Finite-memory continuity of a finite-buffer queue ($|Q_B(x)-Q_B(y)|\le T^2\Vert x-y\Vert$)

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 $S_x(T)\le \lambda T$ 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 $|x_{-k}-y_{-k}|\le (2k-1)\Vert x-y\Vert$ and clipping is Lipschitz continuous with constant one, the two finite recursions started empty differ by at most ${\sum }_{k=1}^T(2k-1)\Vert x-y\Vert =T^2\Vert x-y\Vert$.