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Use arrival rate v during the last t time units and the normal stable rate mu before that, where $\mu <C$. The lookback net input increases from zero to x over the burst, then decreases because the earlier net drift is negative. Its infinite-buffer workload is x, as is the finite-buffer workload map for capacity at least x. For a local convex path cost with density g and $g(\mu )=0$, its cost is $tg(v)=xg(v)/(v-C)$. These paths realize or approximate the optimal workload cost even if an optimizing duration is not attained.

Work deterministically with nonnegative arrivals. Write $S_x(t)={\sum }_{k=1}^t{x}_{-k}$ for integer $t\ge 1$, and $S_x(0)=0$. The relevant metric isThis is the weighted cumulative-input topology for a slotted queue. It is a genuine norm on the ambient space of sequences with finite value: zero cumulative differences force every individual difference to vanish, since $x_{-k}=S_x(k)-S_x(k-1)$. Every input in the specified set $A$ has finite norm, because its long prefixes are bounded by $\lambda t$ and there are only finitely many earlier prefixes.

Use the conventional net-input slot boundary finite-buffer workload map, whose recursion is the clipped Lindley recursionArriving work in a slot is available to that slot's service, and the observed queue state is after service and upper clipping. This fixes the slot convention explicitly. The queue size $Q_B(x)$ is obtained by starting in the distant past and taking the present workload. We first prove that, under the stated strict drift condition, this is uniquely defined independently of the initial state.

Fix $x\in A$ and write $d_0=C-\lambda >0$. Choose a finite integer $T$ so large thatThen $S_x(T)-CT<-B$. Start at time $-T$ with a full buffer $q_{-T}=B$. The trajectory must hit zero by time zero. Otherwise lower clipping is never used, so for nonnegative lost-work amounts ${\ell }_t$ the total recursion would givecontradicting nonnegativity. If zero is attained even at the last update the desired conclusion still holds.

The update $F_B$ is monotone in the initial workload. Couple all starting values in $[0,B]$ using the same arrivals. When the trajectory starting at $B$ reaches zero, all smaller trajectories must also be zero, and they remain identical thereafter. Thus the present queue depends only on the last $T$ slots, not on any earlier state or arrivals. This is finite-memory continuity of a finite-buffer queue. In particular it proves the existence and uniqueness of $Q_B(x)$ without assuming a random stationary law.

The same memory horizon works in a neighborhood of $x$. If $\delta =\Vert x-y\Vert <d_0/2$, thenThe full-buffer trajectory for $y$ therefore also reaches zero within this horizon. Both $Q_B(x)$ and $Q_B(y)$ can consequently be calculated from initial workload zero at time $-T$, even though their respective reset times need not coincide.

Clipping to $[0,B]$ is Lipschitz continuous with constant one, soIterating from the common initial value zero givesThe cumulative-input norm bounds each summand:Summing proves the explicit local estimateGiven $\epsilon >0$, choose $\Vert x-y\Vert <\min \{(C-\lambda )/2,\epsilon /T^2\}$. Then the workload difference is less than $\epsilon$. Since $x\in A$ was arbitrary, the queue size function is continuous on $(A,\Vert \cdot \Vert )$.

The argument also handles the alternative slot ordering that clips arrivals to the buffer before serving them: its update is a composition of monotone one-Lipschitz clipping and service maps, and negative total net input forces a reset. Measuring immediately after arrivals instead adds one finite-coordinate update, which is likewise continuous in this norm. The essential point is strict subcritical drift and loss of dependence on the remote past, rather than incorrectly replacing a finite-buffer history by the minimum of $B$ and an infinite-buffer workload.