Finite-buffer workload map ($0\le {\bar{q}}_B(a)\le \min (q(a),B)$)

With lookback net input $z(s)=a(-s,0]-Cs$, the stable finite-buffer workload is ${\sup }_{t\ge 0}\min ({\sup }_{0\le s\le t}z(s),B+{\inf }_{0\le s\le t}z(s))$. Since $z(0)=0$, it lies between zero and B, and is at most the infinite-buffer supremum $q(a)={\sup }_{s\ge 0}z(s)$. For a constant-rate burst which builds exactly x work from an empty queue, no upper reflection occurs when $x\le B$ and both workloads equal x.

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 ${\sum }_{k=1}^T(2k-1)\Vert x-y\Vert =T^2\Vert x-y\Vert$.

For the local convex action defining the linear workload rate for a local convex action, continuity and the contraction principle for large deviations give workload rate functions. The pointwise bound ${\bar{q}}_B\le q$ makes ${\bar{J}}_B(x)\ge {\inf }_{y\ge x}J(y)=J(x)$ on nonnegative workloads. Constant-rate burst paths produce the reverse bound for x at most B. Values outside the buffer interval are impossible. This statement concerns the rate at the workload value, rather than equality of exact finite-buffer and infinite-buffer stationary distributions.