Transcript System Reliability Analysis - Concepts and Metrics

Systems Engineering Program
Department of Engineering Management, Information and Systems
EMIS 7305/5305
Systems Reliability, Supportability and Availability Analysis
System Reliability Analysis
- Concepts and Metrics
Dr. Jerrell T. Stracener, SAE Fellow
Leadership in Engineering
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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 stress
To 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 Problems
Stress/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 that
an item will operate without failure for a given period of
time 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 be
defined 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 development
Basic 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 Failures
MFHBUM - MFHB Unscheduled Maintenance
Army
MFHBE - Mean Flight Hours Between Events
Navy
MFHBF - Mean Flight Hours Between Failures
MFHBMA - MFHB Maintenance Actions
Automotive Industry
Number 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 system
are in an operational state at the start of the mission.
• Measure
Ps or R(t) - Probability of mission success based on:
Mission Essential Functions
Mission Essential Equipment
Mission Operating Environment
Mission 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 Approach
Donald H. Sanders
McGraw 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:
f (t)  0
for all values of t

and
f
(
t
)
dt

1

0
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Failure Distribution Function
The failure distribution function or, the probability distribution
function is the cumulative proportion of the population failing in
time t, i.e.,
t
Ft   P(T  t )   f ( y)dy
0
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Failure Distribution Function
The failure distribution function, F, has the following
properties:
1. F is nondecreasing, i.e., if 0  t1 < t2 < , then
F(t1)  F(t2),
2. 0  F(t)  1 for all t
3. lim Ft   0
t 0
4.
in general, but here F(0) = 0
lim Ft   1
t 
5. P(a < T  b) = F(b) - F(a)
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Remark
The time to failure distribution has a special name
and symbol in reliability. It is called the unreliability
and is denoted by Q, i.e.
Q(t) = F(t) = P(T  t)
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Failure Densities and Distributions
f(t)
Failure Density
Area = P(t1 < T <t2)
t
0
F(t)
Failure Distribution
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F(t2)
P(t1 < T < t2) = F(t2) - F(t1)
F(t1)
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t1
t2
t
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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 a
proportion, 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
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tp
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Reliability
In terms of the failure density, f, of an item, the
100pth percentile, tp, is
tp
 f t dt  p
0
f(t)
p
0
tp
t
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The Reliability Function
The Reliability of an item is the probability that the item will
survive time t, given that it had not failed at time zero, when
used within specified conditions, i.e.,

Rt   P(T  t )   f ( t )dt  1  F( t )
t
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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. lim
t 
R t   0
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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 reliability
or 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
d
f t    R t 
dt
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Relationship Between h(t), f(t), F(t) and R(t)
f t 
f t 
h t  

R t  1 - Ft 
Remark: The failure rate h(t) is a measure of proneness to
failure as a function of age, t.
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Properties of the Failure Rate
The (instantaneous) failure rate, h, has the following
properties:
1. h(t)  0 , t  0
and
t
2.
lim  h y dy  
t 
0
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The Reliability Function
The reliability of an item at time t may be expressed in terms
of its failure rate at time t as follows:

 0 h ( y ) dy
R ( t )  exp    h ( y)dy   e
 0

t
t
where h(y) is the failure rate
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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.,
t
1
H( t )   h ( y)dy
t0
The average failure rate of an item over an interval of time from
t1 to t2, where t1 < t2, is the number of failures occurring in the
interval (t1, t2), divided by the interval length, t2 - t1
H( t 2 )  H( t1 )
H( t1 , t 2 ) 
t 2  t1
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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 operation
MTBF provides the basis for estimating the number of failures in
a given period of time
Even though an item may be discarded after failure and its mean
life characterized by MTTF, it may be meaningful to
characterize the system reliability in terms of MTBF if the
system is restored after item failure.
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MTTF
MTTF (Mean Time to Failure) or MTBF (Mean Time
Between Failures) may be determined from the
time to failure probability density function by use
of three equivalent methods:
1. definition of MTBF
2. moment generating functions
3. characteristic function
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Relationship Between MTTF and Failure Density
If T is the random time to failure of an item, the
mean time to failure, MTTF, of the item is

ET   MTTF   tf t dt
0
where f is the probability density function of time
to failure, iff this integral exists (as an improper
integral).
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Relationship Between MTTF and Reliability

MTBF  MTTF   R t dt
0
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Reliability “Bathtub Curve”
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Reliability Humor
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