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Use the clipped Lindley recursion $q_{t+1}=\min (B,\max (0,q_t+x_t-C))$ and the weighted cumulative-input topology for a slotted queue. If cumulative past input $S_x(T)\le \lambda T$ eventually with $\lambda <C$, choose a finite $T$ so $S_x(T)-CT<-B$. Starting with a full buffer at time $-T$, the queue must hit zero before the present, since otherwise its final workload is at most $B+S_x(T)-CT<0$. Monotonicity couples all initial workloads by that zero, establishing a finite memory horizon. The same horizon works for sufficiently close input $y$. Since $|x_{-k}-y_{-k}|\le (2k-1)\Vert x-y\Vert$ and clipping is Lipschitz continuous with constant one, the two finite recursions started empty differ by at most ${\sum }_{k=1}^T(2k-1)\Vert x-y\Vert =T^2\Vert x-y\Vert$.

Write $A_n$ for work arriving in slot $n$ and $c$ for the service rate of a queue. For a work-conserving single-server queue, the Lindley recursion is $Q_{n+1}=(Q_n+A_{n+1}-c{)}^{+}$. For stable independent and identically distributed random variables with mean $\mu <c$, its stationary workload supremum has the formDifferent scaling limits of a queue concern different quantities growing large; they need not share either a probability speed or a likely path to overflow.

For large-buffer queue scaling, fix the input law and $c>\mu$ and let the buffer size of a queue $b\to \infty$. Assume finite exponential moments around a positive root ${\theta }_{\ast }$ of ${\Lambda }_A(\theta )=c\theta$. An overflow accumulated in $n\simeq tb$ slots requires an average input about $c+1/t$, so Cramér's theorem assigns logarithmic cost $bt{I}_A(c+1/t)$. Optimizing the duration gives the Cramér-Lundberg workload exponentFor an interior root with ${\Lambda }_A^{\prime }({\theta }_{\ast })>c$, Fenchel–Young inequality verifies the minimization: the lower bound is ${\theta }_{\ast }$ because $I_A(x)\ge {\theta }_{\ast }x-{\Lambda }_A({\theta }_{\ast })$, and equality occurs at $t=[{\Lambda }_A^{\prime }({\theta }_{\ast })-c{]}^{-1}$. The exponential workload bound from cumulative arrival moments gives the upper exponent for every $\theta <{\theta }_{\ast }$; an exponentially tilted positive-drift path or the duration calculation gives the matching lower exponent. In effective bandwidth notation, ${\Lambda }_A({\theta }_{\ast })/{\theta }_{\ast }=c$. This regime measures long accumulation into a large buffer with a fixed positive stability margin.

For many-source queue scaling, superpose $L$ independent copies of an input process, give the server capacity $Lc$, and give it buffer size $Lb$. For each lookback horizon $n$, the average of the $L$ source cumulative arrivals has a large deviation principle with rate function $I_n$. Overflow at that horizon costs $L{I}_n(b+cn)$. Optimizing the lookback horizon suggestsFor independent slots, $I_n(x)=n{I}_A(x/n)$, giving $V(b)={\inf }_{n\ge 1}n{I}_A(c+b/n)$. The integer horizon is important in a slotted system. For temporally dependent sources, use the cumulative-input cumulant-generating function, rather than multiplying a one-slot formula by $n$. The heuristic needs uniform control of the tail of the stationary supremum to justify optimizing an unbounded set of horizons; stability and suitable exponential-moment estimates provide this in the usual models. The growing quantity here is the number of independent sources, while per-source capacity and buffer remain fixed.

For moderate-deviation queue scaling, consider a bufferless queue with aggregate Poisson input of mean $L\lambda$ and capacity $L\lambda +C{L}^{(1+\beta )/2}$, $0<\beta <1$. The spare capacity is much larger than the ordinary central limit theorem scale $\sqrt{L}$ but much smaller than $L$. The Poisson sample-mean rate function near its minimum has expansionWith $h=C{L}^{(\beta -1)/2}$, the cost $L{I}_P(\lambda +h)$ is asymptotic to $C^2{L}^{\beta }/(2\lambda )$. Hence Poisson moderate deviations give $\log \mathbb{P}(\mathrm{overflow})\sim -C^2{L}^{\beta }/(2\lambda )$. The quadratic cost comes from a shrinking neighborhood of the mean, rather than from a fixed macroscopic overload. The regime is useful for a large aggregate whose service margin lies between its typical fluctuation scale and its mean scale.

For a heavy-traffic limit of a queue, instead hold the input law fixed with mean $\mu$ and positive finite variance ${\sigma }^2$, and let $c_{\epsilon }=\mu +\epsilon d$ with $d>0$ and $\epsilon \downarrow 0$. In $t/{\epsilon }^2$ slots the net input has mean about $-dt/\epsilon$ and random fluctuation of size $\sigma \sqrt{t}/\epsilon$. Thus the functional central limit theorem, followed by the Skorokhod reflection map on the half-line, suggestswhere $R$ is reflected Brownian motion with negative drift $-d$ and variance parameter ${\sigma }^2$. The stationary law of negatively drifted reflected Brownian motion has tail $\mathbb{P}(R_{\infty }>b)=e^{-2db/{\sigma }^2}$. When passing the stationary distribution through the limit is justified, this predictsFor dependent input satisfying a suitable invariance principle, replace ${\sigma }^2$ by its long-run variance of a stationary process. This regime is appropriate near critical load and on long time scales; a functional transient limit alone does not automatically justify the stationary-limit interchange.

A different large-service regime holds the input law fixed. With Poisson distribution of mean $\lambda$ and bufferless capacity $c\to \infty$, let $m=\lfloor c\rfloor +1$. The single-point lower bound $\mathbb{P}(X=m)$ and the optimized Chernoff bound for $X\ge m$, using Stirling's formula, give the Poisson large-service overflow asymptoticHere the service rate grows while the input mean does not. A quadratic moderate-deviation approximation would describe the wrong part of the Poisson tail.

To choose a regime, compare the spare capacity, typical fluctuations, number of sources and buffer size before selecting a formula. Use the large-buffer exponent for a fixed stable input and a long overload episode; use the many-source variational cost when independent flows and resources grow together; use moderate deviations for an intermediate safety margin; use the heavy-traffic diffusion when the stability margin tends to zero; and use the fixed-input Poisson tail when service alone becomes very large. Check temporal dependence and exponential-moment assumptions. For heavy-tailed subexponential distributions, a single exceptionally large arrival can dominate overflow, so a finite exponential workload root and a Brownian approximation to the far tail need not apply. These comparisons explain why the same nominal queueing theory system can have different useful asymptotic descriptions.

Work deterministically with nonnegative arrivals. Write $S_x(t)={\sum }_{k=1}^t{x}_{-k}$ for integer $t\ge 1$, and $S_x(0)=0$. The relevant metric isThis is the weighted cumulative-input topology for a slotted queue. It is a genuine norm on the ambient space of sequences with finite value: zero cumulative differences force every individual difference to vanish, since $x_{-k}=S_x(k)-S_x(k-1)$. Every input in the specified set $A$ has finite norm, because its long prefixes are bounded by $\lambda t$ and there are only finitely many earlier prefixes.

Use the conventional net-input slot boundary finite-buffer workload map, whose recursion is the clipped Lindley recursionArriving work in a slot is available to that slot's service, and the observed queue state is after service and upper clipping. This fixes the slot convention explicitly. The queue size $Q_B(x)$ is obtained by starting in the distant past and taking the present workload. We first prove that, under the stated strict drift condition, this is uniquely defined independently of the initial state.

Fix $x\in A$ and write $d_0=C-\lambda >0$. Choose a finite integer $T$ so large thatThen $S_x(T)-CT<-B$. Start at time $-T$ with a full buffer $q_{-T}=B$. The trajectory must hit zero by time zero. Otherwise lower clipping is never used, so for nonnegative lost-work amounts ${\ell }_t$ the total recursion would givecontradicting nonnegativity. If zero is attained even at the last update the desired conclusion still holds.

The update $F_B$ is monotone in the initial workload. Couple all starting values in $[0,B]$ using the same arrivals. When the trajectory starting at $B$ reaches zero, all smaller trajectories must also be zero, and they remain identical thereafter. Thus the present queue depends only on the last $T$ slots, not on any earlier state or arrivals. This is finite-memory continuity of a finite-buffer queue. In particular it proves the existence and uniqueness of $Q_B(x)$ without assuming a random stationary law.

The same memory horizon works in a neighborhood of $x$. If $\delta =\Vert x-y\Vert <d_0/2$, thenThe full-buffer trajectory for $y$ therefore also reaches zero within this horizon. Both $Q_B(x)$ and $Q_B(y)$ can consequently be calculated from initial workload zero at time $-T$, even though their respective reset times need not coincide.

Clipping to $[0,B]$ is Lipschitz continuous with constant one, soIterating from the common initial value zero givesThe cumulative-input norm bounds each summand:Summing proves the explicit local estimateGiven $\epsilon >0$, choose $\Vert x-y\Vert <\min \{(C-\lambda )/2,\epsilon /T^2\}$. Then the workload difference is less than $\epsilon$. Since $x\in A$ was arbitrary, the queue size function is continuous on $(A,\Vert \cdot \Vert )$.

The argument also handles the alternative slot ordering that clips arrivals to the buffer before serving them: its update is a composition of monotone one-Lipschitz clipping and service maps, and negative total net input forces a reset. Measuring immediately after arrivals instead adds one finite-coordinate update, which is likewise continuous in this norm. The essential point is strict subcritical drift and loss of dependence on the remote past, rather than incorrectly replacing a finite-buffer history by the minimum of $B$ and an infinite-buffer workload.

Measure $Q_t$ immediately after service in slot $(t-1,t)$, and let $a_t\ge 0$ denote work eligible for that slot's service. Writing $x^{+}=\max (x,0)$, the Lindley recursion isStart with an empty queue at time $-N$. Repeated substitution gives its time-zero queue workloadwhere the empty sum is zero. The increasing limit defines the minimal causal queue workload:If the mean of the remote past is $\lambda <c$, then $A_n(a)/n\to \lambda$, so $A_n(a)-cn\to -\infty$. The supremum is then finite and is attained at a finite window. The same formula with indices shifted gives $Q_t(a,c)$ and satisfies the Lindley recursion directly: split the supremum into the zero-window term and all positive-window terms.

The work passed downstream in slot $(t-1,t)$ isBecause this work is eligible for downstream service in the same slot, the downstream Lindley recursion isPut $T_t=Q_t+R_t$. If $R_{t-1}+D_t\ge d$, thenIf $R_{t-1}+D_t<d$, then $D_t<d<c$, so the upstream server did not use its full capacity. Consequently $D_t=Q_{t-1}+a_t$ and $Q_t=0$. Also $R_t=0$ and $Q_{t-1}+R_{t-1}+a_t-d<0$. Both cases yieldThis slower-server tandem workload identity uses both $d<c$ and eligibility for same-slot downstream service. It would not describe a system with an obligatory one-slot transfer delay.

Consider a stationary nonnegative work-input process in slotted time, with cumulative work $A_n$ over the last $n$ slots and a work-conserving server of capacity $c$ per slot. Assume its past average tends almost surely to $m<c$. The Lindley recursion then defines the finite stationary queue workloadThis is work remaining, rather than the number of customers when customer service requirements vary. With unit work per customer the two notions coincide. Stationarity makes the distribution of a past cumulative window equal to that of a forward window of the same length.

For a parameter $\theta >0$ and a positive window length $n$, the effective bandwidth isIt measures a traffic rate through exponential moments, so it retains information about rare bursts that the mean alone discards. If the limiting scaled cumulant-generating function exists, putThe finite-window effective bandwidth matters when the relevant overflow window is short or when arrivals are correlated; one should not replace it by its long-window limit without justification.

The following proof gives an actual upper bound for the infinite-horizon queue workload. For $B>0$, if $Q>B$, at least one positive integer $n$ has $A_n-cn>B$. The union bound followed by the Chernoff bound givesAssume each finite-window exponential moment is finite and $\kappa (\theta )<c\theta$. Choose $\eta >0$ such that $\kappa (\theta )+\eta <c\theta$. For all sufficiently large $n$, the summand is at most $e^{-n(c\theta -\kappa (\theta )-\eta )}$. Its tail is a convergent geometric series, and its finite prefix is finite. Denote the resulting sum by $K_{\theta }$. The exponential workload bound from cumulative arrival moments therefore proves, for $b>0$,Optimizing over admissible parameters givesOnly parameters with the stated finite-window moments are included. If there are no such positive parameters, the argument yields only the trivial exponent zero. Infinite ${\theta }_{\ast }$ means that the upper rate is $-\infty$. For a non-strict threshold $Q\ge Lb$, first bound it by $Q>L(b-\epsilon )$ and then let $\epsilon \downarrow 0$. This proof permits dependence between slots. It proves the required upper bound; it does not assert a matching lower bound for every arrival process.

This also gives a closed-set large-deviation upper bound for $Q/L$: set $I_{+}(q)={\theta }_{\ast }q$ for $q>0$, $I_{+}(0)=0$, and $I_{+}(q)=+\infty$ for $q<0$. If a closed set $F$ contains zero, its desired upper bound is the trivial bound zero. If $F$ contains positive points but not zero, its positive part has a smallest point $b>0$, and $\mathbb{P}(Q/L\in F)\le \mathbb{P}(Q\ge Lb)$ gives the bound $-{\inf }_F{I}_{+}$. If it contains no nonnegative points, its probability is zero. This verifies the upper half of a large deviation principle; the actual rate function can be larger when a matching lower bound is unavailable.

For independent and identically distributed slot arrivals $a_t$, let $\kappa (\theta )=\log \mathbb{E}{e}^{\theta a_0}$. Now $\log \mathbb{E}{e}^{\theta A_n}=n\kappa (\theta )$, so effective bandwidth is independent of window length. For $\kappa (\theta )<c\theta$, the geometric-series prefactor is explicitlyThus the familiar workload Chernoff bound for independent increments is a special case, and the largest admissible parameter is often the positive root of $\kappa (\theta )=c\theta$.

For these independent increments there is a sharper argument that works at the root itself when its exponential moment is finite. Along the past read in reverse order, put $S_n=A_n-cn$ and $M_n=e^{\theta S_n}$. If $\mathbb{E}{e}^{\theta (a_0-c)}\le 1$, then $M_n$ is a nonnegative supermartingale starting at one. For the first crossing time ${\tau }_B=\inf \{n\ge 0:S_n\ge B\}$, bounded optional stopping givesLet $N\to \infty$. Strict negative drift makes $S_n\to -\infty$ almost surely, so its supremum is attained and the crossing event agrees with $\{Q\ge B\}$. ConsequentlyThis strengthens the prefactor but uses temporal independence, which was unnecessary for the previous proof.

Several examples show what the effective bandwidth constraint means. Deterministic traffic of $r$ work units per slot has $\kappa (\theta )=\theta r$ and effective bandwidth $r$ at every parameter; if $c\ge r$ there is no stationary backlog. For independent on-off slots carrying $r$ units with probability $p$ and zero otherwise, the Bernoulli effective bandwidth isFor $0<p<1$, it rises from the mean $pr$ to the peak $r$. When $pr<c<r$, strict convexity of $\kappa (\theta )-c\theta$, its negative derivative at zero, and its divergence to infinity give a unique positive root. When $c\ge r$, no slot can increase the queue workload. In the specific case $r=2,c=1,p<1/2$, the workload increases by one with probability $p$ and decreases by one, reflected at zero, with probability $1-p$. Its stationary probabilities satisfyThe service-root equation has $e^{{\theta }_{\ast }}=(1-p)/p$, and thus $\mathbb{P}(Q\ge k)=q^k=e^{-{\theta }_{\ast }k}$. This example realizes the upper exponent exactly.

For independent Poisson slot counts of mean $\nu$, with unit work per arrival, the Poisson effective bandwidth isStability requires $c>\nu$. There is a unique positive root of $\nu (e^{\theta }-1)=c\theta$, again by strict convexity, negative initial derivative after subtracting $c\theta$, and growth to infinity. For independent exponentially distributed work with rate $\beta$, the corresponding expressions areIf $c>1/\beta$, the positive service root lies strictly below $\beta$. The finite moment-generating function domain cannot be ignored.

To see statistical aggregation explicitly, let the entire arrival processes of $k$ sources be independent, though each source may itself have dependence over time. Their cumulative totals satisfyThus the shared queue has an exponential overflow bound when ${\sum }_j{\alpha }_j(\theta )<c$, with the finite-window conditions proved above. One must use a common parameter $\theta$; adding individual service-root exponents is not the rule. For example, an independent Poisson source of mean $\nu$ and an independent on-off source of size $r$ and probability $p$ have aggregate cumulant-generating functionFor $\nu >0$ and $c>\nu +pr$, their positive service root exists and determines the optimized upper exponent. Several independent Poisson sources similarly have the same formula as one source with intensity ${\sum }_j{\nu }_j$.

The connection with large deviation principles also appears in a many-source scaling. For $L$ independent copies of a source, aggregate service $Lc$ and buffer $Lb$, fix a window $n$. Let $X_i(n)$ be each copy's cumulative input in that window. Their total exceeds $L(b+cn)$ only if the empirical mean exceeds $b+cn$. Its Chernoff bound has exponentIf exponential moments exist near zero, Cramér's theorem gives the corresponding finite-window large deviation principle. The threshold exceeds the mean because $m<c$ and $b>0$. A finite union of windows has upper exponential rate at most $-{\min }_n{V}_n(b)$ over those windows. Passing to infinitely many windows requires uniform control of long-window probabilities, not simply an interchange of a limit with an infinite union. A good sample-path large deviation principle in a topology where the queue workload is continuous, as in Question 3, instead permits the contraction principle for large deviations to derive a workload rate function and its closed-set overflow upper bound.

Finally, the mean and peak limits of effective bandwidth quantify the safety margin. With exponential moments near zero, independent slot work of mean $m$ and variance $v$ hasFor bounded work, its large-parameter limit is the essential peak. Stronger tail requirements select larger $\theta$ and hence more capacity; finite-window effective bandwidths also reflect temporal burst correlations. A stable heavy-tailed input can lack every positive exponential moment, in which case this exponential-bound method supplies no positive decay rate. Stability and exponentially small overflow are distinct requirements.

Let U be the total payload arriving in one slot. Conditional on its Poisson packet count, use the exponential moment of each independent payload to get, for $\theta <1$,For net work $Z=U-C\delta$, its cumulant-generating function isThe mean net work is $\delta (\lambda -C)<0$. Under the arrivals-before-service slot convention the Lindley recursion gives stationary queue workload $R_{\delta }\stackrel{d}{=}{\sup }_{m\ge 0}{\sum }_{j=1}^m{Z}_j$. The positive root of the exponential-moment equation is ${\theta }_{\ast }=1-\rho$. Thus $\exp ({\theta }_{\ast }{\sum }_{j=1}^m{Z}_j)$ is a nonnegative martingale. Stop it at the first crossing of r or a finite horizon; its value on the crossing event is at least $e^{{\theta }_{\ast }r}$. Increasing the horizon provesThis is an infinite-horizon bound, not a one-slot Chernoff estimate.

A useful refinement identifies both the exponential rate and the small-slot prefactor. Couple the random walk to the continuous compound Poisson input U(t), and write $W={\sup }_{t\ge 0}(U(t)-Ct)$ and $W_{\delta }={\sup }_{k\ge 0}(U(k\delta )-Ck\delta )$. Their laws are the continuous and slot-boundary stationary queue workloads. Monotonicity of U implies the grid-sampled compound Poisson workload bound: $W_{\delta }\le W\le W_{\delta }+C\delta$, by rounding each t up to the next grid point.

The continuous queue workload law can be obtained directly from the stationary packet count. Conditional on Q=k, the k remaining payloads are independent unit-mean exponentials: unserved payloads have not affected earlier service completions, and the in-service residual uses the memoryless property. HenceThus W has mass $1-\rho$ at zero, and its positive part is exponential with rate $1-\rho$. Combining this stationary workload of an exponential-payload queue with the grid bound gives the explicit estimateIn particular ${\lim }_{r\to \infty }{r}^{-1}\log \mathbb{P}(R_{\delta }\ge r)=-(1-\rho )$, and for small delta the estimate is $\rho e^{-(1-\rho )r}$. The exact continuous-time tail is that latter expression. Another slot convention may shift queue workload by the within-slot input/service timing; its convention must be specified before asserting a prefactor, while this exponential tail rate is unchanged.

Consider two work-conserving slotted queues in tandem, with upstream service $c$ and downstream service $d<c$, where upstream departures can be served downstream in the same slot. Put $D_t=\min \{c,Q_{t-1}+a_t\}$. Then $Q_t=Q_{t-1}+a_t-D_t$ and $R_t=(R_{t-1}+D_t-d{)}^{+}$. If $R_{t-1}+D_t\ge d$, addition gives $(Q_{t-1}+R_{t-1}+a_t-d{)}^{+}$. Otherwise $D_t<d<c$, so the upstream queue empties and both sides of that identity are zero. Thus total workload obeys the Lindley recursion with service $d$. Starting empty in the remote past yields the displayed identity whenever the upstream workload is finite. Both workloads, and hence their difference, are continuous in the weighted cumulative-input topology for a slotted queue under strictly subcritical mean input $\lambda <d<c$.

For independent and identically distributed arrivals with finite mean below service $c$, the strong law of large numbers makes the net cumulative input tend to minus infinity, so the supremum is finite almost surely. Applying the same formula to the past input gives a stationary solution of the Lindley recursion. This representation turns queue workload overflow into an event involving some lookback horizon.