Poisson effective bandwidth ($\alpha (\theta )=\nu (e^{\theta }-1)/\theta$)

For independent Poisson arrival counts of mean $\nu$ per slot, with one unit of work per arrival, the cumulant-generating function is $\nu (e^{\theta }-1)$. Its effective bandwidth is the displayed expression. If $c>\nu$, the equation $\nu (e^{\theta }-1)=c\theta$ has a unique positive root: strict convexity, negative initial derivative after subtracting $c\theta$, and divergence to infinity prove existence and uniqueness. Independent Poisson streams combine by replacing $\nu$ with the sum of their intensities.