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A random access network shares one communication channel among an unbounded collection of possible transmitters. Each packet occupies one slot; exactly one attempt succeeds, an idle slot carries nothing, and a collision delivers none of its attempted packets. Assume perfect slot synchronization, immediate idle/success/collision feedback, no channel errors, and independent random choices by packets. The important quantities are throughput, the size of the retransmission backlog, and delay. A good one-slot success probability alone does not establish stability of the backlog.

For a precise infinite-population slotted ALOHA model, let $A_t$ be independent Poisson random variables of mean $\eta >0$, representing new packets which attempt immediately in slot $t$. Let $B_t$ be the backlog before those arrivals. Each old packet independently retries with a constant probability $p\in (0,1)$, so conditional on $B_t=b$, its retry count is $K_t\sim \mathrm{Binomial}(b,p)$, independent of $A_t$. If $D_t=1_{\{A_t+K_t=1\}}$ records a successful slot, thenThis includes a new packet in the backlog precisely when it has not succeeded; an old packet is removed precisely when it succeeds. The total backlog is a Markov chain in this constant-probability model. Its idle and success probabilities arewith the second term in $S(0)$ interpreted as zero. A success may be either one new packet and no old attempt, or one old attempt and no new packet.

The fluid approximation of slotted ALOHA replaces the binomial retry count by a Poisson count when there are many packets with small retry probability. If $g=\eta +bp$ is total attempt intensity, the approximate success probability is $g{e}^{-g}$. Differentiation givesThis is the limiting ALOHA throughput bound, not an exact upper bound for every finite packet population: one isolated packet which certainly transmits succeeds with probability one. In a fluid model, when $0<\eta <e^{-1}$ the equation $\eta =g{e}^{-g}$ has two roots. The root below one is locally stable for constant $p$, while the root above one is unstable, because the drift derivative is $-p(1-g)e^{-g}$. These approximations explain a low-backlog operating region and a congestion threshold; they do not imply a stationary law for the infinite-population chain.

Indeed, for fixed $p>0$, the exact success probability $S(b)$ tends to zero as $b\to \infty$, so the exact drift tends to $\eta >0$. One can prove instability, rather than relying only on this positive drift. Fix $\theta >0$ and use $V(b)=e^{-\theta b}$. Since $D_t$ is zero or one,The first term is strictly less than one and the second tends to zero. Hence $V(B_{t\wedge \tau })$ is a nonnegative supermartingale outside a sufficiently large finite set, where $\tau$ is the first entrance into that set, say $\{0,\dots ,b_0\}$. The optional sampling theorem for a supermartingale, first at bounded stopping times, givesThis is an exponential-supermartingale escape bound. The chain is irreducible: a no-arrival slot with exactly one old retry can reduce any positive state by one, and Poisson arrivals can reach arbitrarily high states. Therefore the escape probability proves transience and excludes a positive recurrent Markov chain, for every $\eta >0$ and fixed $0<p<1$. A locally stable fluid operating point can consequently coexist with eventual stochastic congestion. At $p=1$, once at least two packets are backlogged, they collide forever, so constant unit retry probability also fails.

A state-dependent retry rule removes this fixed-$p$ congestion mechanism. As an ideal benchmark with exact backlog information and known arrival intensity $0<\eta <e^{-1}$, useThis is backlog-aware ALOHA with arrival-adjusted attempt probabilities. Its total attempt intensity approaches one: the mean old-attempt count is $1-\eta$ and the new-attempt count has mean $\eta$. Substitution in the exact success formula yieldsThus the drift tends to $\eta -e^{-1}<0$. For the Lyapunov function $L(b)=b$, the drift is bounded above by a negative constant outside a finite set, and the expected one-step backlog is finite at every state. Foster theorem therefore gives positive recurrence. This proves a genuine stochastic stability result for this specified, informed controller. Real transmitters generally observe feedback rather than the exact backlog; estimating the backlog and using a probability of order its reciprocal motivates adaptive ALOHA, but an estimate's dynamics must be included in the state and analyzed before claiming the same stability result.

A different nonconstant rule uses a packet's own collision history. In geometric-attempt exponential backoff, a packet at stage $k$ attempts with probability $p_k=p_0{2}^{-k}$, where $0<p_0<1$. A collision advances every attempted packet by one stage, a success removes its packet, and new arrivals enter stage zero. For definiteness, in this version arrivals join at the end of a slot and first become eligible in the next slot. The full Markov chain state is the sequence $(N_0,N_1,\dots )$ of stage populations; the total backlog alone is insufficient. Conditional on that state,These formulas follow from independent attempts and selecting which one packet succeeds. They reduce approximately to $S\simeq G{e}^{-G}$ for many small probabilities with $G={\sum }_k{N}_k{p}_k$. The retry probabilities decrease after failure, preventing repeated aggressive attempts by the same packets. Binary exponential backoff instead chooses uniform random counters from expanding finite windows; that counter model requires residual counters as well as collision stages in its state and is not identical to geometric waiting.

Backoff also exposes a distinction between eventual delivery and acceptable delay. In the simplifying model where each attempt independently fails with a fixed probability $q<1$, the probability of reaching stage $k$ is $q^k$ and its mean waiting time is $1/p_k=2^k/p_0$. The mean delay under independent exponential-backoff failures is thereforeAlthough a packet succeeds almost surely whenever $q<1$, its mean delay can be infinite. In an actual random access network, failures are correlated with the other stage populations, so this fixed-$q$ calculation is a diagnostic approximation, not a proof of network stability. These models show why random-access analysis must track the retry rule, feedback information and full state, and assess throughput, stability and delay separately.

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, $(S_t,N_t)$ 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 areNear $S=1$, 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 ALOHAThis 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:Put $c_0=2/(e-2)$. ThenHence $g$ 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 becomeFor $\kappa \ge 1$,since $g(\kappa )\ge 0$ 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 $\nu \ge 0$, 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 $u\ge 0$. In this region $g(\kappa )\le g(\bar{\kappa })<0$, 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 $n(u)=\kappa (u)s(u)\to 0$ as well. Thus 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 givesNo choice of the three feedback increments can make this fluid backlog drain, since ${\max }_{\kappa \ge 0}\kappa e^{-\kappa }=e^{-1}$. 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.

Use the slotted ALOHA model with one outstanding packet per active station and a collision channel: a slot with exactly one attempt succeeds, whereas an idle slot or a slot with two or more attempts delivers nothing. Let $Y_t$ be independent Poisson random variables of mean $\nu$, independent also of the transmission choices. Arrivals join after the current slot's attempts. Given $(S_t,N_t)=(S,N)$, the attempt count has a binomial distribution with parameters $N$ and $1/S$. Consequently, for $N\ge 1$,For $N=0$, set $p_0=1$ and $p_1=0$. The equation $N_{t+1}=N_t+Y_t-1_{\{Z_t=1\}}$ adds arrivals and removes precisely one packet following a successful slot. The estimate $S_t\ge 1$ controls the common transmission probability and updates using the idle, success, or collision feedback. The conditional distribution of the next state depends only on $(S_t,N_t)$, so this pair is a Markov chain.

Away from the reflecting floor in $S$, its one-step conditional expectations areWhen $N$ and $S$ are large with $N/S\approx \kappa$, the Poisson limit theorem gives $p_0\approx e^{-\kappa }$ and $p_1\approx \kappa e^{-\kappa }$. Treating these approximate drifts as derivatives motivates the fluid approximation of slotted ALOHA:The floor at $S=1$ is negligible in this interior scaling. This calculation motivates the ordinary differential equations; it is not a proof of a stochastic fluid limit.

For $(a,b,c)=(2-e,0,1)$, putThen $g^{\prime }(\kappa )=(e-2+\kappa )e^{-\kappa }>0$ and $g(1)=0$. On the domain $s>0$, introduce time $\tau$ by $d\tau /dt=1/s$. Differentiating $\kappa =n/s$ giveswhere $r(\kappa )=\kappa [1-(e-2+\kappa )e^{-\kappa }]$.

The function $r$ is strictly increasing from zero to infinity, and $r(1)=e^{-1}$. To check the monotonicity explicitly,Indeed, with $d=3-e\in (0,1)$,because its discriminant is $d^2-4d<0$; and $e^{\kappa }\ge 1+\kappa$. Thus the scalar ordinary differential equation for $\kappa$ points toward the unique value ${\kappa }_{\ast }$ satisfying $r({\kappa }_{\ast })=\nu$. Every trajectory with $s(0)>0$, $n(0)\ge 0$ has bounded $\kappa$ and $\kappa (\tau )\to {\kappa }_{\ast }$. Positivity of $s$ follows from $s(\tau )=s(0)\exp ({\int }_0^{\tau }g(\kappa (u))du)$.

If $0\le \nu <e^{-1}$, then ${\kappa }_{\ast }<1$ and $g({\kappa }_{\ast })<0$. Thus $s(\tau )$ decays exponentially for large $\tau$, andThis is finite-time draining of a fluid model. The unmodified equations have $n/s$ undefined at the origin, so the statement that trajectories converge to the origin needs this endpoint interpretation; continuing with an absorbing zero trajectory is an additional fluid-model convention, not a solution of the displayed equations at the origin.

If $\nu >e^{-1}$, then ${\kappa }_{\ast }>1$ and $g({\kappa }_{\ast })>0$. Now $s(\tau )$ grows exponentially, physical time tends to infinity, andIn particular the trajectory escapes to infinity. Even without the ratio analysis, $\dot{n}\ge \nu -e^{-1}>0$, since ${\max }_{\kappa \ge 0}\kappa e^{-\kappa }=e^{-1}$.

For the actual Markov chain, it is essential not to replace the finite-$N$ success probability by $e^{-1}$: a single backlogged packet can succeed with probability one. Instead, for $N\ge 2$, maximizing over $S\ge 1$ gives the ALOHA throughput boundIf $\nu >e^{-1}$, choose $K_0$ and $r_0<\nu$ so that $p_1(N,S)\le r_0$ for all $N>K_0$, uniformly in $S$. For sufficiently small $\theta >0$,because its derivative at zero is $r_0-\nu <0$. Independence of arrivals and attempts giveswhenever $N>K_0$. For any integer $K\ge K_0$, stop at the stopping time ${\sigma }_K=\inf \{t:N_t\le K\}$. The stopped process $e^{-\theta N_{t\wedge {\sigma }_K}}$ is a nonnegative supermartingale. The optional stopping theorem, first at bounded times and then by a limit, yields the exponential-supermartingale escape boundFrom every state with $N\le K$, a Poisson distribution arrival burst has a uniformly positive probability of sending the next backlog to at least $K+m$, for any fixed $m>0$. From there the probability of never revisiting $N\le K$ is at least $1-e^{-\theta m}$. By the Markov property at successive visits, the probability of infinitely many visits to this strip is zero: each visit has a uniform positive chance of being the last. This holds for every integer $K\ge K_0$, soTherefore the Markov chain is transient for every finite choice of $a,b,c$. The proof is uniform in $S$ and does not infer stochastic transience merely from the approximate ordinary differential equations.

Time is divided into slots, and a slot succeeds exactly when one station transmits. With backlog $N$ and independent attempt probability $p$, the success probability is $Np(1-p{)}^{N-1}$. Fluid approximation of slotted ALOHA replaces the large-system binomial probabilities by Poisson probabilities.