Conditional and Marginal Run-Length Quantiles of Gamma CUSUM Charts with an Estimated Scale: Tail Integrability and One-Point MRL Protection

Gamma models are natural for positive, right-skewed quality characteristics and grouped time-between-event observations. Fixed Gamma cumulative sum (CUSUM) charts can detect persistent scale changes, but their deployed performance depends on the in-control (IC) scale estimated from a finite Phase-I sample. This study considers one-sided fixed Gamma CUSUM charts with known shape and a scale estimated once in Phase I. Conditional on this estimate, Phase-II observations follow a Gamma distribution with an effective scale ratio, allowing a one-dimensional Markov approximation to the complete run length (RL) distribution. Integration over the exact estimator law gives the marginal RL distribution while retaining the same estimate throughout each Phase-II path. Although every conditional average run length (ARL) is finite, the marginal ARL can be infinite; the marginal median run length (MRL) remains finite. Explicit integrability conditions characterize this distinction. The marginal MRL, the mean conditional MRL, and the distribution of conditional MRL are treated separately. Pathwise ordering yields the conditional-MRL distribution through scalar estimator crossings and reduces its probability constraint to one evaluation at a known Gamma quantile. Design selection profiles the reference value over a prespecified finite candidate grid, calibrates the corresponding control limits, and enforces finite marginal in-control ARL. The Python implementation reproduces published exponential and fixed-Gamma benchmarks. Across the eligible paired scenarios, conditional-MRL protection increased the probability of meeting the in-control conditional MRL target. The protected profiles had larger target-shift marginal out-of-control (OC) MRLs and larger marginal IC MRLs, so these comparisons do not hold in-control performance fixed. Independent continuous-state simulation assessed the finite-state probabilities at prespecified validation points; no exact-binomial comparison was rejected after Holm adjustment. A protected design whose integer MRL crossing remained sensitive to state resolution was excluded from the paired comparison. Supplementary sensitivity analysis shows reduced protection under downward shape misspecification, while matched-IC comparisons show shift-dependent detection tradeoffs.