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