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ANALYSIS - TOPIC A10 Table A9-1: Redundancy Equation Approximations Summary Redundancy Equations With Repair Without Repair All units are active on-line with equal unit failure rates, (n-q) out of n required for success. Equation 1 n \(X) q+1 Equation 4 fyn-q)/n =" 1 .2 T i=n-q Two active on-line units with different failure and repair rates. One of two required for success. Equation 2 a.1/2 = Equation 5 _ Xa2XB+XAA-B2 V2~XA2+XB2+XAXB One standby off-line unit with n active on-line units required for success. Off-line spare assumed to have a failure rate of zero. On-line units have equal failure rates. Equation 3 Vi/n+1 = n[nM1-P)n]k H+n(P+1)X. Equation 6 nA, *n/n+1 = p+i Key: Xx/y is the effective failure rate of the redundant configuration where x of y units are required for success n = number of active on-line units, n! is n factorial (e.g., 51=5x4x3x2x1 =120, 11=1,01=1) X = failure rate of an individual on-line unit (failures/hour) q = number of on-line active units which are allowed to fail without system failure H = repair rate (ji=1/Mct, where Met is the mean corrective maintenance time in hours) P = probability switching mechanism will operate properly when needed (P=1 with perfect switching) Notes: 1. Assumes all units are functional at the start 2. The approximations represent time to first failure 3. CAUTION: Redundancy equations for repairable systems should not be applied if delayed maintenance is used. 90 ROME LABORATORY RELIABILITY ENGINEER'S TOOLKIT

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