= Weighted cumulative-input topology for a slotted queue
{title2=$d_\#(a,b)=\sup_{n\geq1}|A_n(a)-A_n(b)|/(n+1)$}
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> $Q(a,c)=\sup_{n\geq0}(A_n(a)-cn)$ is locally <Lipschitz continuous>. To prove this, choose $\varepsilon=(c-\lambda)/4$ and $N\geq1$ such that $A_n(a)/n\leq\lambda+\varepsilon$ for $n\geq N$. If $d_\#(a,b)<\varepsilon/2$, then $A_n(b)-cn\leq n(\lambda+2\varepsilon-c)<0$ for all such $n$. Both suprema reduce to $0\leq n<N$, 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>.
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