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Assume independent stationary Compound Poisson process inputs, preemptive-resume priority for voice, first come first served within each class, capacity $C$, a shared buffer of $B$ units of work, packet sizes bounded by $s_{\max }<B$, and voice delay gate $D>0$. Write $B_0=B-s_{\max }$ and $K=CD$. For $n_v,n_d$ flows with per-flow effective bandwidths $a_v,a_d$, the continuous-time exponential workload bound givesprovided $n_v{a}_v({\theta }_T)+n_d{a}_d({\theta }_T)\le C$ and $n_v{a}_v({\theta }_V)\le C$. To prove this, couple with the unthinned infinite-buffer aggregate and voice queues driven by the same offered packets. Dropping work reduces both the total and voice workloads under this service discipline. A buffer-fit failure implies total workload greater than $B_0$, and a voice-gate failure implies voice workload at least $K$. Apply the two bounds and a union bound; Poisson arrivals see time averages converts stationary prearrival probabilities to offered-packet loss fractions. For target losses ${\ell }_d,{\ell }_v$, choose $0<{\eta }_B<\min ({\ell }_d,{\ell }_v)$ and ${\eta }_V={\ell }_v-{\eta }_B$, then setThe two effective bandwidth constraints, together with strict aggregate mean-load stability, define a sufficient integer admission region. Voice loss includes the shared-buffer term, even though data packets have lower service priority.

For the amount of work $A_j(0,t]$ offered by one flow in a time interval of length $t$, its effective bandwidth at parameter $\theta >0$ and timescale $t>0$ isIts units are work per unit time. The parameter $\theta$ measures the price placed on bursts: a stringent rare-overflow target generally requires a larger value of $\theta$ than a loose target. The timescale $t$ describes the duration of the congestion episode. When the limit exists, writeThis is the long-time effective bandwidth. It includes a burstiness cost, rather than just the mean offered rate. For a finite-time flow with the required moments, ${\lim }_{\theta \downarrow 0}{\alpha }_j(\theta ,t)=\mathbb{E}{A}_j(0,t]/t$, and effective bandwidth is increasing in $\theta$. If $A_j(0,t]/t$ is bounded, its limit as $\theta \to \infty$ is the essential supremum of that interval's input rate, the peak-rate limit. The analogous long-time mean limit requires the usual interchange or regularity assumptions.

For independent flows their cumulant-generating functions add, soThe Chernoff bound therefore gives, for any $\theta >0$ in the common domain,Optimizing over $\theta$ produces the relevant Legendre-Fenchel transform. The Cramér theorem states that sample means of independent and identically distributed inputs whose moment-generating function is finite near zero have a large deviation principle, with rate function equal to the Legendre-Fenchel transform of their cumulant-generating function. More generally, the Gärtner–Ellis theorem applies to a convergent scaled log moment-generating function when zero lies inside its domain and the limit is lower semicontinuous and essentially smooth. It gives the same transform as a good rate function. A sample-path version, together with the contraction principle for large deviations for the queue workload, translates input deviations into overflow deviations. The compact-time and infinite-horizon continuity/tail hypotheses must be checked for the chosen traffic model.

A particularly useful rigorous queue bound holds for inputs with stationary increments and independent increments, such that $\mathbb{E}{e}^{\theta A(t)}=e^{t\Lambda (\theta )}$. If $\Lambda (\theta )\le C\theta$, thenis a nonnegative supermartingale, since its conditional increment factor is $e^{(t-s)(\Lambda (\theta )-C\theta )}\le 1$. The maximal inequality for a nonnegative supermartingale on $[0,T]$, followed by $T\to \infty$, gives the continuous-time exponential workload boundHere $W$ is the stationary queue workload of the stable infinite-buffer queue, represented by input from the past. This directly bounds the supremum over time; applying a one-time Chernoff bound alone would not establish it.

For a nonzero Compound Poisson process with bounded positive packet sizes and mean input rate less than $C$, a positive root ${\theta }_{\ast }$ of $\Lambda (\theta )=C\theta$ exists and is unique. Here the root theorem can be made explicit:The upper bound was just proved. For the lower bound, use exponential tilting with density $e^{{\theta }_{\ast }(A(t)-Ct)}$ at time $t$. Under the tilted law the net input has positive mean drift ${\Lambda }^{\prime }({\theta }_{\ast })-C>0$, by strict convexity and the positive-root equation. The strong law of large numbers then makes the first crossing time ${\tau }_b$ finite almost surely under that law. Its overshoot is at most $s_{\max }$. Changing measure at ${\tau }_b$, first on $\{{\tau }_b\le T\}$ and then letting $T\to \infty$, givesDividing the logarithmic bounds by $b$ proves ${\lim }_{b\to \infty }{b}^{-1}\log \mathbb{P}(W\ge b)=-{\theta }_{\ast }$.

Thus the familiar admission rule compares ${\sum }_j{a}_j(\theta )$ with $C$, choosing $\theta$ from the desired buffer-tail exponent. For more general inputs, a logarithmic root theorem requires its own light-tail and renewal or sample-path hypotheses; the displayed supermartingale upper bound needs only the moment and increment assumptions stated above.

The capacity and quality targets alone do not determine flow counts: the per-flow traffic statistics must also be supplied. The following explicit model gives a computable sufficient region, with genuine upper bounds on both loss fractions.

Measure queue workload and buffer size in bits. Denote capacity by $C$, shared buffer size by $B$, voice delay limit by $D$, target voice loss by ${\ell }_v$, and target data loss by ${\ell }_d$, with $0<{\ell }_v,{\ell }_d<1$. Let there be $n_v$ voice flows and $n_d$ data flows. Model each class-$j$ flow, $j\in \{v,d\}$, by a stationary Compound Poisson process with packet rate ${\nu }_j$ and positive independent packet sizes $S_j$, and assume independence across flows and marks. Use a work-conserving preemptive-resume priority queue with a shared buffer, with voice priority and first come first served within each class. This preemptive or fluid approximation makes the predicted voice waiting time exactly $V/C$, where $V$ is the remaining voice queue workload just before arrival. Assume packet sizes are bounded by a known $s_{\max }<B$ and interpret the buffer gate as a packet-fit test; rejecting packets which do not fit is the packet version of the full-buffer rule.

For one flow the Compound Poisson process formula givesFor fixed-size packets of size $s_j$, substitute $e^{\theta s_j}$. Estimate the packet rate and size law from measurements or a declared traffic contract. Also require the strict stability inequalityDefine $B_0=B-s_{\max }>0$ and $K=CD$. Couple the switch with an unthinned infinite-buffer aggregate queue and an unthinned infinite-buffer voice-only queue, both driven by the same offered packets and both served at rate $C$. Under the preemptive service discipline, dropping arrivals can only decrease both the total queue workload and the voice queue workload. Thus total switch queue workload $Q$ and voice queue workload $V$ satisfy $Q\le W_T$ and $V\le W_V$ in the stationary coupling. A packet-fit loss implies $Q>B-S_j\ge B_0$, and a voice-delay rejection implies $V\ge K$.

Apply the continuous-time exponential workload bound to the two comparison queues. For any positive ${\theta }_T,{\theta }_V$ satisfyingthe stationary loss-event probabilities obeyThe second inequality is a union bound: voice can be lost either at the shared buffer or at its delay gate. Data congestion therefore contributes to voice loss despite its lower service priority. Poisson arrivals see time averages applies to each offered packet stream, whose future arrivals are independent of the prearrival state; the stationary event bounds are consequently bounds on the offered-packet loss fractions. Voice packets that are admitted have queueing delay below $D$, because later voice packets join behind them and data work cannot delay them in this model.

To turn the bounds into flow counts, allocate loss budgetsFor example, ${\eta }_B=\min ({\ell }_d,{\ell }_v/2)$ and ${\eta }_V={\ell }_v-{\eta }_B$ also work; equality with ${\ell }_d$ is harmless. The required sufficient effective-bandwidth admission region with a voice delay gate isIt ensures $p_d\le {\ell }_d$ and $p_v\le {\ell }_v$. A simple enumeration isdiscarding negative bounds and pairs failing strict mean-load stability. The formulas assume positive-rate nonzero-size flows. Optimize a chosen revenue or flow-count objective over these integer pairs. Searching over the budget split ${\eta }_B$ can enlarge the sufficient region. For a bit-fluid buffer gate with no packet-fit issue, $B_0$ can be replaced by $B$.

Real nonpreemptive packet service needs an additional residual-service allowance or an explicit priority queue workload analysis: a voice packet may have to wait for the data packet already in service. Burst-correlated voice sources also require richer descriptors, such as measured finite-time effective bandwidths ${\alpha }_j(\theta ,t)$ or a Markov-modulated source model. For such inputs, additivity still holds for independent flows, but the independent-increment supermartingale bound is not automatic; use a valid sample-path large deviation principle or a model-specific overflow bound, with prefactors or a safety margin when using asymptotics. If packet arrivals do not form a Poisson process, loss fractions must be assessed at arrival epochs rather than inferred solely from time-average queue workload probabilities. These qualifications explain both how to calculate an admission region from specified traffic statistics and why the five switch/QoS parameters by themselves cannot supply a numerical answer.