Page: © 2007 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 6Stand-by Redundancy!When primary component fails, standby component is started up.!Stand-by spares are cold spares => unpowered!Switching equipment assumed failure free Let denote the lifetime of the i-th component from the time it is put into operation until its failure.System lifetime: 1Page: © 2007 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 6Stand-by Redundancy!MTTF –gain is linear as a function of the number of components, unlike the case of parallel redundancy–added complexity of detection and switching mechanism2Page: © 2007 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 6M-of-N SystemStarting with N components, we need any M components operable for the system to be operable.Example: TMRWhere is the reliability of the i-th component3Page: © 2007 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 6M-of-N SystemThe probability that exactly j components are not operating isthen4R tNiQ t R tMofNi N iiN M( ) ( ) ( )=!"#$%&'='(0NjQ t RNjNj N jj N j!"#$%&!"#$%&=''( )!!( )!(t) withPage: © 2007 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 6Reliability 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’s5Page: © 2007 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 6Reliability Block Diagram–Example:C1C3C2 C5C4C6C7 C8+ => parallel (1 of N). => series (N of N)6Page: © 2007 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 6K of N system!Example 2-of-3 systemmay abbreviatedraw as parallel C1C3C22-of-37Page: © 2007 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 6Example: Bus-Guardianpass transistorcontrol logic• assume ! for transistor & logic• 50/50 split: fail-on/fail-offTwo failure states for system•QA = failed active (babbling) with•QP = failed passive with bus8Page: © 2007 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 6Example: Bus-Guardianfor eachstage9Page: © 2007 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 6Example: Bus-Guardian!Active Failure–if any one bus guardian is correct then no babble possible–thus we use 1-of-N parallel system model10Page: © 2007 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 6Example: Bus-Guardian–Solution - Parallel»if any one bus guardian is correct then no babble possible»1-of-N parallel system model11Page: © 2007 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 6Example: Bus-Guardiancompute: compute:in general: danger of cancellation => catastrophic results, => legal issues (even though one should realize what the fail rates really mean)12Page: © 2007 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 6Example: Bus-Guardian simplification:3 drivers result in approx. MTTF of twice and not three times that of single driver13e te tAAtt!!= " #= =$$01 0 as with Page: © 2007 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 6Example: Bus-Guardian!Passive Failure–any one of N bus guardians can take out subsystem–thus we use series system modelGiven ! R(t) = e"3#t= 0.970445514Page: © 2007 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 6Example: Bus-Guardian!summary–active failure => parallel => QA –passive failure => series => QP–whole system fails if either mode occurs => seriesQA QP15Page: © 2007 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 6Example: Bus-Guardian!summarySimplex Triplex16Page: © 2007 A.W. Krings CS449/549 Fault-Tolerant Systems Sequence 6What is the unreliability QA?!Two approaches to compute Q(t) at 1000husing we compute ! and use1)2)Now we compute Q(1000) and ...What
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