Reliability and MTBF calculator

Definitions

• Reliability is the probability that an item performs its intended function, under stated conditions, for a stated period of time. All four parts matter; a reliability figure with no time attached means nothing.
• Failure rate λ is failures per unit of operating time: λ = r / T, where r is the number of failures and T the total operating time of all units.
• MTBF (mean time between failures, for repairable items) is T / r = 1 / λ when the failure rate is constant. For non-repairable items the same quantity is called MTTF.

The exponential model and the bathtub curve

With a constant failure rate, reliability over a mission of length t is R(t) = e^−λt = e^−t/MTBF. That holds in the flat middle of the bathtub curve, the useful-life period. It does not hold during infant mortality (falling failure rate, addressed by burn-in and screening) or wear-out (rising failure rate, addressed by replacement before wear-out). A useful check: at t = MTBF, R = e⁻¹ = 0.368. Only about a third of items survive to the MTBF.

Series and parallel systems

• Series: the system works only if every block works. R_sys = R₁ × R₂ × … × R_n. The system is always less reliable than its weakest block. For exponential blocks in series, the failure rates add.
• Active parallel (redundancy): the stage fails only if every unit fails. R = 1 − (1 − R₁)(1 − R₂)…. Two units at 0.90 give 0.99.
• Mixed systems are solved stage by stage: reduce each parallel group to one equivalent block, then multiply the stages in series.

Common traps

• Dividing by the number of units instead of the total unit-hours.
• Reading MTBF as a minimum life or a warranty period.
• Treating a standby unit as active parallel. Standby redundancy depends on the switch and uses a different formula.
• Trusting a point estimate from two or three failures. Report a lower confidence bound for MTBF.

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