Fluid approximation of slotted ALOHA

When backlog $N$ and attempt-control scale $S$ are large with $N/S\approx \kappa$, idle and success probabilities approach $e^{-\kappa }$ and $\kappa e^{-\kappa }$. Expected update increments then motivate a fluid model described by ordinary differential equations.

Let $N_t$ be the backlog in slotted ALOHA, and let stations use attempt probability $1/S_t$ with estimate $S_t\ge 1$. Update the estimate by $a$ after an idle slot, $b$ after a success and $c$ after a collision, clipping its value below at one. These feedback parameters therefore specify an actual controller, rather than only a drift formula.^[ref]

In the interior fluid model, write $s,n>0$ and $\kappa =n/s$. The idle and success probabilities approach $e^{-\kappa }$ and $\kappa e^{-\kappa }$. For $a=b=-A$, $c=2A/(e-2)$ and $A\ge 1$, their weighted mean givesHere $\nu$ is the mean number of arrivals per slot. Since $g(1)=0$ and $g^{\prime }(\kappa )=Ae\kappa e^{-\kappa }/(e-2)>0$, the estimate falls below contention ratio one and rises above it.

Use the ratio time change for a homogeneous fluid model. Its ratio drift is $h=\nu -\kappa e^{-\kappa }-\kappa g$. For $\kappa \ge 1$,because $g\ge 0$ and $Ae{\kappa }^2/(e-2)>\kappa -1$. If $\nu <e^{-1}$, then $h(1)<0$; continuity supplies a trapping upper bound below one. There $g$ is bounded strictly below zero, giving finite-time draining of a fluid model. If $\nu >e^{-1}$, instead $\dot{n}\ge \nu -e^{-1}>0$. These are fluid conclusions; stochastic stability requires a separate argument.