Reliability Block Diagrams

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Reliability Block Diagrams

• A reliability block diagram is a success-oriented
  network describing the function of the system.
• If the system has more than one function, each
  function is considered individually, and separate
  reliability block diagram is established for each
  system function.
• Each component is illustrated by a block. When
  there is a connection between the end points, we
  say that component i is functioning.

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Example

• Consider a pipeline with two independent safety
  valves that are physically installed in series.
  These valves are supplied with a spring-loaded,
  fail-safe, close by hydraulic actuator. The
  valves are opened and held open by hydraulic
  control pressure and is closed automatically by
  spring force whenever the control pressure is
  removed or lost. In normal operation both valves
  are held open. The main function of the valves
  is to act as a safety barrier, i.e., to close and
  stop the flow in the pipeline in case of an
  emergency.

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Example

• It is usually an easy task to convert a fault
  tree to a reliability block diagram. In this
  conversion, we start from the top event and
  replace the gate successfully. An OR-gate is
  replaced by a series structure of the
  components directly beneath the gate, and an
  AND-gate is replaced by a parallel structure of
  the components directly beneath the gate.

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Structure Function

• The state of component i can be described by a
  binary state variable, i.e.,
• Similarly the state of a system can be described
  by a binary function

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Series and Parallel Structures

• Series
• Parallel

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k-out-of-n Structure
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2-out-of-3 Structure
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Coherent Structures

• Definition A system is said to be coherent if
  all its components are relevant and the structure
  function is non-decreasing in each argument.
• Relevant
• Irrelevant
• Non-decreasing structure function

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Definitions
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Example

• Component 2 is irrelevant

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a
b
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Some Theorems for Coherent Structures
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Redundancy at System Level
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Redundancy at Component Level
We obtain a better system by introducing
redundancy at component level than by introducing
redundancy at system level.
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Path Sets and Cut Sets

• A structure of order n consists of n components
  numbered from 1 to n. The set of all components
  is denoted by C.
• A path set P is a set of components in C which by
  functioning ensures that the system is
  functioning. A path set is said to be minimal if
  it cannot be reduced without loosing its status
  as a path set.
• A cut set K is a set of components in C which by
  failing causes that the system to fail. A cut
  set is said to be minimal if it cannot be reduced
  without loosing its status as a cut set.

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Example 1
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Example 2
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Structures Represented by Paths
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Example 2
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Structures Represented by Cuts
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Example 2
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Critical Path

• A critical path vector for component i is a state
  vector
• Such that
• A critical path set corresponding to the critical
  path vector for component i is defined by

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Structural Importance
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Example

• Consider 2-out-of-3 structure

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Example

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Pivotal Decomposition
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Example Bridge Structure
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Structure of Composed Components

• Partition into subsystems is done in such a way
  that each component never appears within more
  than one of the subsystems.

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Some Notations
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Coherent Modules

• Let the coherent structure be given,
  and let
• Then is said to be a coherent module
  of
• if can be written as a function
• where is the structure function of a
  coherent system.
• What we actually do here is to consider all the
  components with the index belonging to A as one
  component with state variable . When
  we interpret the system in this way, the
  structure function can be written as

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Example
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Modular Decomposition

• A modular decomposition of a coherent structure
  is a set of disjoint modules together with an
  organizing structure such that

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Prime Module

• A module that cannot be partitioned into smaller
  modules without letting each component represent
  a module, is called a prime module.
• III represents a prime module. II is not since it
  may be described in Fig 3.35 and hence can be
  partitioned into IIa and IIb.

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