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7. Reliability Modeling

7. Reliability Modeling. Concepts in Probability. Let X denote the life time 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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7. Reliability Modeling

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  1. 7. Reliability Modeling

  2. Concepts in Probability • Let X denote the life time 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

  3. The exponential distribution F (t) = 1-e -λt Distribution function R (t) = e -λt Reliability function f (t) = λe -λtProbability density function • λ is the failure rate for the component and t is the time

  4. Reliability function

  5. Distribution function

  6. Probability density function

  7. Failure rate function h (t) =f (t)/R(t) • The exponential distribution has a constant failure rate: h (t) =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

  8. Proof of h (t) =f (t)/R (t) Cont.

  9. Proof of h (t) =f (t)/R (t) Cont.

  10. Mean Time To Failure (MTTF) • Expected time to failure = - Corresponds to inverse of failure rate • Proof: Let X denote the lifetime for the component

  11. MTTF for the exponential distribution ,

  12. 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.

  13. Reliability Block Diagrams • Series systems • Parallel systems • m-of-n systems • Hybrid

  14. Series system with n components • The reliability of the series system is • Iffthe component failures are independent.

  15. 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.

  16. MTTF for a series system • The failure rate for the series system is • MTTF is

  17. Example 1 - Series system • The reliability of the system is:

  18. Parallel system with n components • The reliability of the parallel system is: • where Fparis the failure probability (distribution function) for the parallel system and Fithe failure probability for component i

  19. Examples • The reliability for parallel systems consisting of 2 or 3 identical components

  20. The reliability for parallel systems

  21. 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

  22. MTTF for parallel systems

  23. 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?

  24. Example 1, A parallel system Reliability block diagram • The system reliability is:

  25. Example 2, A parallel system Reliability block diagram • The system reliability is:

  26. Coverage • Failure detection: requires concurrent detection. • Need redundancy. • Switchover: • good state loaded in U2. • Process restarted

  27. Imperfect Coverage

  28. 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

  29. Reliability for TMR and Simplex systems

  30. MTTF for a TMR-system • The MTTF for the TMR-system is only 5/6 of the MTTF for the simplex System!

  31. 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

  32. 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 Cidenote 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

  33. Reliability Block Diagram • Example • S = (C1+C2+C3)(C4C5)(C6+C7C8) • - Parallel (1 of N) • - Series (N of N)

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