Reliability Application

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Systems Engineering Program
Department of Engineering Management, Information
and Systems
EMIS 7370/5370 STAT 5340 PROBABILITY AND
STATISTICS FOR SCIENTISTS AND ENGINEERS
Reliability Application
Dr. Jerrell T. Stracener, SAE Fellow
Leadership in Engineering
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An Application of Probability to Reliability
Modeling and 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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What is Reliability?

• Reliability is defined as the probability that an
  item will perform its intended function 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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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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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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The Reliability Function
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Reliability
Relationship between failure density and
reliability
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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.
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The Reliability Function
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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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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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The Exponential Model (Weibull Model with ß 1)

• Definition
• A random variable T is said to have the
  Exponential
• Distribution with parameters ?, where ? gt 0, if
  the
• failure density of T is
• , for t ? 0
• , elsewhere

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Probability Distribution Function

• Weibull W(b, q)
• , for t ? 0
• Where F(t) is the population proportion failing
  in time t
• Exponential E(q) W(1, q)

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The Exponential Model
Remarks The Exponential Model is most often
used in Reliability applications, partly
because of mathematical convenience due to a
constant failure rate. The Exponential Model is
often referred to as the Constant Failure Rate
Model. The Exponential Model is used during the
Useful Life period of an items life, i.e.,
after the Infant Mortality period before
Wearout begins. The Exponential Model is most
often associated with electronic equipment.
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Reliability Function

• Probability Distribution Function
• Weibull
• Exponential

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The Weibull Model - Distributions
Reliability Functions
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Mean Time Between Failure - MTBF
Weibull Exponential
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The Gamma Function ?
Values of the Gamma Function
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Percentiles, tp
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Failure Rates - Weibull
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Failure Rates - Exponential

• Failure Rate
• Note
• Only for the Exponential Distribution
• Cumulative Failure

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The Weibull Model - Distributions
Failure Rates
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The Binomial Model - Example Application 1

• Problem -
• Four Engine Aircraft
• Engine Unreliability Q(t) p 0.1
• Mission success At least two engines survive
• Find RS(t)

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The Binomial Model - Example Application 1

• Solution -
• X number of engines failing in time t
• RS(t) P(x ? 2) b(0) b(1) b(2)
• 0.6561 0.2916 0.0486 0.9963

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Series Reliability Configuration

• Simplest and most common structure in reliability
  analysis.
• Functional operation of the system depends on the
  successful operation of all system components
  Note The electrical or mechanical configuration
  may differ from the reliability configuration
• Reliability Block Diagram
• Series configuration with n elements E1, E2,
  ..., En
• System Failure occurs upon the first element
  failure

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Series Reliability Configuration with Exponential
Distribution

• Reliability Block Diagram
• Element Time to Failure Distribution
• with failure rate , for i1,
  2,, n
• System reliability
• where

is the system failure rate

• System mean time to failure

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Series Reliability Configuration

• Reliability Block Diagram
• Identical and independent Elements
• Exponential Distributions
• Element Time to Failure Distribution
• with failure rate
• System reliability

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Series Reliability Configuration
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• Parallel Reliability Configuration
  Basic Concepts
• Definition - a system is said to have parallel
  reliability configuration if the system function
  can be performed by any one of two or more paths
• Reliability block diagram - for a parallel
  reliability configuration consisting of n
  elements, E1, E2, ... En

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• Parallel Reliability
  Configuration
• Redundant reliability configuration - sometimes
  called a redundant reliability configuration.
  Other times, the term redundant is used only
  when the system is deliberately changed to
  provide additional paths, in order to improve the
  system reliability
• Basic assumptions
• All elements are continuously energized
  starting at time t 0
• All elements are up at time t 0
• The operation during time t of each element can
  be described
• as either a success or a failure, i.e. Degraded
  operation or
• performance is not considered

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Parallel Reliability
Configuration System success - a system having
a parallel reliability configuration operates
successfully for a period of time t if at least
one of the parallel elements operates for time t
without failure. Notice that element failure does
not necessarily mean system failure.
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• Parallel Reliability
  Configuration
• Block Diagram
• System reliability - for a system consisting of
  n elements, E1, E2, ... En

if the n elements operate independently of each
other and where Ri(t) is the reliability of
element i, for i1,2,,n
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• System Reliability Model - Parallel
  Configuration
• Product rule for unreliabilities

• Mean Time Between System Failures

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Parallel Reliability Configuration
s
pR(t)
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Parallel Reliability Configuration with
Exponential Distribution

• Element time to failure is exponential with
  failure rate ?
• Reliability block diagram
• Element Time to Failure Distribution
• with failure rate for I1,2.

E1
E2

• System reliability

• System failure rate

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Parallel Reliability Configuration with
Exponential Distribution

• System Mean Time Between Failures
• MTBFS 1.5 ?

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Example
A system consists of five components connected
as shown. Find the system reliability, failure
rate, MTBF, and MTBM if TiE(?) for i1,2,3,4,5
E2
E1
E3
E4
E5
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Solution
This problem can be approached in several
different ways. Here is one approach There are
3 success paths, namely, Success
Path Event E1E2 A E1E3 B E4E5 C Then
Rs(t)Ps P(A)P(B)P(C)-P(AB)-P(AC)-P
(BC)P(ABC) P(A)P(B)P(C)-P(A)P(B)-P(
A)P(C)-P(B)P(C) P(A)P(B)P(C)
P1P2P1P3P4P5-P1P2P3-P1P2P4P5 -P1P3P4P5P1P2P
3P4P5 assuming independence and where PiP(Ei)
for i1, 2, 3, 4, 5
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Since Pie-?t for i1,2,3,4,5 Rs(t) hs(t)
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MTBFs