Phase Throughput Examples - Revision history

Lisa Hacker at 21:02, 18 September 2023

2023-09-18T21:02:40Z

← Older revision		Revision as of 21:02, 18 September 2023
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	<noinclude>{{Banner BlockSim Examples}}		<noinclude>{{Banner BlockSim Examples}}
	''These examples also appear in the [~~[Reliability_Phase_Diagrams_(RPDs)\|~~System ~~Analysis Reference~~]~~] book~~.'' </noinclude><br>		''These examples also appear in the [https://help.reliasoft.com/reference/system_analysis System analysis reference].'' </noinclude><br>
	=== Constant Throughput Example ===		=== Constant Throughput Example ===
	{{:Constant Throughput Example}}		{{:Constant Throughput Example}}

Kate Racaza: Replaced content with '{{Banner BlockSim Examples}} ''These examples also appear in the System Analysis Reference book.''
=== Const…'

2012-08-22T02:35:19Z

Replaced content with '<noinclude>{{Banner BlockSim Examples}} ''These examples also appear in the System Analysis Reference book.'' </noinclude><br> === Const…'

Show changes

Richard House: /* Variable Throughput Example */

2012-08-10T06:06:15Z

Variable Throughput Example

← Older revision		Revision as of 06:06, 10 August 2012
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	\end{align}</math>		\end{align}</math>

	It can be seen that throughput from phase P1 sent to block A increases with time. But the capacity of block A remains constant at 10 units per hour during the entire phase P1. Thus, it becomes important to know the point of time when the number of units sent by P1 exceed the capacity of block A and A starts accumulating a backlog. This can be obtained by solving the throughput model for phase P1 ( <math>y = x\,\!</math> ) using the capacity of block A by substituting <math>y\,\!</math> as the block capacity as shown next:		It can be seen that throughput from phase P1 sent to block A increases with time. But the capacity of block A remains constant at 10 units per hour during the entire phase P1. Thus, it becomes important to know the point of time when the number of units sent by P1 exceed the capacity of block A and A starts accumulating a backlog. This can be obtained by solving the throughput model for phase P1 <math>(y = x)\,\!</math> using the capacity of block A by substituting <math>y\,\!</math> as the block capacity as shown next:

	::<math>\begin{align}		::<math>\begin{align}

Richard House: /* Variable Throughput Example */

2012-08-10T06:05:45Z

Variable Throughput Example

← Older revision		Revision as of 06:05, 10 August 2012
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	:2. Phase P2 from 10 to 20 hours		:2. Phase P2 from 10 to 20 hours

	::-The throughput of phase P2 follows the model <math>y = ax + b\,\!</math> with <math>a = 3\,\!</math> and <math>b = 0\,\!</math> . The throughput from P2 goes to block A. The capacity of block A in this phase is 6 units per hour. The point of time when the number of units sent by P2 equals the capacity of block A can be obtained by substituting the capacity of block A into the model equation ( <math>y = 3x\,\!</math> ) as shown next:		::-The throughput of phase P2 follows the model <math>y = ax + b\,\!</math> with <math>a = 3\,\!</math> and <math>b = 0\,\!</math> . The throughput from P2 goes to block A. The capacity of block A in this phase is 6 units per hour. The point of time when the number of units sent by P2 equals the capacity of block A can be obtained by substituting the capacity of block A into the model equation <math>(y = 3x)\,\!</math> as shown next:

	::<math>\begin{align}		::<math>\begin{align}

Richard House: /* Variable Throughput Example */

2012-08-10T06:03:55Z

Variable Throughput Example

← Older revision		Revision as of 06:03, 10 August 2012
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	:3. Phase P3 from 20 to 30 hours		:3. Phase P3 from 20 to 30 hours
	::-The throughput of phase P3 follows the model <~~span class="texhtml"~~>''y'' = ~~''a''''x'''''<b>~~ + ~~''b''</~~b> </~~span~~> with <~~span class="texhtml"~~>''a'' = 0.5</~~span~~> and <~~span class="texhtml"~~>''b'' = 0</~~span~~> . A maximum throughput capacity of 4 units per hour is also set for this phase. The throughput from this phase goes to block A. The capacity of block A in this phase is 15 units per hour. Thus it can be seen that the throughput sent by P3 will never exceed the capacity of block A in this phase. It can also be seen that at some point of time the throughput of phase P3 will reach the maximum throughput capacity of 4 units per hour and thereafter a constant throughput of 4 units per hour will be sent by phase P3. This point of time can be calculated by solving the linear throughput model for the phase <~~span class="texhtml"~~>''y'' = 0.~~5''x''~~</~~span~~> by substituting <~~span class="texhtml"~~>''y''</~~span~~> as the maximum throughput capacity as shown next:		::-The throughput of phase P3 follows the model <math>y = ax + b\,\!</math> with <math>a = 0.5\,\!</math> and <math>b = 0\,\!</math> . A maximum throughput capacity of 4 units per hour is also set for this phase. The throughput from this phase goes to block A. The capacity of block A in this phase is 15 units per hour. Thus it can be seen that the throughput sent by P3 will never exceed the capacity of block A in this phase. It can also be seen that at some point of time the throughput of phase P3 will reach the maximum throughput capacity of 4 units per hour and thereafter a constant throughput of 4 units per hour will be sent by phase P3. This point of time can be calculated by solving the linear throughput model for the phase <math>y = 0.5x\,\!</math> by substituting <math>y\,\!</math> as the maximum throughput capacity as shown next:

	::<math>\begin{align}		::<math>\begin{align}
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	\end{align}</math>		\end{align}</math>

	Thus, at time <~~span class="texhtml"~~>''x'' = 8</~~span~~> hours of phase P3 (or at the time of 28 hours) the number of units sent per hour by phase P3 will reach the constant value of 4 units per hour. In the first eight hours of the phase the throughput will remain variable. The total number of units sent by P3 to block A during this time is given by:		Thus, at time <math>x = 8\,\!</math> hours of phase P3 (or at the time of 28 hours) the number of units sent per hour by phase P3 will reach the constant value of 4 units per hour. In the first eight hours of the phase the throughput will remain variable. The total number of units sent by P3 to block A during this time is given by:

	::<math>\begin{align}		::<math>\begin{align}

Richard House: /* Variable Throughput Example */

2012-08-10T06:01:23Z

Variable Throughput Example

← Older revision		Revision as of 06:01, 10 August 2012
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	\end{align}</math>		\end{align}</math>

	It can be seen that throughput from phase P1 sent to block A increases with time. But the capacity of block A remains constant at 10 units per hour during the entire phase P1. Thus, it becomes important to know the point of time when the number of units sent by P1 exceed the capacity of block A and A starts accumulating a backlog. This can be obtained by solving the throughput model for phase P1 ( <~~span class="texhtml"~~>''y'' = ''x''</~~span~~> ) using the capacity of block A by substituting <~~span class="texhtml"~~>''y''</~~span~~> as the block capacity as shown next:		It can be seen that throughput from phase P1 sent to block A increases with time. But the capacity of block A remains constant at 10 units per hour during the entire phase P1. Thus, it becomes important to know the point of time when the number of units sent by P1 exceed the capacity of block A and A starts accumulating a backlog. This can be obtained by solving the throughput model for phase P1 ( <math>y = x\,\!</math> ) using the capacity of block A by substituting <math>y\,\!</math> as the block capacity as shown next:

	::<math>\begin{align}		::<math>\begin{align}
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	\end{align}</math>		\end{align}</math>

	Thus, at time <~~span class="texhtml"~~>''x'' = 10</~~span~~> hours the number of units supplied per hour by phase P1 will equal the capacity of block A. At this point of time A will function to its full capacity with no excess capacity and no backlog. Beyond this point the units sent by P1 will exceed the capacity of A and A will start accumulating a backlog. Since the duration of P1 is 10 hours, backlog at A will not be a concern in this phase because up to this time the capacity of A exceeds the units sent by P1. Thus the total units sent by P1 to block A during the phase duration of 10 hours is:		Thus, at time <math>x = 10\,\!</math> hours the number of units supplied per hour by phase P1 will equal the capacity of block A. At this point of time A will function to its full capacity with no excess capacity and no backlog. Beyond this point the units sent by P1 will exceed the capacity of A and A will start accumulating a backlog. Since the duration of P1 is 10 hours, backlog at A will not be a concern in this phase because up to this time the capacity of A exceeds the units sent by P1. Thus the total units sent by P1 to block A during the phase duration of 10 hours is:

	::<math>\begin{align}		::<math>\begin{align}
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	:2. Phase P2 from 10 to 20 hours		:2. Phase P2 from 10 to 20 hours

	::-The throughput of phase P2 follows the model <~~span class="texhtml"~~>''y'' = ~~''a''''x'''''<b>~~ + ''b''</~~b> </span~~> with <~~span class="texhtml"~~>''a'' = 3</~~span~~> and <~~span class="texhtml"~~>''b'' = 0</~~span~~> . The throughput from P2 goes to block A. The capacity of block A in this phase is 6 units per hour. The point of time when the number of units sent by P2 equals the capacity of block A can be obtained by substituting the capacity of block A into the model equation ( <~~span class="texhtml"~~>''y'' = ~~3''x''~~</~~span~~> ) as shown next:		::-The throughput of phase P2 follows the model <math>y = ax + b\,\!</math> with <math>a = 3\,\!</math> and <math>b = 0\,\!</math> . The throughput from P2 goes to block A. The capacity of block A in this phase is 6 units per hour. The point of time when the number of units sent by P2 equals the capacity of block A can be obtained by substituting the capacity of block A into the model equation ( <math>y = 3x\,\!</math> ) as shown next:

	::<math>\begin{align}		::<math>\begin{align}
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	\end{align}</math>		\end{align}</math>

	Thus, at <~~span class="texhtml"~~>''x'' = 2</~~span~~> hours of phase P2 (or at the total simulation time of 12 hours), the number of units sent per hour by phase P2 equals the capacity of block A. Thus, during the first two hours of phase P2, A will have excess capacity. The total number of units sent by phase P2 during the first two hours of the phase are:		Thus, at <math>x = 2\,\!</math> hours of phase P2 (or at the total simulation time of 12 hours), the number of units sent per hour by phase P2 equals the capacity of block A. Thus, during the first two hours of phase P2, A will have excess capacity. The total number of units sent by phase P2 during the first two hours of the phase are:

	::<math>\begin{align}		::<math>\begin{align}

Richard House: /* Variable Throughput Example */

2012-08-10T05:57:53Z

Variable Throughput Example

← Older revision		Revision as of 05:57, 10 August 2012
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	:1. Phase P1 from 0 to 10 hours		:1. Phase P1 from 0 to 10 hours
	:::a) The throughput from phase P1 follows a time-varying model given by <~~span class="texhtml"~~>''y'' = ~~''a''''x'''''<b>~~ + ''b''</~~b> </span~~> with parameters <~~span class="texhtml"~~>''a'' = 1</~~span~~> and <~~span class="texhtml"~~>''b'' = 0</~~span~~> . Thus, during the first hour the throughput sent by phase P1 to block A is:		:::a) The throughput from phase P1 follows a time-varying model given by <math>y = ax + b\,\!</math> with parameters <math>a = 1\,\!</math> and <math>b = 0\,\!</math> . Thus, during the first hour the throughput sent by phase P1 to block A is:
	::<math>\begin{align}		::<math>\begin{align}
	Phase throughput from 0 to1hr= & \frac{a}{2}(x_{2}^{2}-x_{1}^{2})+b({{x}_{2}}-{{x}_{1}}) \\		Phase throughput from 0 to1hr= & \frac{a}{2}(x_{2}^{2}-x_{1}^{2})+b({{x}_{2}}-{{x}_{1}}) \\

Richard House: /* Variable Throughput Example */

2012-08-10T05:43:54Z

Variable Throughput Example

← Older revision		Revision as of 05:43, 10 August 2012
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	=== Variable Throughput Example ===		=== Variable Throughput Example ===

	To examine variable throughput in phase diagrams, consider the phase diagram used above in the constant throughput example. All phase properties and linked RBDs remain unchanged for this illustration except that all the three phases now use variable throughput models. For the first phase P1 the variable throughput model used is <math>y = x\,\!</math> (which is the linear model <math>y = ax+ b\,\!</math> with parameters <math>a = 1\,\! </math> and <math>b = 0\,\!</math>). For the second phase P2 the model used is <math>y = 3x\,\!</math> (which is the linear model <math>y = ax+ b\,\! </math> with parameters <math>a = 3\,\! </math> and <math>b = 0\,\! </math>). The third phase uses <~~span class="texhtml"~~>''y''= 0.~~5''x''~~ </~~span~~> as the variable throughput model (which is the linear model <~~span class="texhtml"~~> ''y''= ~~''a'' ''x''~~+ ''b''</~~span~~> with parameters <~~span class="texhtml"~~>''a'' = 0.5</~~span~~> and <~~span class="texhtml"~~> ''b'' = 0</~~span~~>). Phases P1 and P2 have a value of 500 units per hour set for the maximum throughput capacity. This relatively larger value of throughput is set for the two phases so that the phases do not reach their maximum throughput value during the simulation. Phase P3 has a value of 4 units per hour set as the maximum throughput capacity. As in the previous example, all phases have a duration of ten hours. For the sake of simplicity it is assumed that the blocks can never fail and that items are routed equally to each path. The ignore backlog option is left unchecked for all blocks in the phase diagram.		To examine variable throughput in phase diagrams, consider the phase diagram used above in the constant throughput example. All phase properties and linked RBDs remain unchanged for this illustration except that all the three phases now use variable throughput models. For the first phase P1 the variable throughput model used is <math>y = x\,\!</math> (which is the linear model <math>y = ax+ b\,\!</math> with parameters <math>a = 1\,\! </math> and <math>b = 0\,\!</math>). For the second phase P2 the model used is <math>y = 3x\,\!</math> (which is the linear model <math>y = ax+ b\,\! </math> with parameters <math>a = 3\,\! </math> and <math>b = 0\,\! </math>). The third phase uses <math>y = 0.5x\,\!</math> as the variable throughput model (which is the linear model <math>y = ax + b\,\!</math> with parameters <math>a = 0.5\,\!</math> and <math>b = 0\,\!</math>). Phases P1 and P2 have a value of 500 units per hour set for the maximum throughput capacity. This relatively larger value of throughput is set for the two phases so that the phases do not reach their maximum throughput value during the simulation. Phase P3 has a value of 4 units per hour set as the maximum throughput capacity. As in the previous example, all phases have a duration of ten hours. For the sake of simplicity it is assumed that the blocks can never fail and that items are routed equally to each path. The ignore backlog option is left unchecked for all blocks in the phase diagram.

	[[Image:11.21.png\|center\|250px]]		[[Image:11.21.png\|center\|250px]]

Richard House: /* Variable Throughput Example */

2012-08-10T05:41:33Z

Variable Throughput Example

← Older revision		Revision as of 05:41, 10 August 2012
Line 73:		Line 73:
	=== Variable Throughput Example ===		=== Variable Throughput Example ===

	To examine variable throughput in phase diagrams, consider the phase diagram used above in the constant throughput example. All phase properties and linked RBDs remain unchanged for this illustration except that all the three phases now use variable throughput models. For the first phase P1 the variable throughput model used is <math>y = x\,\!</math> (which is the linear model <math>y = ax+ b\,\!</math> with parameters <~~span class="texhtml"~~ >''a''= 1 </~~span~~> and <~~span class="texhtml"~~>''b''= 0)</~~span~~>. For the second phase P2 the model used is <~~span class="texhtml"~~>''y''= ~~3''x''~~</~~span~~> (which is the linear model <~~span class="texhtml"~~>''y''= ~~''a'' ''x''~~+ ''b'' </~~span~~> with parameters <~~span class="texhtml~~>''a''= 3 </~~span~~> and <~~span class="texhtml"~~>''b''= 0 </~~span~~>). The third phase uses <span class="texhtml">''y''= 0.5''x'' </span> as the variable throughput model (which is the linear model <span class="texhtml"> ''y''= ''a'' ''x''+ ''b''</span> with parameters <span class="texhtml">''a'' = 0.5</span> and <span class="texhtml"> ''b'' = 0</span>). Phases P1 and P2 have a value of 500 units per hour set for the maximum throughput capacity. This relatively larger value of throughput is set for the two phases so that the phases do not reach their maximum throughput value during the simulation. Phase P3 has a value of 4 units per hour set as the maximum throughput capacity. As in the previous example, all phases have a duration of ten hours. For the sake of simplicity it is assumed that the blocks can never fail and that items are routed equally to each path. The ignore backlog option is left unchecked for all blocks in the phase diagram.		To examine variable throughput in phase diagrams, consider the phase diagram used above in the constant throughput example. All phase properties and linked RBDs remain unchanged for this illustration except that all the three phases now use variable throughput models. For the first phase P1 the variable throughput model used is <math>y = x\,\!</math> (which is the linear model <math>y = ax+ b\,\!</math> with parameters <math>a = 1\,\! </math> and <math>b = 0\,\!</math>). For the second phase P2 the model used is <math>y = 3x\,\!</math> (which is the linear model <math>y = ax+ b\,\! </math> with parameters <math>a = 3\,\! </math> and <math>b = 0\,\! </math>). The third phase uses <span class="texhtml">''y''= 0.5''x'' </span> as the variable throughput model (which is the linear model <span class="texhtml"> ''y''= ''a'' ''x''+ ''b''</span> with parameters <span class="texhtml">''a'' = 0.5</span> and <span class="texhtml"> ''b'' = 0</span>). Phases P1 and P2 have a value of 500 units per hour set for the maximum throughput capacity. This relatively larger value of throughput is set for the two phases so that the phases do not reach their maximum throughput value during the simulation. Phase P3 has a value of 4 units per hour set as the maximum throughput capacity. As in the previous example, all phases have a duration of ten hours. For the sake of simplicity it is assumed that the blocks can never fail and that items are routed equally to each path. The ignore backlog option is left unchecked for all blocks in the phase diagram.

	[[Image:11.21.png\|center\|250px]]		[[Image:11.21.png\|center\|250px]]

Richard House: /* Variable Throughput Example */

2012-08-10T05:38:55Z

Variable Throughput Example

← Older revision		Revision as of 05:38, 10 August 2012
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	=== Variable Throughput Example ===		=== Variable Throughput Example ===

	To examine variable throughput in phase diagrams, consider the phase diagram used above in the constant throughput example. All phase properties and linked RBDs remain unchanged for this illustration except that all the three phases now use variable throughput models. For the first phase P1 the variable throughput model used is <~~span class="texhtml"~~> ''y'' = ''x''</~~span~~> (which is the linear model <~~span class="texhtml"~~>''y'' = ~~''a'' ''x''~~+ ''b''</~~span~~> with parameters <span class="texhtml" >''a''= 1 </span> and <span class="texhtml">''b''= 0)</span>. For the second phase P2 the model used is <span class="texhtml">''y''= 3''x''</span> (which is the linear model <span class="texhtml">''y''= ''a'' ''x''+ ''b'' </span> with parameters <span class="texhtml>''a''= 3 </span> and <span class="texhtml">''b''= 0 </span>). The third phase uses <span class="texhtml">''y''= 0.5''x'' </span> as the variable throughput model (which is the linear model <span class="texhtml"> ''y''= ''a'' ''x''+ ''b''</span> with parameters <span class="texhtml">''a'' = 0.5</span> and <span class="texhtml"> ''b'' = 0</span>). Phases P1 and P2 have a value of 500 units per hour set for the maximum throughput capacity. This relatively larger value of throughput is set for the two phases so that the phases do not reach their maximum throughput value during the simulation. Phase P3 has a value of 4 units per hour set as the maximum throughput capacity. As in the previous example, all phases have a duration of ten hours. For the sake of simplicity it is assumed that the blocks can never fail and that items are routed equally to each path. The ignore backlog option is left unchecked for all blocks in the phase diagram.		To examine variable throughput in phase diagrams, consider the phase diagram used above in the constant throughput example. All phase properties and linked RBDs remain unchanged for this illustration except that all the three phases now use variable throughput models. For the first phase P1 the variable throughput model used is <math>y = x\,\!</math> (which is the linear model <math>y = ax+ b\,\!</math> with parameters <span class="texhtml" >''a''= 1 </span> and <span class="texhtml">''b''= 0)</span>. For the second phase P2 the model used is <span class="texhtml">''y''= 3''x''</span> (which is the linear model <span class="texhtml">''y''= ''a'' ''x''+ ''b'' </span> with parameters <span class="texhtml>''a''= 3 </span> and <span class="texhtml">''b''= 0 </span>). The third phase uses <span class="texhtml">''y''= 0.5''x'' </span> as the variable throughput model (which is the linear model <span class="texhtml"> ''y''= ''a'' ''x''+ ''b''</span> with parameters <span class="texhtml">''a'' = 0.5</span> and <span class="texhtml"> ''b'' = 0</span>). Phases P1 and P2 have a value of 500 units per hour set for the maximum throughput capacity. This relatively larger value of throughput is set for the two phases so that the phases do not reach their maximum throughput value during the simulation. Phase P3 has a value of 4 units per hour set as the maximum throughput capacity. As in the previous example, all phases have a duration of ten hours. For the sake of simplicity it is assumed that the blocks can never fail and that items are routed equally to each path. The ignore backlog option is left unchecked for all blocks in the phase diagram.

	[[Image:11.21.png\|center\|250px]]		[[Image:11.21.png\|center\|250px]]

Phase Throughput Examples - Revision history

Lisa Hacker at 21:02, 18 September 2023

Kate Racaza: Replaced content with '{{Banner BlockSim Examples}} ''These examples also appear in the System Analysis Reference book.'' === Const…'

Richard House: /* Variable Throughput Example */

Richard House: /* Variable Throughput Example */

Richard House: /* Variable Throughput Example */

Richard House: /* Variable Throughput Example */

Richard House: /* Variable Throughput Example */

Richard House: /* Variable Throughput Example */

Richard House: /* Variable Throughput Example */

Richard House: /* Variable Throughput Example */

Kate Racaza: Replaced content with '{{Banner BlockSim Examples}} ''These examples also appear in the System Analysis Reference book.''
=== Const…'