= Finite-memory continuity of a finite-buffer queue
{title2=$|Q_B(x)-Q_B(y)|\leq T^2\|x-y\|$}
Use the clipped <Lindley recursion> $q_{t+1}=\min(B,\max(0,q_t+x_t-C))$ and the <weighted cumulative-input topology for a slotted queue>. If cumulative past input $S_x(T)\leq\lambda T$ eventually with $\lambda<C$, choose a finite $T$ so $S_x(T)-CT<-B$. Starting with a full buffer at time $-T$, the queue must hit zero before the present, since otherwise its final workload is at most $B+S_x(T)-CT<0$. Monotonicity couples all initial workloads by that zero, establishing a finite memory horizon. The same horizon works for sufficiently close input $y$. Since $|x_{-k}-y_{-k}|\leq(2k-1)\|x-y\|$ and clipping is <Lipschitz continuous> with constant one, the two finite recursions started empty differ by at most $\sum_{k=1}^T(2k-1)\|x-y\|=T^2\|x-y\|$.
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