Exponential workload bound from cumulative arrival moments ($\mathbb{P}(Q>B)\le e^{-\theta B}{\sum }_{n\ge 1}{e}^{K_n(\theta )-\theta cn}$)

Let a slotted queue have stationary input, constant service $c$, and finite stationary queue workload $Q={\sup }_{n\ge 0}(A_n-cn)$. Write $K_n(\theta )=\log \mathbb{E}{e}^{\theta A_n}$ for its past cumulative arrivals. The union bound and Chernoff bound give the displayed inequality for $B>0$. It requires no temporal independence. If each $K_n(\theta )$ is finite and $K_n(\theta )/n\to \kappa (\theta )<c\theta$, the series is finite: its tail is bounded by a geometric series. Consequently ${\mathrm{limsup}}_{L\to \infty }{L}^{-1}\log \mathbb{P}(Q>Lb)\le -\theta b$ for $b>0$. The condition is that the asymptotic effective bandwidth $\kappa (\theta )/\theta$ be less than service capacity.