We consider a discrete-time leaky integrator of the form Hₙ = λ H₍-₁ + xₙ with 0 < λ < 1, xₙ ∈ −1, 0, 1, and a hard state constraint |Hₙ| ≤ K for all n. This system models a finite-capacity accumulator with exponential forgetting: past inputs decay but cannot accumulate beyond K. We prove that the long-term average input A = limsup₍→∞ (1/N) Σ xₙ is bounded above by Aₘax = ( (1−λ) / (1+λ) ) K, and that this bound is sharp. The proof uses the telescoping identity Σ xₙ = (1−λ) Σ Hₙ + HN − λ H-₁ together with the state constraint |Hₙ| ≤ K. The result is equivalent to the capacity of a token-bucket regulator in communication networks. We provide a self-contained derivation and discuss analogous structures in biological, physical, and mathematical systems where finite capacity limits sustainable throughput.
Goss, Jr., Matthew J. (Sat,) studied this question.