= Bernoulli effective bandwidth
{c}
{title2=$\alpha(\theta)=\theta^{-1}\log(1-p+pe^{\theta r})$}
Independent slots carry either zero work or $r>0$ units, the latter with probability $p$. Their <effective bandwidth> is the displayed expression, with small-parameter limit $pr$ and large-parameter limit $r$ when $0<p<1$. For $pr<c<r$, the positive root of $\log(1-p+pe^{\theta r})=c\theta$ determines the optimized exponential overflow bound. In the special case $r=2,c=1,p<1/2$, the stationary <queue workload> is geometric with ratio $p/(1-p)$, and the root is $\theta_* =\log((1-p)/p)$, giving exactly $\mathbb P(Q\geq k)=e^{-\theta_*k}$ for integers $k\geq0$.
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