6. Reliability Modeling Reliable System Design 2010 by: Amir M. Rahmani

matlab1.ir 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

matlab1.ir 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

matlab1.ir Reliability function

matlab1.ir Distribution function

matlab1.ir Probability density function

matlab1.ir 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

matlab1.ir Mean Time To Failure (MTTF) Expected time to failure = - Corresponds to inverse of failure rate Proof: Let X denote the lifetime for the component

matlab1.ir MTTF for the exponential distribution,

matlab1.ir 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.

matlab1.ir Reliability Block Diagrams Series systems Parallel systems m-of-n systems Combinational systems

matlab1.ir Series system with n components The reliability of the series system is Iff the component failures are independent.

matlab1.ir 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.

matlab1.ir MTTF for a series system The failure rate for the series system is MTTF is

matlab1.ir Example 1 - Series system The reliability of the system is:

matlab1.ir Parallel system with n components The reliability of the parallel system is: where F par is the failure probability (distribution function) for the parallel system and F i the failure probability for component i

matlab1.ir Examples The reliability for parallel systems consisting of 2 or 3 identical components

matlab1.ir The reliability for parallel systems

matlab1.ir 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

matlab1.ir MTTF for parallel systems

matlab1.ir 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?

matlab1.ir Example 1, A parallel system Reliability block diagram The system reliability is:

matlab1.ir Example 2, A parallel system Reliability block diagram The system reliability is:

matlab1.ir Coverage Failure detection: requires concurrent detection. Need redundancy. Switchover: good state loaded in U2. Process restarted

matlab1.ir Imperfect Coverage

matlab1.ir Example m-of-n systems Obtain the reliability for a TMR system ( 2-of-3 system). Let R denote the reliability for one module. R TMR = P (all modules are functioning) + P (two modules are functioning) = R 3 + 3R 2 · (1 – R ) = 3R 2 – 2R 3 If the lifetimes of the modules are exponentially distributed, we obtain: R = e -λt => R TMR =3e -2λt – 2e -3λt

matlab1.ir Reliability for TMR and Simplex systems

matlab1.ir MTTF for a TMR-system The MTTF for the TMR-system is only 5/6 of the MTTF for the simplex System!

matlab1.ir 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

matlab1.ir 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 C i 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

matlab1.ir Reliability Block Diagram Example S = (C 1 +C 2 +C 3 )(C 4 C 5 )(C 6 +C 7 C 8 ) - Parallel (1 of N) - Series (N of N)

matlab1.ir Combinational systems Example Consider component or components that eliminating or considering the default mode of graph, could be changed into series or parallel combinations.

matlab1.ir Example: C 2 ( Two state: C 2 is ok and C 2 fail) - If C 2 is ok : (C 1 || C 5 ).(C 4 || C 3 ) S 1 =(C 1 +C 5 )(C 4 +C 3 ) R t1 =R 2.[(1-(1-R 1 )(1-R 5 )).(1-(1-R 4 )(1-R 3 ))]

matlab1.ir - If C 2 fail: (C 1.C 4 ) || (C 5. C 3 ) S 2 =(C 1.C 4 ) + (C 5.C 3 ) R t2 =(1-R 2 ).[1-(1-R 1 R 4 )(1-R 5 R 3 )] The system reliability is: =R t1 +R t2

matlab1.ir Karno Table Example: A parallel system The system reliability is: =(1-R 1 )R 2 +R 1 (1-R 2 )+R 1.R 2 =R 1 +R 2 -R 1 R 2