsfail

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Example 1

• Always Keep in Mind “Redundancy Has a Cost”

# of Components in Parallel R Wt. Benefit/ Cost

1 0.95 5 lb -

2 0.9975 10 lb .0475 / 5 lbs

3 0.999875 15 lb .002375 / 5 lbs

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27

Example 2

• Exponential Subsystem Lifetime, Identical Subsystems

R t eat n( ) = − −[ ]−1 1 λ

θλ

θa a

i

n

i

n

R t dti i

= = ===

∞

∑∑∫ ( )*1

110

e. g., if n = 3 and θλ

θ

= 1

= 1000 hours

= 1000

1 +

10002

+ 1000

3 = 1000 + 500 + 333.33 = 1833.33

a

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Special Configurations

• K-out-of-n Configuration

– Systems works only if at least k of n components areworking. Assume identical components with reliabilityR(t):

• If

R tn

iR t R ts

i k

nn i( ) ( ) ( ) ( )= [ ] −[ ]

=

−∑ i 1

R T e ei

tt

i k

n

( ) = = =−−

=∑λ θ θ θ

exponential, then s

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29

Special Configurations (cont)

• Combined Series-Parallel

– Key:Treat Components in parallel as single component,then expand

( )( )[ ]CBaBUCas RRRRRR −−−== 111*

( )( )[ ] ( )( )[ ]DCB

CUDAUBs

RRR

RRR

−−−−==

1111R-1-1

*

A

See pages 354 - 355

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Availability Measurement

• Inherent Availability (Ideal Support Environment)

• Does not include preventive maintenance, logistics delay, oradministrative delay.

• Achieved Availability ( Ideal Support Environment)

– MTBM = mean time between any maintenance action, corrective or preventive

cti MMTBF

MTBFA

+=

MMTBM

MTBMAa +

=M = mean active maintenance time

= weighted average of correctiveand preventive maintenance time.

Mct = mean corrective maintenance time

= mean time to repair (MTTR)

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31

Availability Measurement

Operational Availability ( Actual Support Environment)

MDTMTBMMTBM

A+

=σ

MDT = mean downtime = weighted average of activemaintenance (current and previous) and delays (logistical andadministrative.

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Comments on Availability

• Availability is a function of both:

– Reliability of a prime item

– The logistics support subsystem

• Equipment designer can exert little control over supportoperations, but can design in:

– Built-in diagnostics

– Easy access

– Rapid disconnect / connect

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33

Comments on Availability (cont)

• The proper balance of R&M must be decided in earlystages, when flexibility is great.

• Discussion of availability is always in some context:

– Actual failure or not

– Which mission, what is critical to success

– Maintenance crew, equipment, spares availability

34

Reliability Techniques in SystemDesign Phase

• Conceptual Design Phase

– Assignment of system reliability goal based on:

• Mission analysis

• Cost analysis

• Technical Limits

• Preliminary Design:

– Block Diagram Models

– Estimation of Ri(t) Functions

– Study of failure points, solutions

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35

Reliability Techniques in SystemDesign Phase(cont)

• Preliminary Design Phase (Cont.)

– Definition of Success/ Failure criteria

– Budgeting/ Revision of Reliability Requirements

• Detail Design:

– Material and Parts Selection

– Standardization

– Test and Evaluation

– Requirements for Suppliers

– Series-Parallel Recommendations

– De-rating

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Standardization

• Standardization:

– Means selection of components and materials whosereliability characteristics are known, as well as theirdegradation under stress and aging. This indirectlyeases the burden on spare parts inventories, by havingsame component used in several systems

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37

De-rating

• De-rating:- Use part in application below its rated value

– A type of overdesign to provide reliability margin

• Steps:

– Identify operating interval

– Select de-rating % ( see RCA Corp. Table)

– Calculate de-rated value of component to be used

• Example: ceramic capacitor for 100v (max) application- RCA recommends 70% de-rating- X (0.7) = 100, X = 142.85 v minimum requirement forcomponent

38

Binomial Expansion to ExplainParallel-Redundant Systems

• Consider 3 Identical Components in Parallel

– P = Probability of Operation of Each

– Q = Probability of Failure of Each

(P + Q) = P +

= P

3 3

3

3

1

3

2

3

3

3 3

2 1 2 0 3

2 1 2 3

+

+

+ + +

P Q P Q P Q

P Q P Q Q

All 3up

2 up,one failed

One up,two failed

All 3down

P (System operating) 1 1 1

3 3

3 3

3 2 1 2

− = − −

+ +

Q P

P P Q P Q

( )

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39

• Let PA=PB=PC=PD = 0.9Which configuration is more reliable? Why?

Binomial Expansion to ExplainParallel-Redundant Systems

A B

DC

A B

DC

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Parallel Redundancy Has ItsDrawbacks

• Limitations• Each subsystem must have a “switch” to assure its failure doesn’t

disable the remaining components

• Sometimes necessary to “disconnect” failed system

• Redundancy increases weight, volume, cost and sometimescomplexity. The failure sensing device may be unreliable

• Alternatives to Redundancy• Reduce number of parts

• Simplify

• Improve reliability level of parts used, especially at critical“nodes”

• Burn-in of Parts

• On-board spares, repairs

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41

Standby Redundancy

• Assume “cold” standby, not energized until failuredetected in original component

• Assume reliability of “decision switch” is 100%

• Lifetime variable is T=T1+…+Tn

• Standby always more reliable than simple parallel, ifswitch is 100% Reliable

DS

1

2

n

42

• Assume lifetime variable is as follows:

Standby Redundancy (Cont,)

T = T T + - - - - + T

T each exponential, is gamma ( , n)

E (T) = n

= n

1 2 n

i

2

+

=

=

∑∑

E T E T

V T V T

If t

V T

i

i

( ) ( )

( ) ( )

( )

λ

λ

λ

n = 2 R(t) = P(system life > t / one standby) = e

n = 3 R(t) = P(system life > t / two standbys) = e

- t

- t

λ λ

λ λ λ

λ

λ λ

+

+ +

−

− −

( )

( )( )

!

t e

t et

e

t

t t2

2

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43

Benefits of Computerized Reliability Models

• Helps keep track of reliability relationships

– Across levels of design– Within a given level

• Rapid Sensitivity Analysis

– Is overall R goal even feasible

– Study effect of different R allocations

– Study effects of configuration changes on R

– Study effects of substituting different components

– Perform “worst-case” analysis

• Can be adapted to multiple missions -in essence, one model for each set ofmission equipment/conditions

• Can be used to evaluate proposed modifications to existing system

44

Analytical Methods to Support Reliability Estimation andAssist in Design Decisions

• Stress-Strength Analysis

• Critical-Useful-Life Analysis

• For Complex Systems ( radar, missiles, computers)

– Failure Mode and Effect Analysis

– Worst-Case Analysis

– Sneak-Circuit Analysis

• Safety Analysis Techniques

– Fault-Tree Analysis

– Task and Error Analysis

– Hazard Analysis

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45

Discussion of Stress-StrengthAnalysis

• Measures Resistance to Stress (strength)

• Examples: operating wattage versus rated wattageOperating temperature vrs rated temperaturepounds/square inch

• Includes:

– Stress distribution, especially maximum stress

– Stress causes, timing, frequency

– Stress testing, such as metal fatigue tests

46

Discussion of Critical-Useful-LifeAnalysis

• Critical-Useful-Life Analysis:

– Identification of critical item list and requirements ofeach of these items for a preventive maintenance,corrective maintenance, and replacement.

Includes studies of how to eliminate critical itemsthrough redesign

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47

Discussion of FMEA

• Failure Mode and Effect Analysis:

– Identification of all possible failure modes ofequipment, the possible causes and the possibleimmediate/ ultimate effects on the system andoperation

• Formal documentation in words not diagrams

• Estimation of probability of occurrence

• Classify each failure by criticality

• Describe corrective action alternatives

48

Discussion of Worst-Case Analysis

• Worst Case Analysis:

– Examining how the performance of an electrical circuit(or other device) will change over time as a result ofdrift in part characteristics. Provides guidance on howto allow for part parameter variation in design

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49

Discussion of Sneak-Circuit Analysis

• Sneak-Circuit Analysis:

– Use of math models to identify any unanticipatedperformance signal paths in a circuit that may degradeperformance or introduce failure.

50

Reliability Prediction at Part, Circuit,and Subsystem Level

• Based On:– Similar equipment--Extrapolate. Not very accurate.

– Number and complexity of “active element groups”--these are controllers orconverters of energy

– part types, counts, failure rates are combined into an estimate of systemreliability

– Prediction based on testing, such as stress tests

• Used For:– Higher-level reliability prediction

– As input to maintenance and logistic support analysis

– Comparison with requirement, where are we over/ under reliability

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51

Reliability Degradation Studies/Action

• Determine and correct potential/ actual adverse effects dueto:

– Storage, packing, transportation, handling

– Unpacking, assembly, set-up

– Preventive and corrective maintenance

• Carelessness

• Wrong tools and equipment

• Didn’t follow/ know proper procedure

52

Reliability Test and Evaluation

• To answer question : “will the mature system achieve itsMTBF requirement in operation ?”

• Should be part of an integrated test plan to test entire spec.

• Type I Tests:– are early enough in design process so that design changes are fairly cheap

• Type II and III Tests must :– Follow approved procedures ( first drafts of tech manuals and training

courses)

– Use test and support equipment that was specified in the maintenanceconcept and detailed in LSA

– Be provided with ( test ) supply support

– Be carefully planned, instrumented, documented, analyzed

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53

Type II Reliability Testing

• Evaluation of prototype and early production models, usingproducer personnel

• Includes:

– Reliability qualification tests, to determine

• MTBF

• MTBM

• Failure sequences, detection, performance degradation

• Maintenance procedure adequacy

• Maintenance induced failures

– Production sampling acceptance tests

54

Types of Type 2 Tests

• Sequential Qualification Tests

– Environmental test chambers

– Environmental test cycle, equipment duty cycle

– Multiple identical test items

– Statistics-based accept-reject test plan

• Producer’s Risk α

• Consumer’s Risk β }usually range from

.05 to .25 (negotiated)

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55

Types of Type 2 Tests (cont)

• Reliability Acceptance Testing- Plot MTBF versus time,look for growth/decline

• Reliability Life Testing- To determine failure distribution

– Continuous (Steady)

• Fixed Time, Count Failures

• Fixed number of Failures, Count Time

– Step-Stress (Accelerated) Testing

• Step up stress until all units fail

• Aids in planning burn in

56

Type 3 Testing

• Definition- Operational Testing Using:– A group of production units

– Designated field test sight

– Representative mix of mission profiles

– User personnel (first trained)

– 1st sets of support equipment; spares

• Uniqueness– All elements of the system are operational and evaluated together

– Where the true R, M, A and other performance measures are knownfor first time, rather than estimated via models plus some type 1 & 2test data