RBDs and Analytical System Reliability - Revision history

Richard House: /* Reliability of Nonidentical k-out-of-n Independent Components */

2016-01-05T18:52:10Z

Reliability of Nonidentical k-out-of-n Independent Components

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	Using the Analytical Quick Calculation Pad, the reliability can be calculated to be 0.9586. The following figure shows the returned result. Note that you are not required to enter a mission end time for this system into the Analytical QCP because all of the components are static and thus the reliability results are independent of time.		Using the Analytical Quick Calculation Pad, the reliability can be calculated to be 0.9586. The following figure shows the returned result. Note that you are not required to enter a mission end time for this system into the Analytical QCP because all of the components are static and thus the reliability results are independent of time.
	[[Image:Fig 4.12.PNG\|center\|~~500px~~\|<div align="center">Reliability results for Example 9 </div>]]		[[Image:Fig 4.12.PNG\|center\|600px\|<div align="center">Reliability results for Example 9 </div>\|link=]]

	'''Example 10'''		'''Example 10'''

Richard House: /* Identical Block Simplification Option ("Use IBS") */

2016-01-05T17:03:11Z

Identical Block Simplification Option ("Use IBS")

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	[[Image:Fig 4.31.PNG\|center\|400px\|<div align="center"> BlockSim's complete solution for Complex RBD</div>]]		[[Image:Fig 4.31.PNG\|center\|400px\|<div align="center"> BlockSim's complete solution for Complex RBD</div>\|link=]]

	When using IBS, the resulting equation is invalidated if any of the block properties (e.g., failure distributions) have changed since the equation was simplified based on those properties.		When using IBS, the resulting equation is invalidated if any of the block properties (e.g., failure distributions) have changed since the equation was simplified based on those properties.

	[[Image:Fig 4.32.PNG\|center\|400px\|<div align="center"> BlockSim's symbolic solution, with the IBS option selected for Complex RBD]]		[[Image:Fig 4.32.PNG\|center\|400px\|<div align="center"> BlockSim's symbolic solution, with the IBS option selected for Complex RBD\|link=]]

Richard House: /* Symbolic Solutions in BlockSim */

2016-01-05T17:02:07Z

Symbolic Solutions in BlockSim

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	Several algebraic solutions in BlockSim were used in the prior examples. BlockSim constructs and displays these equations in different ways, depending on the options chosen. As an example, consider the complex system shown next.		Several algebraic solutions in BlockSim were used in the prior examples. BlockSim constructs and displays these equations in different ways, depending on the options chosen. As an example, consider the complex system shown next.

	[[Image:1 thru 11 network_new.png\|center\|~~450px~~\|Complex RBD]]		[[Image:1 thru 11 network_new.png\|center\|500px\|Complex RBD\|link=]]

	When BlockSim constructs the equation internally, it does so in what we will call a symbolic mode. In this mode, portions of the system are segmented. For this example, the symbolic (internal) solution is shown next and composed of the terms shown in the following equations.		When BlockSim constructs the equation internally, it does so in what we will call a symbolic mode. In this mode, portions of the system are segmented. For this example, the symbolic (internal) solution is shown next and composed of the terms shown in the following equations.

	[[Image:BS4-30_new.png\|center\|550px\|BlockSim's symbolic solution]]		[[Image:BS4-30_new.png\|center\|550px\|BlockSim's symbolic solution\|link=]]

Richard House: /* RBDs for Failure Modes and Other Applications */

2016-01-05T17:00:57Z

RBDs for Failure Modes and Other Applications

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	The following figure shows the diagram for this configuration. Analysis of this diagram follows the same principles as the ones presented in this chapter and can be performed in BlockSim, if desired.		The following figure shows the diagram for this configuration. Analysis of this diagram follows the same principles as the ones presented in this chapter and can be performed in BlockSim, if desired.

	[[Image:chp4image35.png\|center\|~~400px~~\|<div align="center"> Reliability block diagram for Example 17. </div>]]		[[Image:chp4image35.png\|center\|500px\|<div align="center"> Reliability block diagram for Example 17. </div>\|link=]]

	'''Another Example for Failure Mode Analysis'''		'''Another Example for Failure Mode Analysis'''
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	The diagram is shown next:		The diagram is shown next:

	[[Image:chp4image36.png\|center\|~~400px~~\|<div align="center"> Reliability block diagram for Example 18.</div>]]		[[Image:chp4image36.png\|center\|500px\|<div align="center"> Reliability block diagram for Example 18.</div>\|link=]]

	The reliability equation, as obtained from BlockSim is:		The reliability equation, as obtained from BlockSim is:

Richard House: /* Mirrored Blocks */

2016-01-05T16:59:09Z

Mirrored Blocks

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	To better illustrate this consider the following block diagram:		To better illustrate this consider the following block diagram:

	[[Image:Mirror_block.PNG\|center\|~~250px~~\|]]		[[Image:Mirror_block.PNG\|center\|300px\|link=]]

	In this diagram <math>Bm\,\!</math> is a mirrored block of <math>B\,\!</math>. Since <math>B\,\!</math> and <math>Bm\,\!</math> are identical, this diagram is equivalent to a diagram with <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math> in series, as shown next:		In this diagram <math>Bm\,\!</math> is a mirrored block of <math>B\,\!</math>. Since <math>B\,\!</math> and <math>Bm\,\!</math> are identical, this diagram is equivalent to a diagram with <math>A\,\!</math>, <math>B\,\!</math> and <math>C\,\!</math> in series, as shown next:

	[[Image:EqvMirror.PNG\|center\|~~350px~~\|]]		[[Image:EqvMirror.PNG\|center\|300px\|link=]]

	'''Example: Using Mirrored Blocks'''		'''Example: Using Mirrored Blocks'''
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	In the diagram shown below, electricity can flow in both directions. Successful system operation requires at least one output (O1, O2 or O3) to be working. Create a block diagram for this system.		In the diagram shown below, electricity can flow in both directions. Successful system operation requires at least one output (O1, O2 or O3) to be working. Create a block diagram for this system.

	[[Image:chp4image33.png\|center\|~~300px~~\|<div align="center"> Electrical network diagram. </div>]]		[[Image:chp4image33.png\|center\|400px\|<div align="center"> Electrical network diagram. </div>\|link=]]

	The bidirectionality of this system can be modeled using mirrored blocks. The RBD is shown next, where blocks 5A, 7A and 1A are duplicates (or mirrored blocks) of 5, 7 and 1 respectively.		The bidirectionality of this system can be modeled using mirrored blocks. The RBD is shown next, where blocks 5A, 7A and 1A are duplicates (or mirrored blocks) of 5, 7 and 1 respectively.

	[[Image:chp4image34.png\|center\|~~350px~~\|<div align="center"> RBD of Electrical network diagram</div>]]		[[Image:chp4image34.png\|center\|450px\|<div align="center"> RBD of Electrical network diagram</div>\|link=]]

	= RBDs for Failure Modes and Other Applications =		= RBDs for Failure Modes and Other Applications =

Richard House: /* Multi Blocks */

2016-01-05T16:54:06Z

Multi Blocks

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	By using multi blocks within BlockSim, a single block can represent multiple identical blocks in series or in parallel configuration. This technique is simply a way to save time when creating the RBD and to save space within the diagram. Each item represented by a multi block is a separate entity with identical reliability characteristics to the others. However, each item is not rendered individually within the diagram. In other words, if the RBD contains a multi block that represents three identical components in a series configuration, then each of those components fails according to the same failure distribution but each component may fail at different times. Because the items are arranged reliability-wise in series, if one of those components fails, then the multi block fails. It is also possible to define a multi block with multiple identical components arranged reliability-wise in parallel or ''k''-out-of-''n'' redundancy. The following figure demonstrates the use of multi blocks in BlockSim.		By using multi blocks within BlockSim, a single block can represent multiple identical blocks in series or in parallel configuration. This technique is simply a way to save time when creating the RBD and to save space within the diagram. Each item represented by a multi block is a separate entity with identical reliability characteristics to the others. However, each item is not rendered individually within the diagram. In other words, if the RBD contains a multi block that represents three identical components in a series configuration, then each of those components fails according to the same failure distribution but each component may fail at different times. Because the items are arranged reliability-wise in series, if one of those components fails, then the multi block fails. It is also possible to define a multi block with multiple identical components arranged reliability-wise in parallel or ''k''-out-of-''n'' redundancy. The following figure demonstrates the use of multi blocks in BlockSim.

	[[Image:chp4image30.png\|center\|~~350px~~\|Illustrating multi-blocks.]]		[[Image:chp4image30.png\|center\|500px\|Illustrating multi-blocks.\|link=]]

	= Mirrored Blocks =		= Mirrored Blocks =

Richard House: /* Inherited Subdiagrams */

2016-01-05T16:53:22Z

Inherited Subdiagrams

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	Within BlockSim, a subdiagram block inherits some or all of its properties from another block diagram. This allows the analyst to maintain separate diagrams for portions of a system and incorporate those diagrams as components of another diagram. With this technique, it is possible to generate and analyze extremely complex diagrams representing the behavior of many subsystems in a manageable way. In the first figure below, Subdiagram Block A in the top diagram represents the series configuration of the subsystem reflected in the middle diagram, while Subdiagram Block G in the middle diagram represents the series configuration of the subsubsystem in the bottom diagram.		Within BlockSim, a subdiagram block inherits some or all of its properties from another block diagram. This allows the analyst to maintain separate diagrams for portions of a system and incorporate those diagrams as components of another diagram. With this technique, it is possible to generate and analyze extremely complex diagrams representing the behavior of many subsystems in a manageable way. In the first figure below, Subdiagram Block A in the top diagram represents the series configuration of the subsystem reflected in the middle diagram, while Subdiagram Block G in the middle diagram represents the series configuration of the subsubsystem in the bottom diagram.

	[[Image:chp4image27.png\|center\|~~350px~~\|Illustration of subdiagrams.]]		[[Image:chp4image27.png\|center\|400px\|Illustration of subdiagrams.\|link=]]

	'''Example: Calculating the Reliability with Subdiagrams'''		'''Example: Calculating the Reliability with Subdiagrams'''
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	For this example, obtain the reliability equation of the system shown below.		For this example, obtain the reliability equation of the system shown below.

	[[Image:BS4.23_New.png\|center\|~~500px~~\|An RBD of two computer systems in series where each computer system inherits its diagram from another subdiagram]]		[[Image:BS4.23_New.png\|center\|550px\|An RBD of two computer systems in series where each computer system inherits its diagram from another subdiagram\|link=]]

	The system reliability equation is:		The system reliability equation is:

Richard House: /* Load Sharing and Standby Redundancy */

2016-01-05T16:51:51Z

Load Sharing and Standby Redundancy

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	Units in load sharing redundancy exhibit different failure characteristics when one or more fail. In the figure below, blocks 1, 2 and 3 are in a load sharing container in BlockSim and have their own failure characteristics. All three must fail for the container to fail. However, as individual items fail, the failure characteristics of the remaining units change since they now have to carry a higher load to compensate for the failed ones. The failure characteristics of each block in a load sharing container are defined using both a life distribution and a life-stress relationship that describe how the life distribution changes as the load changes.		Units in load sharing redundancy exhibit different failure characteristics when one or more fail. In the figure below, blocks 1, 2 and 3 are in a load sharing container in BlockSim and have their own failure characteristics. All three must fail for the container to fail. However, as individual items fail, the failure characteristics of the remaining units change since they now have to carry a higher load to compensate for the failed ones. The failure characteristics of each block in a load sharing container are defined using both a life distribution and a life-stress relationship that describe how the life distribution changes as the load changes.

	[[Image:Fig_4.20.PNG\|center\|~~350px~~\|<div align="center">An RBD with a load sharing container. </div>]]		[[Image:Fig_4.20.PNG\|center\|500px\|<div align="center">An RBD with a load sharing container. </div>\|link=]]

	Standby redundancy configurations consist of items that are inactive and available to be called into service when/if an active item fails (i.e., the items are on standby). Within BlockSim, a container block with other blocks inside is used to better achieve and streamline the representation and analysis of standby configurations. The container serves a dual purpose. The first is to clearly delineate and define the standby relationships between the active unit(s) and standby unit(s). The second is to serve as the manager of the switching process. For this purpose, the container can be defined with its own probability of successfully activating standby units when needed.		Standby redundancy configurations consist of items that are inactive and available to be called into service when/if an active item fails (i.e., the items are on standby). Within BlockSim, a container block with other blocks inside is used to better achieve and streamline the representation and analysis of standby configurations. The container serves a dual purpose. The first is to clearly delineate and define the standby relationships between the active unit(s) and standby unit(s). The second is to serve as the manager of the switching process. For this purpose, the container can be defined with its own probability of successfully activating standby units when needed.
	The next figure includes a standby container with three items in standby configuration where one component is active while the other two components are idle. One block within the container must be operating, otherwise the container will fail, leading to a system failure (since the container block is part of a series configuration in this example).		The next figure includes a standby container with three items in standby configuration where one component is active while the other two components are idle. One block within the container must be operating, otherwise the container will fail, leading to a system failure (since the container block is part of a series configuration in this example).

	[[Image:Fig_4.21.PNG\|center\|~~350px~~\|<div align="center">An RBD with a standby container.</div>]]		[[Image:Fig_4.21.PNG\|center\|500px\|<div align="center">An RBD with a standby container.</div>\|link=]]

	Because both of these concepts are better understood when time dependency is considered, they are addressed in more detail in [[Time-Dependent_System_Reliability_(Analytical)\|Time-Dependent System Reliability (Analytical)]].		Because both of these concepts are better understood when time dependency is considered, they are addressed in more detail in [[Time-Dependent_System_Reliability_(Analytical)\|Time-Dependent System Reliability (Analytical)]].

Richard House: /* Difference Between Physical and Reliability-Wise Arrangement */

2016-01-05T16:49:57Z

Difference Between Physical and Reliability-Wise Arrangement

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	The RBD is:		The RBD is:

	[[Image:chp4image24.png\|center\|~~350px~~\|]]		[[Image:chp4image24.png\|center\|500px\|link=]]

	In this example it can be seen that even though the three components were physically arranged in parallel, their reliability-wise arrangement is in series.		In this example it can be seen that even though the three components were physically arranged in parallel, their reliability-wise arrangement is in series.

Richard House: /* Difference Between Physical and Reliability-Wise Arrangement */

2016-01-05T16:49:09Z

Difference Between Physical and Reliability-Wise Arrangement

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	Consider the following circuit:		Consider the following circuit:

	[[Image:BS3sigma.png\|center\|~~250px~~\|]]		[[Image:BS3sigma.png\|center\|350px\|link=]]

	The equivalent resistance must always be less than <math>1.2\Omega \,\!</math>.		The equivalent resistance must always be less than <math>1.2\Omega \,\!</math>.