Repairable m-of-n Redundancy and Standby Reliability

Content

Repairable m-of-n redundancy and standby reliability

Compare active repair, finite-mission m-of-n reliability, and ideal cold standby with transparent assumptions.

Identical active units with repair

count

count

Falex
/10⁶ h

h

How to use

  1. Choose Active repair, Mission reliability, or Cold standby.
  2. For active repair, enter units required m, total active units n, each unit’s constant failure rate in failures per million hours, and mean corrective repair time in hours.
  3. For mission reliability, enter m, n, mission time, and either an exponential failure rate or Weibull shape β and characteristic life η. Enter zero renewal time to omit scheduled-renewal MTBF.
  4. For cold standby, enter total units, mission time, and active-unit exponential failure rate. This mode always means one operating unit plus ideal dormant spares.
  5. Select Calculate to update the canonical result, secondary metrics, and reliability trend. Select Reset to restore the documented active-repair example.
  6. Interpret active-repair MTBCF as a long-run critical-failure screening result; interpret mission and standby results as survival probability over the entered time.
  7. Correct warnings for nonpositive rates/times, noninteger counts, or m greater than n. Compare the approximation’s λMct value with the stated small-rate limit.
  8. Do not use these models for dependent or unequal units, common-cause failures, imperfect switching, standby failures, multiple/priority repair crews, repair distributions that matter, or unvalidated age-dependent field behavior.

Equations used

Let n be total units, m the minimum functioning units, q = n − m tolerated failures, λ the energized-unit failure rate in h⁻¹, Mct the mean corrective repair time in h, μ = 1/Mct, and πj the steady probability of j failed units.

λeff = [n! / ((m−1)! q!)] λq+1 Mctq; MTBCFeff = 1/λeff
πj = πj−1[(n−j+1)λ/(jμ)]; Σπj = 1; A∞ = Σj=0..qπj; νcritical = πqmλ

For mission time t, p(t) is one-unit survival: pexp(t) = exp(−λt), or pW(t) = exp[−(t/η)β].

Rsys(t) = Σk=m..n C(n,k)p(t)k[1−p(t)]n−k
MTBFeff(T) = ∫0..TRsys(t)dt / [1−Rsys(T)]
Rcold(t) = exp(−λt) Σj=0..n−1(λt)j/j!

Definitions, units, and limits: probabilities are dimensionless and shown as percentages; rates are entered and displayed per 10⁶ h but evaluated in h⁻¹. The repair approximation requires λ ≪ μ; λMct is shown as a screening ratio. Scheduled renewal is perfect and instantaneous. Cold standby is only ideal 1-of-n with no switching or dormant failures. Sign convention: times and rates are positive magnitudes and counts are non-negative integers.

Method notes and references

Active-repair results show both the established small-λMct approximation and a normalized one-repair-channel birth-death steady state. Mission reliability treats failures as non-repairable during the mission; scheduled renewal restarts the complete system at T. These models exclude common-cause dependence, coverage loss, and logistics delay.