Systems Reliability Modeling

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Systems ReliabilityModeling AnalysisSeries
Configurations

• Systems Reliability, Supportability and
  Availability Analysis

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System Reliability Models

• Series Configurations
• Parallel or Redundant Configurations
• Active Parallel
• r-out-of-n
• Standby
• Series-Parallel and Parallel-Series
  Configurations
• General

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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
• There are two basic types of reliability
  configurations
• - Series
• - 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 Component, an Assembly, an
  Equipment, a Line Replaceable Unit (LRU), a
  Subsystem or a System
• Block A logical representation of an Element.
• Reliability Block Diagram A logical
  representation of a System, Subsystem, or
  Assembly in terms of its Elements.

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Series Reliability Configuration
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System Reliability Models - 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
• Block Diagram For Series Reliability
  Configuration with n
• elements E1, E2, ..., En
• Since a single path exists, the failure of any
  element in the
• system interrupts the path and causes the system
  to fail.

E1
E2
En
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System Reliability Models - Series Configuration
Product Rule

• 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, ...
• And Ei-1 have survived for time t, for i 1, 2,
  ..., n
• if the n elements are independent

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System Reliability Models - Series Configuration

• General time to element failure distributions
• System reliability
• Where Hi(t) is the cumulative failure rate of
  element i, for
• i 1, 2, ... n
• System mean time to failure

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Series Reliability Configuration with Exponential
Distribution

• Reliability Block Diagram
• Exponential distributions of element time to
  failure
• Ti E(?i) for i 1, 2, ... n
• System reliability
• Where the system failure rate is
• System mean time to failure

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Series Reliability Configuration with Exponential
Distribution

• Reliability Block Diagram
• Exponential distributions of element time to
  failure
• Ti E(?) for i 1, 2, ... n
• System reliability
• 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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Series Reliability Configuration with Weibull
Distribution

• Reliability Block Diagram
• Weibull distribution of element time to failure
• Ti W(?i,?i) for i 1, 2, ...
  n
• System reliability
• System failure rate

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System Reliability Models - Series Configuration

• System cumulative failure rate
• System mean time to failure rate

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System Reliability Models - Series Configuration

• Weibull distribution of element time to failure
• Ti W(?,?) for i 1, 2, ... n
• System reliability
• System failure rate

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System Reliability Models - Series Configuration

• System cumulative failure rate
• System mean time to failure

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Example Series Reliability Configuration