Statistical Analysis
Gamma Waiting-Time Quantile & SLA Buffer Formula (GAMMA.INV Wilson-Hilferty Cube-Root)
Calculate multi-stage Gamma and Erlang waiting-time quantiles (GAMMA.INV) using exact exponential inversion for shape k = 1 and the Wilson-Hilferty cube-root transformation for multi-stage queues.
Sizing SLA time budgets across k sequential processing stages
When an order passes through k independent exponential stages each with mean duration theta, total cycle time follows Gamma(k, theta) with mean k * theta and variance k * theta^2. Whereas GAMMA.DIST evaluates the probability of finishing before a fixed deadline, capacity planners need the inverse quantile GAMMA.INV(p, k, theta): the deadline met by p% of workflows.
For k = 1, the inverse is exact: -theta * LN(1 - p). For multi-stage pipelines (k >= 2), the Wilson-Hilferty cube-root transformation (X / (k * theta))^(1/3) ~ Normal(1 - 1/(9k), 1/(9k)) maps any standard normal quantile z_p directly into an accurate Gamma quantile.
| Shape k | Scale theta (min) | Target z_p | Formula Path | SLA Deadline (min) |
|---|---|---|---|---|
| 9 stages | 2.0 min | z = 0 (50th pct) | 18 * (80/81)^3 | 17.3415 |
| 9 stages | 2.0 min | z = 1 (84.13% SLA) | 18 * (89/81)^3 | 23.8774 |
| 9 stages | 2.0 min | Mean k * theta | 9 * 2.0 | 18.0000 |
| 1 stage | 10.0 min | p = 0.50 | -10 * LN(0.5) | 6.9315 |
Wilson-Hilferty Gamma Quantile Formula
Combine exact k = 1 logarithmic inversion with the Wilson-Hilferty cube-root formula theta * k * MAX(0, 1 - 1/(9*k) + z_p * SQRT(1/(9*k)))^3.
=LET(shape_k, B2, scale_th, B3, z_quant, B4, wh_var, 1 / (9 * shape_k), ROUND(scale_th * shape_k * MAX(0, 1 - wh_var + z_quant * SQRT(wh_var)) ^ 3, 4))SLA Buffer Above Expected Mean Cycle Time
Subtract expected cycle time k * theta from the high-percentile quantile to isolate the contingency buffer required for queue variability.
=ROUND(E3 - E2, 4)Try the example
Use the original values or change them to test your understanding.