Weighted cumulative-input topology for a slotted queue ($d_{\#}(a,b)={\sup }_{n\ge 1}|A_n(a)-A_n(b)|/(n+1)$)

For one-sided past inputs, let $A_n(a)={\sum }_{j=0}^{n-1}{a}_{-j}$ and $A_0=0$. The displayed metric controls cumulative differences over every past window, including the latest arrival. On the affine class $A_n(a)/n\to \lambda <c$, the queue workload $Q(a,c)={\sup }_{n\ge 0}(A_n(a)-cn)$ is locally Lipschitz continuous. To prove this, choose $\epsilon =(c-\lambda )/4$ and $N\ge 1$ such that $A_n(a)/n\le \lambda +\epsilon$ for $n\ge N$. If $d_{\#}(a,b)<\epsilon /2$, then $A_n(b)-cn\le n(\lambda +2\epsilon -c)<0$ for all such $n$. Both suprema reduce to $0\le n<N$, and their difference is at most $(N+1)d_{\#}(a,b)$. Ignoring the latest coordinate makes this assertion false for a post-slot workload of a queue.