= Solution
In the <slotted ALOHA> model, a station retains one packet until it successfully transmits. Conditional on $(S_t,N_t)=(S,N)$, take fresh independent trials with a <Bernoulli distribution> of parameter $1/S$ for the $N$ backlogged stations. Let arrivals be independent across slots and independent of these trials, with <Poisson distribution> of mean $\nu$. New arrivals join the next slot's backlog. An idle slot serves nobody, exactly one attempt serves one packet, and a collision serves nobody. Consequently the backlog changes by arrivals minus the success indicator.
Under these <independence> assumptions, \b[$(S_t,N_t)$ is a time-homogeneous <Markov chain>]: the feedback <probabilities> and the independent arrival law depend only on its current state. Merely specifying Poisson arrival marginals would not suffice. For example, take $S_t=1$, $a=b=c=0$, $N_0=0$, and arrivals $Y_0=H$, $Y_1=K$, $Y_2=H$ for <independent random variables> $H,K$ with <Poisson distributions>. Histories with $(N_1,N_2)=(0,2)$ and $(2,2)$ both have current backlog two, but the next backlog is respectively two and four. The usual model therefore includes fresh arrivals as an assumption.
Write $p_0=(1-1/S)^N$ and $p_1=(N/S)(1-1/S)^{N-1}$. Set $p_1=0$ for $N=0$, and use their exact <binomial distribution> <probabilities> at $S=1$. Away from the clipping boundary, the conditional drifts are
$$
\mathbb E[\Delta N\mid S,N]=\nu-p_1,\qquad
\mathbb E[\Delta S\mid S,N]=ap_0+bp_1+c(1-p_0-p_1).
$$
Near $S=1$, the latter must use $\max(1,S+d)-S$ for each update $d$, rather than $d$ itself.
When both coordinates are large with $N/S\to\kappa$, the <Poisson limit theorem> approximates the attempt count, which has a <binomial distribution>, by a <Poisson distribution> of mean $\kappa$. Thus $p_0\to e^{-\kappa}$ and $p_1\to\kappa e^{-\kappa}$. Rescaling state by $K$ and slot time by $K$ motivates the <fluid approximation of slotted ALOHA>
$$
\dot s=g(\kappa)=(a-c)e^{-\kappa}+(b-c)\kappa e^{-\kappa}+c,\qquad
\dot n=f(\kappa)=\nu-\kappa e^{-\kappa},\qquad\kappa=n/s.
$$
This is an interior <fluid approximation of slotted ALOHA>, not the exact conditional drift at a clipped boundary.
Here is one explicit set of sufficient conditions, independent of the arrival rate within the subcritical range:
$$
\boxed{a=b=-A,\qquad c=\frac{2A}{e-2},\qquad A\geq1,\qquad0\leq\nu<e^{-1}}.
$$
Put $c_0=2/(e-2)$. Then
$$
g(\kappa)=A\{c_0-(1+c_0)(1+\kappa)e^{-\kappa}\},\quad
g(1)=0,\quad g'(\kappa)=A(1+c_0)\kappa e^{-\kappa}>0\ (\kappa>0).
$$
Hence $g$ is negative below one and positive above one: the controller decreases the attempt denominator when offered contention is too small, and increases it when contention is too large.
Use the <ratio time change for a homogeneous fluid model>, $du/dt=1/s$. The equations become
$$
\frac{d\kappa}{du}=h(\kappa):=f(\kappa)-\kappa g(\kappa),\qquad
\frac{d\log s}{du}=g(\kappa).
$$
For $\kappa\geq1$,
$$
h'(\kappa)=e^{-\kappa}(\kappa-1)-g(\kappa)-A(1+c_0)\kappa^2e^{-\kappa}<0,
$$
since $g(\kappa)\geq0$ and $A(1+c_0)>1$. Also $h(1)=\nu-e^{-1}<0$. Therefore $h$ is negative on $[1,\infty)$, and by continuity on $[\bar\kappa,\infty)$ for some $\bar\kappa<1$. Its value at zero is $\nu\geq0$, so nonnegative backlog is preserved.
The ratio remains bounded by $K=\max(\kappa(0),1)$ and eventually enters $[0,\bar\kappa]$: on the compact interval $[\bar\kappa,K]$, its <derivative> is bounded above by a strictly negative number. Once inside it cannot cross upward. The bounded ratio and smooth coefficients make the transformed equations exist for all $u\geq0$. In this region $g(\kappa)\leq g(\bar\kappa)<0$, so $s(u)$ decreases at least exponentially in $u$. The original time satisfies $t(u)=\int_0^u s(v)\,dv$ and has a finite limit $T$; boundedness of $\kappa$ gives $n(u)=\kappa(u)s(u)\to0$ as well. Thus \b[every nonnegative interior fluid trajectory drains to the origin in finite fluid time]. At the origin the ratio equation is undefined; the usual stopped fluid trajectory is held there afterward. A state with $s=0<n$ enters the interior under the continuous limiting boundary drift $\dot s=c>0$; the same argument then applies.
This proof supplies sufficient conditions, not a characterization of every stabilizing triplet. It also does not assert the <positive recurrent Markov chain> property of the original stochastic chain solely from the heuristic ODE.
If $\nu>e^{-1}$, the <ALOHA throughput bound> gives
$$
\boxed{\dot n\geq\nu-e^{-1}>0,\qquad n(t)\geq n(0)+(\nu-e^{-1})t}.
$$
No choice of the three feedback increments can make this fluid backlog drain, since $\max_{\kappa\geq0}\kappa e^{-\kappa}=e^{-1}$. For the displayed controller, the supercritical fluid trajectory is global and grows instead of draining. At $\nu=e^{-1}$ the ray $n=s>0$ consists of stationary fluid states, explaining why the strict load inequality matters.
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