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 $W\le W_{\delta }+C\delta$. This supplies explicit stationary-tail brackets and shows that time discretization preserves the workload's exponential tail rate.