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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.

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.

A multiclass queue must specify its service discipline as well as its arrival and service rates. Here is an ordered-queue model that includes the usual class-independent exponential single-server queue. Let class-$r$ arrivals be independent Poisson processes of rates ${\alpha }_r$, and let a class-$r$ service requirement have the exponential distribution of rate ${\mu }_r$. A state is a finite word $x=(c_1,\dots ,c_n)$ of customer classes.

Choose service fractions ${\gamma }_i(n)\ge 0$ with ${\sum }_{i=1}^n{\gamma }_i(n)=1$, and insertion probabilities ${\delta }_i(n)\ge 0$ with ${\sum }_{i=1}^n{\delta }_i(n)=1$. The customer at position $i$ departs at rate ${\mu }_{c_i}{\gamma }_i(n)$; a class-$r$ arrival is inserted at position $i$ at rate ${\alpha }_r{\delta }_i(n+1)$. All rates leading to the same word are added. For class-dependent ${\mu }_r$, impose the symmetric service discipline ${\delta }_i(n)={\gamma }_i(n)$. If all ${\mu }_r=\mu$, arbitrary class-independent $\gamma ,\delta$ are permitted, including first come first served with insertion at the tail and service at the head.

Put ${\rho }_r={\alpha }_r/{\mu }_r$ and $\rho ={\sum }_r{\rho }_r$. For $\rho <1$, the equilibrium word law isFor the symmetric service discipline, insertion and deletion satisfy detailed balance for a continuous-time Markov chain becauseFor the common-service-rate case, use time reversal of a continuous-time Markov chain instead: the reversed queue inserts with probabilities $\gamma$ and serves with fractions $\delta$. A forward insertion gives a reverse deletion rate $\mu {\delta }_i$, and a forward deletion gives a reverse insertion rate ${\alpha }_r{\gamma }_i$. Forward and reverse total rates both equal ${\sum }_r{\alpha }_r+\mu 1_{\{n>0\}}$, proving global balance for a continuous-time Markov chain.

Summing the weights of all words of length $n$ gives $(1-\rho ){\rho }^n$, so the law is normalized. If $n_r$ counts class-$r$ customers and $n={\sum }_r{n}_r$, the multinomial coefficient counting class orderings givesFor processor sharing, ${\gamma }_i(n)={\delta }_i(n)=1/n$, the class-count vector itself is a Markov chain, with arrival rate ${\alpha }_r$ and departure rate ${\mu }_r{n}_r/n$. Under first come first served, the ordered word is generally necessary. Class-dependent service rates combined with first come first served do not in general have the displayed simple equilibrium law; the discipline assumption matters.

A two-class queue has one total buffer but serves the high-priority class before the other. A preemptive-resume service discipline interrupts low-priority work immediately and later resumes it without wasting completed work. Under first come first served within the high-priority class, a new high-priority packet waits $V/C$ when $V$ is the high-priority workload immediately before its arrival. The total workload determines shared-buffer overflow, while $V$ determines the high-priority waiting time. An additional admission gate rejects high-priority packets when $V/C$ exceeds a prescribed delay limit. Finite packets require a fit test against the remaining buffer space, and nonpreemptive service adds residual low-priority service time.

Queueing theory studies how jobs accumulate while waiting for limited service: customers at a desk, requests at a computer, or calls competing for a connection. A queue model specifies how jobs arrive, their service times, the number of servers, the buffer size of a queue, and the service discipline. Its practical questions include the queue population, queueing delay, customer sojourn time, queue throughput, and server utilization.

Average spare capacity does not guarantee short waits. Even when the service rate of a queue exceeds the arrival rate, bursts of arrivals and unusually long service times can leave work waiting. The M/M/1 utilization and delay example shows that an 80 percent server utilization can turn a 12-minute mean service time into a 60-minute mean customer sojourn time. As capacity is approached, congestion grows sharply; reserving spare capacity can be much more valuable than its idle fraction suggests.

Two complementary ideas make queueing theory especially useful. Little's law converts a measured queue population and queue throughput into a mean customer sojourn time, using a counting identity rather than a particular arrival law. Poisson arrivals see time averages answers a different question: when do arriving customers see the same state frequencies as an observer sampling over time? That requires an exogenous Poisson process and a system that cannot anticipate its future arrivals. Neither result by itself determines all queueing delays; detailed models such as the M/M/1 queue and M/G/1 queue supply the remaining information.

The queue workload of a queue is the total remaining service work of all present customers, including residual work in service. With a work-conserving server of rate $C$ and stationary input $A$, the stationary queue workload is represented byThis is the amount by which past offered work exceeds available service, maximized over possible starts of a busy period. It is the reflection of net input at zero. Under first come first served, an arriving infinitesimal customer waits $W/C$; a priority customer instead waits for the remaining work ahead of it under its service discipline.