= Effective-bandwidth admission region with a voice delay gate
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> gives
$$
p_d\leq e^{-\theta_TB_0},\qquad
p_v\leq e^{-\theta_TB_0}+e^{-\theta_VK},
$$
provided $n_va_v(\theta_T)+n_da_d(\theta_T)\leq C$ and $n_va_v(\theta_V)\leq C$. 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 set
$$
\theta_T=B_0^{-1}\log(1/\eta_B),\qquad
\theta_V=K^{-1}\log(1/\eta_V).
$$
The 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.
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