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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.

Consider an M/M/1 queue with an exogenous Poisson process of arrivals of rate $\lambda$, one server, unlimited waiting space, first come first served service, and independent and identically distributed service times with an exponential distribution of rate $\mu$, independent of arrivals. There is no blocking or abandonment, so offered rate and admitted queue throughput coincide: ${\lambda }_{\mathrm{eff}}=\lambda$. The memoryless property makes the queue population a birth-death process. For $0<\lambda <\mu$, the stationary distribution of an M-M-1 queue isThis is a geometric distribution on the nonnegative integers. Its mean, together with Little's law, givesHere $L,W$ count the whole system, and $L_q,W_q$ count only waiting. These distinctions and formulas are derived in MIT 2.854, M/M/1 Queue notes, pages 1-2.

Take $\lambda =4$ customers per hour and $\mu =5$ customers per hour. The mean service time is $1/5$ hour, or 12 minutes, and the server utilization is $\rho =4/5=0.8$. Computing each quantity explicitly,Thus Little's law checks both boundaries: $4=4\cdot 1$ and $3.2=4\cdot 0.8$. The mean number in service is $L-L_q=0.8$, rather than one, because the server is idle 20 percent of the time. By PASTA, an arriving customer sees four customers already in the system on average. Under first come first served, the memoryless property gives mean residual service time $1/\mu$ for the customer in service; clearing those four customers takes $4/5=0.8$ hour on average, independently confirming the mean queueing delay.

Holding $\mu =5$ per hour fixed, increasing arrivals from $4$ to $4.5$ per hour raises server utilization from $0.8$ to $0.9$, but doubles mean customer sojourn time from one to two hours. At $\lambda =4.9$ per hour, server utilization is $0.98$, $L=49$, $W=10$ hours, $L_q=48.02$, and $W_q=9.8$ hours. Both mean population and mean duration diverge as $\lambda$ approaches $\mu$ from below. The formula describes an explosive congestion penalty near capacity even though all arrivals still eventually receive service at every fixed subcritical rate. At or above capacity there is no stationary distribution for this unlimited M/M/1 queue, so these finite stationary formulas no longer apply.

The inverse dependence of mean customer sojourn time on spare capacity, $W=1/(\mu -\lambda )$, makes server utilization a poor standalone measure of service quality. Adding capacity at a nearly saturated M/M/1 queue can sharply reduce queueing delay. Stanley B. Gershwin develops this capacity interpretation in Markov Processes and Queues, MIT 2.854, 2016, slides 71-72.

An ordered state $x=(c_1,\dots ,c_n)$ records the classes and positions of customers at a single server. Class-$r$ arrivals have rate ${\alpha }_r$ and exponential service parameter ${\mu }_r$. Under a symmetric service discipline, its stationary distribution is $(1-\rho ){\prod }_i({\alpha }_{c_i}/{\mu }_{c_i})$, where $\rho ={\sum }_r{\alpha }_r/{\mu }_r<1$. If every ${\mu }_r=\mu$, class-independent insertion and service position rules need not be equal: time reversal of a continuous-time Markov chain interchanges them and proves the same law. Aggregating words gives $(1-\rho )n!{\prod }_r[({\alpha }_r/{\mu }_r{)}^{n_r}/n_r!]$ by the multinomial coefficient. With class-dependent service rates, first come first served need not have this product law.

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.

The queueing delay is the time between a customer's arrival and the start of its service, excluding its own service time. Under first come first served with service rate $C$, its queueing delay is the queue workload ahead of it divided by $C$. The customer sojourn time also includes the customer's own service. In a preemptive priority queue with a shared buffer, a voice customer's initial waiting time depends only on the voice queue workload already present.

Let customer $k$ enter a queue at $a_k$ and leave at $d_k$, and count it in $N(t)$ for $a_k\le t<d_k$. On a finite observation interval $[0,T]$, the integral of the queue population is exactly the sum of the lengths of the overlaps of customer visits with that interval, as displayed above; $x_{+}=\max (x,0)$ is the positive part. This deterministic identity includes visits already in progress at time zero and visits still unfinished at time $T$. It needs no probability distribution for arrivals or service times. The area argument is also developed in Richard C. Larson and Amedeo R. Odoni, Urban Operations Research, section 4.4.

For a fully computed example, start empty and observe through $T=6$ minutes. Three customers arrive at $a=(0,1,4)$ minutes, begin service at $b=(0,2,4)$ minutes, and depart at $d=(2,3,5)$ minutes. These times describe a single first come first served server with service times $(2,1,1)$ minutes. Their customer sojourn times are $(2,2,1)$ minutes, and their queueing delays are $(0,1,0)$ minutes. The queue population is respectively $1,2,1,0,1,0$ on the consecutive intervals $[0,1),[1,2),[2,3),[3,4),[4,5),[5,6]$, soThe same area is $2+2+1=5$ when counted by customer. Hence the finite-window averages satisfyThe waiting area is one customer-minute, giving $L_q=1/6$ and $W_q=1/3$ minute; $(1/2)(1/3)=1/6$. The server works for four minutes, so its server utilization is $4/6=2/3$ and $L-L_q=5/6-1/6=2/3$. Repeating this six-minute pattern indefinitely makes all these values long-run averages, without a Poisson process of arrivals.

An observation window that cuts visits short requires the overlap terms, rather than full customer sojourn times. Starting empty, let $A(T)$ count admissions before $T$ and write $w_k=d_k-a_k$. ThenFor the example cut at $T=2.5$ minutes, the area is $1+2+0.5=3.5$ customer-minutes, whereas the full durations of the two admitted customers sum to $4$; the remaining half-minute of the second visit is $R(2.5)=0.5$. A precise sufficient sample path condition for the limiting Little's law is $A(T)/T\to {\lambda }_{\mathrm{eff}}\in (0,\infty )$, $A(T{)}^{-1}{\sum }_{a_k<T}{w}_k\to W<\infty$, and $R(T)/T\to 0$. Dividing the identity by $T$ then gives $L={\lambda }_{\mathrm{eff}}W$. Equating admission and departure rates additionally uses $N(T)/T\to 0$, by the balance $N(T)=A(T)-D(T)$ for an initially empty queue, where $D(T)$ counts all departures.

A service discipline specifies which customers receive the server's capacity and where newly arriving customers join the queue. First come first served and processor sharing give different sample paths even when their equilibrium populations agree.

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.