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

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.

There is an indexing issue in the printed topology. Its cumulative sums omit $a_0$, whereas the post-slot workload of a queue in part (a) includes $a_0$. To see the problem, take all $a_t=\lambda$ for $t\le -1$, with $\lambda <c$, and compare two inputs with $a_0=\lambda$ and $a_0=c+1$. Their distance according to the printed cumulative-sum expression is zero, yetThus the post-slot workload is not continuous in the literal printed topology. On two-sided sequences that expression is a seminorm, since it also ignores future coordinates; the fixed-mean input class is not a vector space (if signed inputs are allowed, it is an affine set).

The appropriate repair for the post-slot convention is to use one-sided past inputs including time zero and the weighted cumulative-input topology for a slotted queue:This is a finite metric on the fixed-mean class, since cumulative differences are $o(n)$; equality of all cumulative sums implies equality of all past coordinates. It is equivalent to augmenting the printed cumulative-difference distance by $|a_0-b_0|$: their cumulative sums differ only by this extra coordinate and a one-step shift.

Here is the full corrected continuity proof. Fix $a$ with $A_n(a)/n\to \lambda <c$, and let $\epsilon =(c-\lambda )/4$. Choose $N\ge 1$ such that $A_n(a)/n\le \lambda +\epsilon$ whenever $n\ge N$. If $d_{\#}(a,b)<\epsilon /2$, then, for all such $n$,Both queue workloads therefore maximize over the same finite collection $0\le n<N$, containing the value zero. For any two finite lists, the difference of their maxima is at most the maximum absolute difference of corresponding entries. Thusin this neighborhood. In particular the queue workload is locally Lipschitz continuous and hence continuous.

An alternative indexing repair retains the printed topology but interprets the time-zero function as the workload before slot $(-1,0)$, namely ${\sup }_{n\ge 0}\{{\sum }_{j=1}^n{a}_{-j}-cn\}$. The same finite-window argument proves its continuity. One must then shift the time labels consistently in the later recursions. The remaining answers use the post-slot convention and $d_{\#}$.

The indexing defect also affects the downstream claim. For the two inputs in part (b), now choose $\lambda <d<c$. The constant input has zero downstream queue workload, while the input with $a_0=c+1$ hasTheir printed distance is zero, so the literal post-slot downstream continuity assertion is false as well.

In the corrected weighted cumulative-input topology for a slotted queue, apply the finite-window proof of part (b) separately with services $d$ and $c$. Both have strictly negative long-window drift because $\lambda <d<c$. There are a neighborhood of $a$ and finite constants $C_d,C_c$ such thatThis proves continuity, and indeed local Lipschitz continuity, for the intended downstream function.

Suppose a family of random arrival paths $a^{(L)}$ has a large deviation principle in this corrected input space, at speed $L$ with good rate function $\mathcal{I}$. The contraction principle for large deviations then gives the downstream good rate functionIt is infinity for $r<0$. The path large deviation principle, its topology, and its subcritical domain must actually be established for the chosen stochastic model: a collection of finite-dimensional large deviation principles alone does not control infinitely long workload windows. Nor should upstream departures be assumed independent; the path contraction avoids that unjustified assumption.

Consider two work-conserving slotted queues in tandem, with upstream service $c$ and downstream service $d<c$, where upstream departures can be served downstream in the same slot. Put $D_t=\min \{c,Q_{t-1}+a_t\}$. Then $Q_t=Q_{t-1}+a_t-D_t$ and $R_t=(R_{t-1}+D_t-d{)}^{+}$. If $R_{t-1}+D_t\ge d$, addition gives $(Q_{t-1}+R_{t-1}+a_t-d{)}^{+}$. Otherwise $D_t<d<c$, so the upstream queue empties and both sides of that identity are zero. Thus total workload obeys the Lindley recursion with service $d$. Starting empty in the remote past yields the displayed identity whenever the upstream workload is finite. Both workloads, and hence their difference, are continuous in the weighted cumulative-input topology for a slotted queue under strictly subcritical mean input $\lambda <d<c$.