6. reliability modeling reliable system design 2010 by: amir m. rahmani
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6. Reliability Modeling
Reliable System Design 2010by: Amir M. Rahmani
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Concepts in Probability
Let X denote the lifetime for a component.
F (t) = P ( X ≤ t ) Distribution function
R (t) = 1 – F (t) = P (X > t ) Reliability function
f (t ) = d F (t )/d t Probability density function
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The exponential distribution
F (t) = 1-e -λt Distribution function
R (t) = e -λt Reliability function
f (t) = λe -λt Probability density function
λ is the failure rate for the component and t is the time
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Reliability function
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Distribution function
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Probability density function
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Failure rate function
The exponential distribution has a constant failure rate:
f (t)/R(t)= λe -λt /e -λt = λ
Failure rate:• - Fraction of “units_failing/(Total_unit× time)”
Ex: 1000 units, 3 failed in 2 hours• -Failure rate = 3/(1000×2) = 1.5×10-3 failure per hour
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Mean Time To Failure (MTTF)
Expected time to failure = - Corresponds to inverse of failure rate
Proof: Let X denote the lifetime for the component
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MTTF for the exponential distribution
,
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Example MTTF
What is the probability that a component with an exponentially distributed lifetime will survive the expected lifetime?
Only 37% of the components survive the expected lifetime.
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Reliability Block Diagrams
Series systems Parallel systems m-of-n systems Combinational systems
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Series system with n components
The reliability of the series system is
Iff the component failures are independent.
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Series system of components with exponentially distributed lifetimes
The failure rate of the series system is equal to the sum of the component failure rates.
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MTTF for a series system The failure rate for the series system is
MTTF is
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Example 1 - Series system
The reliability of the system is:
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Parallel system with n components
The reliability of the parallel system is:
where Fpar is the failure probability (distribution function) for the parallel system and Fi the failure probability for component i
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Examples The reliability for parallel systems consisting of 2
or 3 identical components
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The reliability for parallel systems
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MTTF for a parallel system with n components
Let X denote the lifetime of the system
Make the substitution
We then obtain:
Note that MTTF increases slowly with the number components in the system
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MTTF for parallel systems
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Example - MTTF for parallel systems
What is the MTTF for a parallel system consisting of 4 components if the MTTF for one component is 1/λ?
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Example 1, A parallel system
Reliability block diagram The system reliability is:
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Example 2, A parallel system
Reliability block diagram
The system reliability is:
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Coverage
Failure detection: requires concurrent detection.• Need redundancy.
Switchover:• good state loaded in U2.• Process restarted
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Imperfect Coverage
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Example m-of-n systems
Obtain the reliability for a TMR system ( 2-of-3 system). Let R denote the reliability for one module.
RTMR = P (all modules are functioning) +
P (two modules are functioning)
= R3 + 3R2 · (1 – R ) = 3R2 – 2R3
If the lifetimes of the modules are exponentially distributed, we obtain:
R = e-λt => RTMR =3e-2λt – 2e-3λt
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Reliability for TMR and Simplex systems
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MTTF for a TMR-system
The MTTF for the TMR-system is only 5/6 of the MTTF for the simplex System!
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m-of-n systems
In general, the reliability for an m-of-n system is:
Example1: 2-of-3 system
Example2: 2-of-4 system
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Reliability Block Diagram
Series Parallel Graph• – a graph that is recursively composed of series and
parallel structures.• – therefore it can be “collapsed” by applying series
and/or parallel reduction• – Let Ci denote the condition that component i is
operable• 1 = up, 0 = down
• – Let S denote the condition that the system is operable• 1 = up, 0 = down
• – S is a logic function of C’s
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Reliability Block Diagram
Example
S = (C1+C2+C3)(C4C5)(C6+C7C8)• - Parallel (1 of N)• - Series (N of N)
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Combinational systems
Example
• Consider component or components that eliminating or considering the default mode of graph , could be changed into series or parallel combinations.
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Example: C2
( Two state: C2 is ok and C2 fail)
- If C2 is ok:
(C1 || C5).(C4 || C3)
• S1=(C1+C5)(C4+C3)
• Rt1=R2.[(1-(1-R1)(1-R5)).(1-(1-R4)(1-R3))]
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- If C2 fail:
(C1.C4) || (C5. C3)
• S2=(C1.C4) + (C5.C3)
• Rt2=(1-R2).[1-(1-R1R4)(1-R5R3)]
• The system reliability is: =Rt1+Rt2
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Karno Table Example: A parallel system
• The system reliability is: =(1-R1)R2+R1(1-R2)+R1.R2
=R1+R2-R1R2