Effective bandwidth ($\alpha (s,t)=\log \mathbb{E}[e^{sX(t)}]/(st)$)

Effective bandwidth describes the exponential-moment traffic rate at a specified time scale and tail parameter. Independent sources have additive effective bandwidths. The Chernoff bound connects their sum to overflow bounds. For a single time unit, the small-parameter limit is the expected value and the large-parameter limit is the essential supremum, with appropriate moment or extended-value conventions.

For independent identically distributed traffic increments and constant service $C$, let $a(s)=\mathbb{E}[e^{s(X-C)}]$. If $a(s)<1$, apply the Chernoff bound at each candidate workload window and sum a geometric series. This bounds the stationary random-walk supremum. It explains the effective bandwidth criterion $\alpha (s)<C$ and distinguishes a queue's infinite-horizon overflow event from a single-window traffic event.

For nonnegative traffic with exponential moments near zero, its effective bandwidth has expansion $\mathbb{E}X+s\mathrm{Var}(X)/2+O(s^2)$. 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 $\alpha (s)\ge a+s^{-1}\log \mathbb{P}(X>a)$. Unbounded traffic has an infinite large-parameter limit when all positive exponential moments exist; a finite moment domain must be respected.

For a normal distribution $N(\lambda ,{\sigma }^2)$, the logarithm of its moment-generating function is $s\lambda +s^2{\sigma }^2/2$. Dividing by $s>0$ gives its effective bandwidth. Independent demands add these bandwidths. For total mean $m$ and positive total variance $v$, optimizing a Chernoff bound with target $e^{-\gamma }$, $\gamma >0$, yields the sufficient margin $C\ge m+\sqrt{2\gamma v}$.