Convexity properties of loss and overflow functions

We show that the fluid loss ratio in a fluid queue with finite buffer $b$ and constant link capacity $c$ is always a jointly convex function of $b$ and $c$. This generalizes prior work [6] which shows convexity of the $(b,c)$ trade-off for large number of i.i.d. multiplexed sources, using the large deviations rate function as approximation for fluid loss. Our approach also leads to a simpler proof of the prior result, and provides a stronger basis for optimal measurement-based control of resource allocation in shared resource systems.