Bernoulli effective bandwidth ($\alpha (\theta )={\theta }^{-1}\log (1-p+p{e}^{\theta r})$)

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 $\log (1-p+p{e}^{\theta r})=c\theta$ 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 ${\theta }_{\ast }=\log ((1-p)/p)$, giving exactly $\mathbb{P}(Q\ge k)=e^{-{\theta }_{\ast }k}$ for integers $k\ge 0$.