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• The MTTR (Mean Time To Repair) for a system is the average time to repair a system. This is not simple to determine and often is based on experimental estimates.
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number of repairs |
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time------------------------------------------------------------period for all repairs |
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• The MTTF and MTTR both measure the time that the system is running between repairs, and the time the system is down for repairs. But, they must be combined for the more useful measure MTBF (Mean Time Before Failure),
MTBF = MTTF + MTTR
• The difference between MTBF and MTTR is often small, but when critical the difference must be observed.
8.1.3 The Theory of System Reliability
•Fault Coverage is the probability that a system will recover from a failure. This can be derived approximately by examining the design, and making reliable estimates. This number will be difficult to determine exactly because it is based on real, and often unpredictable phenomenon.
•Reliability can be determined with individual system components as a function of probabilities. The two main categories of systems are series, and parallel (redundant). In the best case a high reliability system would have many parallel systems in series.
•In terms of design, a system designer must have an intuitive understanding of the concept of series/parallel functions.
•We can consider a series system where if any of the units fails, then the system becomes inoperative. Here the reliabilities of each of the system components is chained (ANDed) together.
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Rs( t) = ( R1( t) ) ( R2( t) )… ( Rn( t) ) = ∏ Ri( t)
i = 1
where,
Rs( t) = the reliability of a series system at time t
Ri( t) = the reliability of a unit at timet
Now, consider the exponential failure law presented before. If each element in a system observes this law, then we can get an exact value of reliability.
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• We can also consider a parallel system. If any of the units fails the system will continue to operate. Failure will only come when all of the modules fail. Here we are concerned with complements of the chained unreliabilities.
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R1( t) ,λ 1
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R2( t) ,λ 2
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where,
Qs( t) = the unreliability of a parallel system at time t
Qi( t) = the unreliability of a module at time t
Rp( t) = the reliability of a parallel system at time t
Ri( t) = the unreliability of a module at time t
• also consider the case of a parallel system that requires ‘m’ of ‘n’ identical modules to be functional, such as a hybrid system, or a voting system that needs two out of three functional units. The student will consider the binomial form of the probabilities.