Multiclass single-server queue

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.

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.

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.