Logistics Systems Engineering

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NTU SY-521-N
SMU SYS 7340
Logistics Systems Engineering Systems Reliability
Modeling Analysis
Dr. Jerrell T. Stracener, SAE Fellow
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• System Reliability Models
• The reliability definitions, concepts and models
  presented apply at any level of a system, from a
  single discrete component up to and including the
  entire system.
• Systems reliability deals with the reliability of
  the end-item system and is based on the system
  configuration and component failure rates as well
  intended service usage.

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• System Reliability Models
• There are two basic types of reliability
  configurations
• Series
• Parallel or Redundant

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• Terminology and Notation
• Path A physical means for accomplishing a given
  function.
• Element The basic system level under discussion.
  An element may be a Function, Component, an
  Assembly, an Equipment, a Line Replaceable Unit
  (LRU), a Subsystem or a System.
• Block A logical representation of an Element.

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• Terminology and Notation
• Reliability Block Diagram A logical
  representation of a System,Subsystem, or Assembly
  in terms of its Elements.

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• Series 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

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• Series Reliability Configuration
• Series Reliability configuration with n elements
  E1, E2, ..., En
• Block Diagram
• Systems Reliability

E1
E2
En
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Remark Since a single path exists, the failure
of any element in the system interrupts the path
and causes the system to fail.
where Ri(t) is the reliability of the ith element
and the n elements are independent
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Reliability System Configuration

• Series Configuration
• Rs(t) P(A1) P(A2A1) P(A3A1A2) ... P(AnA1A2
  ... An-1)
• Where RS(t) is system reliability, i.e. The
  probability of system success for time t, given
  that the system was up at t 0 and P(AiA1 A2
  ... Ai-1) is the conditional probability of event
  A occurring (i.e., element Ei survives for time
  t), given that events A1, A2, ... And Ai-1 have
  occurred, i.e. Elements E1, E2,

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Reliability System Configuration

• Series Model Example
• What is the reliability of the following system
  given that
• E1 0.9400
• E2 0.9500
• E3 0.9800

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Reliability System Configuration

• Series Model Example
• Solution
• Use the product rule
• Rs(t) R1(t)?R2(t)?R3(t)
• (0.9400) (0.9500) (0.9800)
• 0.8751

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• Series Configuration
• System failure rate
• System mean time to failure

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• Series Configuration
• Exponential distributions of element time to
  failure Ti ?(?i) for i 1, 2, ... n
• System reliability
• where is the system failure rate
• System mean time to failure

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• Series Configuration
• Exponential distribution of element time to
  failure special case each element has the same
  distribution Ti ?(?) for i 1, 2, ... n
• System reliability

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• Series Configuration
• System mean time to failure
• Which is the same as the expected time to
• the first failure, E(T1), when n identical
• items are put into service

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• System Reliability Models - Parallel
  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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• System Reliability Models - Parallel
  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.
• System reliability - for a system consisting of
  n elements, E1, E2, ... En

where Ri(t) is the reliability of element i. if
the n elements operate independently of each other
,
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• Parallel Model Example
• What is the reliability of the following system
  given that
• R1(t) 0.9400
• R2(t) 0.9500
• R3(t) 0.9800

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• Parallel Model Example
• Solution
• RS(t) 1 - (1 - R1(t)) (1 - R2(t)) (1 - R3(t))
• 1-(1-0.9400)(1-0.9500)(1-0.9800)
• 1 - (0.0600)(0.0500)(0.0200)
• 0.9999

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• Parallel Configuration

n elements
E1
E1
E1
E1
E2
E2
m elements
E2
E2
Em
Em
Em
Em
Rs(t) 1 - (1-p)m
Rs(t) 1 - (1-pn)m
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• Parallel Configuration

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• System Reliability Models - r-out-of-n
  Reliability Configuration
• Definition - a system containing n elements, out
  of which at least
• r are required for system success, is the so
  called r-out-of-n
• reliability configuration
• Remark - the r-out-of-n reliability
  configuration is a general
• configuration. If r 1, the configuration is a
  parallel configuration.
• If r n, the configuration is a series
  configuration.
• Example - a piece of stranded wire, with n
  strands, which at least
• r strands are necessary to support the required
  load

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Reliability System Configuration

• Standby Model Example
• What is the reliability of the following system
  given that
• l1 l2 0.01
• t 100

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Reliability System Configuration

• Standby Model Example
• Solution
• RS(t) (1 ?t)e-?t
• 1 (0.01)(100)e-(0.01)(100)
• 1 (1)e-1
• (2) (0.3679)
• 0.7358

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• System Reliability Models - Standby
  Configuration Conclusions
• As the number of redundant paths increases, the
  mission reliability
• approaches the reliability of the
  monitor/switching device.
• When the failure rates of the path, the
  switching devices, and the
• monitor/switching device are equal, standby
  redundancy with two
• paths results in a mission reliability
  considerably less than that of a
• single non-redundant path.
• For systems where the switching device and
  monitor failure rates
• are less than the path failure rate, the greatest
  increase in reliability
• occurs when one redundant path is added to a
  single path.

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• System Reliability Models - Conclusions
  continued
• For a given path and switching device failure
  rate, reliability
• improvement increases rapidly as the monitor
  failure rate
• decreases and the number of redundant paths
  increases. The same
• is true if the monitor failure rate is held
  constant and the switching
• device failure rate decreases.
• Significant improvement in mission reliability
  through
• redundancy results from the utilization of
  switching devices and
• monitors that are much more reliable than the
  path being switched.

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• Configuration Considerations in Design
• Series Configuration - Relative to Redundant
  Configuration
• Simpler
• Increases Basic Reliability
• Reduces Support Resources
• Decreases Mission Reliability
• Redundant Configuration - Relative to Series
  Configuration
• More Complex - Increases Weight
• Requires More Testability
• Increases Support Resources
• Decreases Basic Reliablity
• Increases Mission Reliability

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Conclusion