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Series/Parallel System Reliability

Combine identical block reliabilities into a series chain, a parallel (redundant) bank, or a k-out-of-n voting group.

InputSeries: R = Rⁿ , Parallel: R = 1 − (1−R)ⁿ , k-of-n: R = Σ_{i=k}^{n} C(n,i)·Rⁱ·(1−R)^{n−i}

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The engineering

Reliability block diagrams reduce a system to the pattern of how failures propagate. In series everything must survive, so reliabilities multiply and the chain is always weaker than its worst link. In parallel only one path must survive, so the redundant bank is stronger than any single block — you multiply the unreliabilities instead.

The k-out-of-n case sits between them: a triple-redundant sensor voting 2-of-3 tolerates one failure but not two. Rule of thumb — adding series stages punishes you fast (ten 0.99 blocks give only 0.904), while a single parallel spare on a 0.9 block jumps the pair to 0.99. Note this uses one identical R per block; mixed blocks need the full RBD product by hand.

Where this math comes from

System reliability math grew out of World War II and the missile-age push to make complex electronics predictable rather than merely hoped-for. By the 1950s the U.S. military was formalizing reliability prediction, and the reliability block diagram became the standard shorthand for turning a schematic into a probability.

Dimitri Kececioglu, working at the University of Arizona from the 1960s onward, codified this material into the reliability-engineering curriculum — his multi-volume Reliability Engineering Handbook lays out the series, parallel, and k-out-of-n formulas exactly as this card evaluates them, and trained a generation of reliability engineers.

  1. 1952U.S. DoD AGREEAdvisory Group on Reliability of Electronic Equipment formalizes reliability prediction.
  2. 1957RADC / MIL-HDBKHandbook methods make block-diagram reliability standard engineering practice.
  3. 1965Dimitri KececiogluBuilds the University of Arizona reliability program and codifies the RBD formulas.
  4. 1991Dimitri KececiogluReliability Engineering Handbook published — the reference for these expressions.

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