= Finite-buffer workload map
{title2=$0\leq\bar q_B(a)\leq\min(q(a),B)$}
With lookback net input $z(s)=a(-s,0]-Cs$, the stable finite-buffer workload is $\sup_{t\geq0}\min(\sup_{0\leq s\leq t}z(s),B+\inf_{0\leq s\leq t}z(s))$. Since $z(0)=0$, it lies between zero and B, and is at most the infinite-buffer <supremum> $q(a)=\sup_{s\geq0}z(s)$. For a constant-rate burst which builds exactly x work from an empty queue, no upper reflection occurs when $x\leq B$ and both workloads equal x.
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