stracener_emis 7305/5305_spr08_01.24.08 1 system reliability analysis - concepts and metrics dr....

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Stracener_EMIS 7305/5305_Spr08_01.24.08

System Reliability Analysis- Concepts and Metrics

Dr. Jerrell T. Stracener, SAE Fellow

Leadership in Engineering

EMIS 7305/5305Systems Reliability, Supportability and Availability Analysis

Systems Engineering ProgramDepartment of Engineering Management, Information and Systems

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Reliability Definitions and Concepts

• Figures of merit

• Failure densities and distributions

• The reliability function

• Failure rates

• The reliability functions in terms of the failure rate

• Mean time to failure (MTTF) and mean time between failures (MTBF)

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Reliability Concepts, Principles and Methodology

• Hardware

• Software

• Operator

• Service

• Product

• Production/Manufacturing Processes and Equipment

• Product and Customer Support

• Systems

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What is Reliability?

• To the user of a product, reliability is problem free operation

• Reliability is a function of stressTo understand reliability, understand stress on hardware

- where its going to be used- how its going to be used- what environment it is going to be used in

To efficiently achieve reliability, rely on analytical understanding of reliability and less on understanding reliability through testing

Field ProblemsStress/Design, Parts and Workmanship

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Definitions of Reliability

• Reliability is a measure of the capability of a system, equipment or component to operate without failure when in service.

• Reliability provides a quantitative statement of the chance thatan item will operate without failure for a given period oftime in the environment for which it was designed.

• In its simplest and most general form, reliability is the probability of success.

• To perform reliability calculations, reliability must first bedefined explicitly. It is not enough to say that reliability is a probability. A probability of what?

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More Definitions of Reliability

• Reliability is defined as the probability that an item will perform its intended unction for a specified interval under stated conditions. In the simplest sense, reliability means how long an item (such as a machine) will perform its intended function without a breakdown.

• Reliability: the capability to operate as intended, whenever used, for as long as needed.

Reliability is performance over time, probability that something will work when you want it to.

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Definitions of Reliability

• Essential elements needed to define reliability are:– What does it do?

• System, subsystem, equipment or component functions

– What is satisfactory performance?• Figures of merit @ System• Allocations &/or derived @ subsystem, equipment & component

– How long does it need to function?Life: required number of operational units (time, sorties, cycles, etc)

– What are conditions under which it operates?• Environment• Operation• Maintenance• Support

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Reliability Figures of Merit

• Basic or Logistic Reliability

MTBF - Mean Time Between Failures

measure of product support requirements

• Mission Reliability

Ps or R(t) - Probability of mission success

measure of product effectiveness

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Basic Reliability

• Design and developmentBasic reliability is a measure of serial reliability or logistics reliability and reflects all elements in a system

• Measures

Air Force MFHBF - Mean Flight Hours Between FailuresMFHBUM - MFHB Unscheduled Maintenance

Army MFHBE - Mean Flight Hours Between Events

Navy MFHBF - Mean Flight Hours Between FailuresMFHBMA - MFHB Maintenance Actions

Automotive IndustryNumber of defects per 100 vehicles

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Mission Reliability

• Mission Reliability is defined as the probability that a system will perform its mission essential functions during a specified mission, given that all elements of the systemare in an operational state at the start of the mission.

• Measure

Ps or R(t) - Probability of mission success based on:

Mission Essential FunctionsMission Essential EquipmentMission Operating EnvironmentMission Length

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Basic Elements of Reliability Modeling & Analysis

• Reliability is a probability

• Therefore a working knowledge of probability, random variables and probability distributions is required for:

- Development of reliability models

- Performing reliability analyses

• An understanding of the concepts of probability is required for design and support decisions

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Reliability Humor: Statistics

“If I had only one day left to live, I would live it in my statistics class --it would seem so much longer.”

From: Statistics A Fresh ApproachDonald H. SandersMcGraw Hill, 4th Edition, 1990

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Failure Density Function

associated with a continuous random variable T, the time to failure of an item, is a function f, called the probability density function, or in reliability, the failure density. The function f has the following properties:

for all values of t

and

0)t(f

1dt)t(f0

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Failure Distribution Function

The failure distribution function or, the probability distributionfunction is the cumulative proportion of the population failing in time t, i.e.,

dy)y(f)tT(PtFt

0

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Failure Distribution Function

The failure distribution function, F, has the followingproperties:

1. F is nondecreasing, i.e., if 0 t1 < t2 < , thenF(t1) F(t2),

2. 0 F(t) 1 for all t

3. in general, but here F(0) = 0

4.

5. P(a < T b) = F(b) - F(a)

0tFlim0t

1tFlimt

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Remark

The time to failure distribution has a special nameand symbol in reliability. It is called the unreliabilityand is denoted by Q, i.e.

Q(t) = F(t) = P(T t)

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Failure Densities and Distributions

Failure Density

Failure Distribution

f(t)

t

Area = P(t1 < T <t2)

F(t)

t

F(t2)

F(t1)

t2t1

P(t1 < T < t2) = F(t2) - F(t1)

1

0

0

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Percentile

The 100pth percentile, 0 < p < 1, of the time to failure probability distribution function, F, is the time, say tp, within which aproportion, p, of the items has failed, i.e., tp is the value of t such that

F(tp) = P(T tp) = p

or tp = F-1(p)F(t)

p

tp

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Reliability

In terms of the failure density, f, of an item, the100pth percentile, tp, is

pdttfpt

0

t

p

f(t)

0 tp

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The Reliability Function

The Reliability of an item is the probability that the item willsurvive time t, given that it had not failed at time zero, when used within specified conditions, i.e.,

)tT(PtR

t

)t(F1dt)t(f

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Properties of the Reliability Function

1. R is a non-increasing function, i.e., if 0 t1 < t2 < , then

R(t1) R(t2)

2. 0 R(t) 1 for all t

3. R(t) = 1 at t = 0

4. 0tRlimt

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Properties of the Reliability Function

The probability of failure in a given time interval, t1

to t2, can be expressed in terms of either reliabilityor unreliability functions, i.e.,

P(t1 < T < t2) = R(t1) - R(t2)

= Q(t2) - Q(t1)

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Reliability

Relationship between failure density and reliability

tRdt

dtf

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Relationship Between h(t), f(t), F(t) and R(t)

Remark: The failure rate h(t) is a measure of proneness to failure as a function of age, t.

tF-1

tf

tR

tfth

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Properties of the Failure Rate

The (instantaneous) failure rate, h, has the followingproperties:

1. h(t) 0 , t 0

and

2.

t

0t

dyyhlim

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The Reliability Function

The reliability of an item at time t may be expressed in termsof its failure rate at time t as follows:

where h(y) is the failure rate

t

0dy)y(ht

0

edy)y(hexp)t(R

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Cumulative Failure Rate

The cumulative failure rate at time t, H(t), is the cumulative number of failures at time t, divided by the cumulative time, t, i.e.,

The average failure rate of an item over an interval of time fromt1 to t2, where t1 < t2, is the number of failures occurring in the interval (t1, t2), divided by the interval length, t2 - t1

t

0

dy)y(ht

1)t(H

12

1221 tt

)t(H)t(H)t,t(H

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Mean Time to Failure and Mean Time Between Failures

Mean Time to Failure (or Between Failures) MTTF (or MTBF)is the expected Time to Failure (or Between Failures)

Remarks:MTBF provides a reliability figure of merit for expected failure

free operationMTBF provides the basis for estimating the number of failures in

a given period of timeEven though an item may be discarded after failure and its mean

life characterized by MTTF, it may be meaningful tocharacterize the system reliability in terms of MTBF if thesystem is restored after item failure.

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MTTF

MTTF (Mean Time to Failure) or MTBF (Mean TimeBetween Failures) may be determined from thetime to failure probability density function by useof three equivalent methods:

1. definition of MTBF2. moment generating functions3. characteristic function

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Relationship Between MTTF and Failure Density

If T is the random time to failure of an item, themean time to failure, MTTF, of the item is

where f is the probability density function of timeto failure, iff this integral exists (as an improperintegral).

0

dtttfMTTFTE

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Relationship Between MTTF and Reliability

0

dttRMTTFMTBF

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Reliability “Bathtub Curve”

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Reliability Humor