= Mean and peak limits of effective bandwidth
{title2=$\alpha(0+)=\mathbb EX,\quad\alpha(\infty)=\operatorname{ess\,sup}X$}
For nonnegative traffic with exponential moments near zero, its <effective bandwidth> has expansion $\mathbb EX+s\operatorname{Var}(X)/2+O(s^2)$. For bounded traffic with essential supremum $M$, it increases toward $M$: the exponential moment is at most $e^{sM}$, and any positive probability of $X>a$ gives $\alpha(s)\geq a+s^{-1}\log\mathbb P(X>a)$. Unbounded traffic has an infinite large-parameter limit when all positive exponential moments exist; a finite moment domain must be respected.
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