When Little's Law Fails as an Estimator: Finite-Sample Variance, Heavy-Traffic Pathology, and Reliability Diagnostics in Queueing Systems

Little's Law (L = λW) is exact, yet finite-sample estimation routinely exhibits 40–50% errors at high utilization. We argue these discrepancies reflect estimation pathology, not model failure, and develop a unified theory identifying three distinct failure modes. First, estimand mismatch: under non-stationary arrivals, time-average and arrival-average queue lengths diverge by up to 28%, causing systematic bias even with infinite samples. Second, variance explosion: estimator variance scales as Θ((1−ρ)⁻³/T)—approximately five orders of magnitude larger at ρ = 0.99 than at moderate utilization. We prove this scaling is fundamental via minimax lower bounds: no estimator can achieve better mean squared error than Ω((1−ρ)⁻³/T), regardless of construction. Third, distributional breakdown: heavy-tailed service causes CLT failure, with coefficient of variation exceeding 250%. We provide finite-sample concentration bounds with explicit constants, anytime-valid confidence sequences for adaptive stopping, and practical diagnostics (median-mean ratio, CV thresholds) to detect which pathology is operating. Extensive Monte Carlo validation confirms all predictions. The unifying insight is that apparent Little's Law violations are predictable and diagnosable—the law itself is exact; only estimation can be unreliable.