## Example 367

A series system consists of three components, where the probability of failure of each individual component is pf = 0.01. What is the probability of failure of the system if the failure modes are independent? If they are perfectly correlated?

If the three failure modes are independent, so the correlation is p = 0.0, the failure probability of the system is pfseries = 1-0 -(1.0 - 0.01)(1.0 - 0.01)(1.0 - 0.01) = 0.0297

If the three failure modes are perfectly correlated, so the correlation is p = 1.0, the failure probability of the system is pfjseries = 0.01.

### 36.4.2 Parallel Systems

A parallel system, also called a redundant or fail-safe system, requires every individual member to fail for the system to fail. A parallel system is at least as safe as its most reliable member. The probability of failure of a parallel system is the probability of an intersection of failure events:

where f]za=1 {ga(X) < 0} is the intersection of all z failure mode events. The possible range of values for this system failure probability are (Ang and Tang 1984)

where the lower bound results from mutual independence and the upper bound from perfect correlation. These bounds are often too wide to provide a useful solution. An alternative approach is to reduce all random variables to their equivalent normal distributions and solve the n-dimensional joint standardized distribution integral (Thoft-Christensen and Murotsu 1986): f i f i f i i pf = ... - 1-e-1=2{b}W 1{b}T d{b}

where {|} = {|a, lb, ..., |z}, psys is the system correlation matrix, and z is the number of members in the parallel system.

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