= 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 $W\leq W_\delta+C\delta$. When the workload is finite <almost surely>, these inequalities give stationary-tail brackets. If its logarithmic exponential tail rate exists, adding the bounded shift $C\delta$ does not change that rate.
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