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Serial/Parallel System MTBF

Roll up component failure rates into a system MTBF for series or active-parallel redundancy.

InputSeries: λ_sys = Σλᵢ, MTBF = 1/λ_sys , Parallel(n identical): MTBF = (1/λ)·Σ_{k=1}^{n} 1/k , R_series = ∏Rᵢ, R_parallel = 1 − ∏(1−Rᵢ)

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The engineering

In a series system every component must survive, so failure rates add and the system MTBF is always lower than the weakest part — this is the default for any chain of boxes with no redundancy. In active parallel the system lives until the last unit dies, and for identical exponential units the MTBF grows only as the harmonic series, so a second unit buys you 50% more MTBF, a third only another 33%.

Watch the diminishing returns: doubling from one to two parallel units is the big win; past three or four the added MTBF is tiny relative to the extra hardware, weight, and interconnect failures. The reliability R(t) numbers are usually a better decision metric than MTBF for a fixed mission — a system with high MTBF can still have poor short-mission reliability if you have many series parts.

Where this math comes from

System reliability math grew out of WWII electronics, where vacuum-tube gear failed so often that whole missions were scrubbed. The constant-failure-rate (exponential) model that lets you simply add series rates was formalized in the 1950s by reliability engineers at the RAND Corporation, ARINC, and the military, giving planners a way to predict field failures before deployment.

The US military codified it in MIL-HDBK-217, first issued in 1962, which gave part-level failure rates and the series/parallel combination rules used here. Successive revisions through 217F (1991) became the de facto handbook for electronics reliability prediction, and though officially retired its methods still underpin most FIT and MTBF rollups.

  1. 1952US DoD AGREE groupAdvisory Group on Reliability of Electronic Equipment formalizes exponential-model reliability prediction.
  2. 1957ARINC / RAND engineersPublish series/parallel network reliability combination methods.
  3. 1962US Department of DefenseIssues MIL-HDBK-217 with part failure rates and system rollup rules.
  4. 1991US Air Force RADCReleases MIL-HDBK-217F, the last major revision still widely referenced.

See the full timeline of the math behind every calculator →

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