K-out-of-n Systems: Difference between revisions

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Since the time to failure is exponential, system reliability is calculated via Equation 4.23 on page 78 by substituting the given data as:
Since the time to failure is exponential, system reliability is calculated via Equation 4.23 on page 78 by substituting the given data as:


::<math>R(t) = \sum^{n}_{i=k}C^{i}_{5} e^{-\lambda i t} (1-e^{-\lambda t})^{n-i}\,\!</math>
::<math>R(t) = \sum^{n}_{i=k}C^{i}_{n} e^{-\lambda i t} (1-e^{-\lambda t})^{n-i}\,\!</math>


::<math>R(8760) = \sum^{5}_{i=3}C^{i}_{5} e^{-2.7 \times 10^{-5} \times 8760i} (1-e^{-2.7 \times 10^{-5} \times 8760})^{5-i} = 0.9336\,\!</math>
::<math>R(8760) = \sum^{5}_{i=3}C^{i}_{5} e^{-2.7 \times 10^{-5} \times 8760i} (1-e^{-2.7 \times 10^{-5} \times 8760})^{5-i} = 0.9336\,\!</math>
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Each server is modeled with an exponential distribution with a lambda of 2.7 × 10-5 failures per hour.
Each server is modeled with an exponential distribution with a lambda of 2.7 × 10<sup>-5</sup> failures per hour.




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The reliability of the web host after one year of continuous operation (8760 hours) is calculated in the QCP as 93.36%, which is the same as the result calculated in the reference book.
The reliability of the web host after one year of continuous operation (8760 hours) is calculated in the QCP as 93.36%, which is the same as the result calculated in the reference book.


[[Image:kn_qcprel.png|center]]
[[Image:kn_qcprel.png|center|500px]]




Mean time between failures (MTBF) is estimated to be 29,004 hours.
The mean time between failures (MTBF) is estimated to be 29,004 hours.


[[Image:kn_qcpmtfbf.png|center]]
[[Image:kn_qcpmtbf.png|center|500px]]




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We can also estimate the results by using the simulation tool in BlockSim. The simulation settings are shown below.  
We can also estimate the results by using the simulation tool in BlockSim. The simulation settings are shown below.  


[[Image:kn_sim.png|center]]
[[Image:kn_sim.png|center|500px]]




The point reliability after 8760 hours of use is estimated to be 93.28%.
The point reliability after 8760 hours of use is estimated to be 93.45%.


[[Image:kn_simqcprel.png|center]]
[[Image:kn_simqcprel.png|center|500px]]




And the mean time to failure (MTTF) is estimated to be 29,105 hours.
And the mean time to failure (MTTF) is estimated to be 29,048 hours.


[[Image:kn_simqcpmttf.png|center]]
[[Image:kn_simqcpmttf.png|center|500px]]

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K-out-of-n Systems

This example validates the results for k-out-of-n systems in BlockSim's analytical and simulation diagrams.


Reference Case

This data set is from example 4.6 on page 78 in the book Life Cycle Reliability Engineering by Dr. Guangbin Yang, John Wiley & Sons, 2007.


Data

A web host has 5 independent and identical servers connected in parallel, and at least 3 of them must be operational for the web service to be operational. Each server has an exponential life distribution model with a lambda of 2.7 × 10-5 failures per hour. Mean time between failures (MTBF) and the reliability of the web host after one year of continuous operation are calculated.


Result

It is assumed that the server is repaired immediately to a good-as-new condition if failed; therefore, the MTBF is the same as the MTTF and is calculated by using Equation 4.24 on page 78:

[math]\displaystyle{ MTTF = \frac{1}{\lambda}\sum^{n}_{i=k}\frac{1}{i} = \frac{1}{2.7 \times 10^{-5}}\sum^{5}_{i=3}\frac{1}{i} = 2.9 \times 10^{4}\,\! }[/math]   hours


Since the time to failure is exponential, system reliability is calculated via Equation 4.23 on page 78 by substituting the given data as:

[math]\displaystyle{ R(t) = \sum^{n}_{i=k}C^{i}_{n} e^{-\lambda i t} (1-e^{-\lambda t})^{n-i}\,\! }[/math]
[math]\displaystyle{ R(8760) = \sum^{5}_{i=3}C^{i}_{5} e^{-2.7 \times 10^{-5} \times 8760i} (1-e^{-2.7 \times 10^{-5} \times 8760})^{5-i} = 0.9336\,\! }[/math]


Results in BlockSim

In BlockSim, the server system RBD is configured as shown below.

Kn rbd.png


Each server is modeled with an exponential distribution with a lambda of 2.7 × 10-5 failures per hour.


Analytical Proof

The reliability of the web host after one year of continuous operation (8760 hours) is calculated in the QCP as 93.36%, which is the same as the result calculated in the reference book.

Kn qcprel.png


The mean time between failures (MTBF) is estimated to be 29,004 hours.

Kn qcpmtbf.png


Simulation Proof

We can also estimate the results by using the simulation tool in BlockSim. The simulation settings are shown below.

Kn sim.png


The point reliability after 8760 hours of use is estimated to be 93.45%.

Kn simqcprel.png


And the mean time to failure (MTTF) is estimated to be 29,048 hours.

Kn simqcpmttf.png
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