System Reliability Analysis - Concepts and Metrics

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System Reliability Analysis- Concepts and
Metrics

• Systems Reliability, Supportability and
  Availability Analysis

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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 _at_ System
• Allocations /or derived _at_ 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

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Basic Reliability

• 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
• for all values of t
• and

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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.,

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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 lt t2 lt ?,
  then F(t1) ? F(t2),
• 2. 0 ? F(t) ? 1 for all t
• 3. in general, but here F(0) 0
• 4.
• 5. P(a lt 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)
Area P(t1 lt T ltt2)
t
0
F(t)
1
F(t2)
P(t1 lt T lt t2) F(t2) - F(t1)
F(t1)
t
0
t2
t1
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Percentile

• The 100pth percentile, 0 lt p lt 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
tp
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Reliability

• In terms of the failure density, f, of an item,
  the 100pth percentile, tp, is

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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.,

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Properties of the Reliability Function

• R is a non-increasing function, i.e.,
• if 0 ? t1 lt t2 lt ?, then
  R(t1) ? R(t2)
• 2) 0 ? R(t) ? 1 for all t
• 3) R(t) 1 at t 0
• 4)

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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 lt T lt t2) R(t1) - R(t2)
• F(t2) F(t1)

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Reliability

• Relationship between failure density and
  reliability

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Failure Rate

• 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
• 2.

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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
• 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.,
• The average failure rate of an item over an
  interval of time from t1 to t2, where t1 lt t2, is
  the number of failures occurring in the interval
  (t1, t2), divided by the interval length, t2 - 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
• 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
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Reliability Bathtub Curve
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Reliability Humor
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Example

• If f(t) ?e-?t for t ? 0,
• Verify that f(t) is a failure density and derive
  the mathematical expression for
• b. R(t)
• c. MTBF
• d. h(t) and H(t)
• e. tp
• f. Show that P(T gt t1 t2 T gt t1) P(T gt t2)

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c. or d.
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• Since
• , since

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f.
But so that
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Following the same argument
so
therefore
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