Grid-sampled compound Poisson workload bound ($(W-C\delta {)}^{+}\le W_{\delta }\le W$)

For nondecreasing compound Poisson input $U$, let $W={\sup }_{t\ge 0}(U(t)-Ct)$ and $W_{\delta }={\sup }_{k\ge 0}(U(k\delta )-Ck\delta )$. 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 $W\le W_{\delta }+C\delta$. 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.