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# Risk Analysis and Management Fault tolerant systems Nuclear Power Plant Safety

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2. A fault-tolerant nuclear reactor protection system consists of 3 processors and 6 memories. The system will fail if (a) any 2 of the 3 processors fail, or (b) any 2 of the 6 memories fail. The nominal failure rates of these processors and memories, provided by the manufacturers, are 1 per 10,000 hours and 1 per 2,000 hours respectively.

(a) Calculate the probability that the system will fail in 3,000 hours.
(b) Supposing no failure of any component occurs in 5,000 hours of operation. What would be your revised estimate of failure rates of the components in your system?

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#### Solution Preview

(a) Calculate the probability that the system will fail in 3,000 hours.

The probability that a processor will fail one or more times during 3,000 hours can be
calculated using Poisson distribution as follows,
pproc = 1 - p(r=0,&#61548; = 10-4 h-1, T = 3000 h) = 0.2592

The probability that 2 or more out of 3 processors will fail in 3,000 hours can be calculated using Binominal distribution
P2-out-of-3 proc = p(r = 2,N=3,p=0.2592) + p(r = 3,N=3,p=0.2592) = 0.1667

Similarly, the probability that a memory will fail in 3,000 hours is
pmem = 1 - p(r=0,&#61548; = 5 x 10-4 h-1, T = 3000 h) = 0.7769

The probability that 2 or more out of 6 memories will fail in 3,000 ...

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