= Stable fluid controller for slotted ALOHA
{title2=$a=b=-A,\quad c=2A/(e-2),\quad A\geq1$}
This feedback family in <slotted ALOHA> gives $g(\kappa)=A\{2/(e-2)-[e/(e-2)](1+\kappa)e^{-\kappa}\}$, which changes sign at contention ratio one. For $0\leq\nu<e^{-1}$, the ratio drift $h=\nu-\kappa e^{-\kappa}-\kappa g$ is strictly negative on $[1,\infty)$ and on some interval beginning below one. The <ratio time change for a homogeneous fluid model> traps the ratio in a region where $g<0$ and proves finite-time draining. For $\nu>e^{-1}$, backlog drift is at least $\nu-e^{-1}>0$ regardless of this controller. These are statements about the interior <fluid model>, not a stand-alone theorem of stochastic stability.
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