 Sublinear path space for fluid queues

The useful infinite-horizon path space for a stable queue isFor $\delta ,\sigma >0$, the workload functional $R(f)={\sup }_{t\ge 0}(\sigma f(t)-\delta t)$ is finite and continuous. Fix $f$, and choose $T\ge 1$ so that $|\sigma f(t)|\le \delta t/4$ for $t\ge T$. If $\Vert g-f{\Vert }_{\mathrm{sl}}<\delta /(4\sigma )$, then $\sigma g(t)-\delta t<0$ for $t\ge T$. Both suprema can thus be taken over $[0,T]$, whereUniform convergence on bounded time intervals alone does not give this continuity: a pulse escaping to later times can produce a large workload while tending locally to zero.