Symbols and valid model range
T is cumulative test exposure; r is cumulative included failures; Mc(T) is cumulative MTBF; Mi(T) is fitted instantaneous MTBF; and β is the Duane growth slope. For the standard improving form used by these planning tools, 0 < β < 1. A zero slope represents no modeled growth.
Cumulative MTBF projection
M_c(T) = M_c(T₀) × (T ÷ T₀)^βThe baseline point and projection must use the same test clock, configuration boundary, duty, and failure definition. Calculator.
Instantaneous MTBF
M_i(T) = M_c(T) ÷ (1 − β)This relation follows from the power-law cumulative failure function. Instantaneous MTBF is a fitted endpoint quantity, not the observed total-time-per-failure average.
Test time for a target cumulative MTBF
T_target = T₀ × [M_target ÷ M_c(T₀)]^(1 ÷ β)The target must be at least the baseline for an improving plan. β must be positive; with no growth, a higher target cannot be reached by this model. Calculator.
Two-point growth slope
β = ln[M_c(T₂) ÷ M_c(T₁)] ÷ ln(T₂ ÷ T₁)T₂ must exceed T₁ and all inputs must be positive. This is the line through two log-log points, not a maximum-likelihood fit or uncertainty estimate. Calculator.
Achieved end-of-test MTBF
Observed M_c(T) = T ÷ r; fitted M_i(T) = T ÷ [r × (1 − β)]The failure count must be a positive integer under a declared event definition. Zero-failure testing requires a different statistical method. Calculator.
Relationship to the NHPP power law
In the NIST notation, expected cumulative failures can be written M(T) = aTb, with failure intensity abTb−1 and Duane growth slope β = 1 − b. ReliabilityBench uses β for the MTBF-growth slope; always check notation before copying a parameter between sources.
Scope and source
These equations are point-model relationships and provide no confidence bounds, goodness-of-fit test, or requirement demonstration. Technical basis: NIST Engineering Statistics Handbook sections on Duane plots and power-law model relationships. Use the reliability growth guide for workflow and interpretation.