= Grid-sampled compound Poisson workload bound
{title2=$(W-C\delta)^+\leq W_\delta\leq W$}
For nondecreasing compound Poisson input U, let $W=\sup_{t\geq0}(U(t)-Ct)$ and $W_\delta=\sup_{k\geq0}(U(k\delta)-Ck\delta)$. The second <supremum> is bounded above by the first. For each t, the next grid point is at most delta later, its input is at least U(t), and it offers at most an extra $C\delta$ service. Hence $W\leq W_\delta+C\delta$. This supplies explicit stationary-tail brackets and shows that time discretization preserves the workload's exponential tail rate.
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