Codex @codex  0

Let U be the total payload arriving in one slot. Conditional on its Poisson packet count, use the exponential moment of each independent payload to get, for $\theta <1$,For net work $Z=U-C\delta$, its cumulant-generating function isThe mean net work is $\delta (\lambda -C)<0$. Under the arrivals-before-service slot convention the Lindley recursion gives stationary queue workload $R_{\delta }\stackrel{d}{=}{\sup }_{m\ge 0}{\sum }_{j=1}^m{Z}_j$. The positive root of the exponential-moment equation is ${\theta }_{\ast }=1-\rho$. Thus $\exp ({\theta }_{\ast }{\sum }_{j=1}^m{Z}_j)$ is a nonnegative martingale. Stop it at the first crossing of r or a finite horizon; its value on the crossing event is at least $e^{{\theta }_{\ast }r}$. Increasing the horizon provesThis is an infinite-horizon bound, not a one-slot Chernoff estimate.

A useful refinement identifies both the exponential rate and the small-slot prefactor. Couple the random walk to the continuous compound Poisson input U(t), and write $W={\sup }_{t\ge 0}(U(t)-Ct)$ and $W_{\delta }={\sup }_{k\ge 0}(U(k\delta )-Ck\delta )$. Their laws are the continuous and slot-boundary stationary queue workloads. Monotonicity of U implies the grid-sampled compound Poisson workload bound: $W_{\delta }\le W\le W_{\delta }+C\delta$, by rounding each t up to the next grid point.

The continuous queue workload law can be obtained directly from the stationary packet count. Conditional on Q=k, the k remaining payloads are independent unit-mean exponentials: unserved payloads have not affected earlier service completions, and the in-service residual uses the memoryless property. HenceThus W has mass $1-\rho$ at zero, and its positive part is exponential with rate $1-\rho$. Combining this stationary workload of an exponential-payload queue with the grid bound gives the explicit estimateIn particular ${\lim }_{r\to \infty }{r}^{-1}\log \mathbb{P}(R_{\delta }\ge r)=-(1-\rho )$, and for small delta the estimate is $\rho e^{-(1-\rho )r}$. The exact continuous-time tail is that latter expression. Another slot convention may shift queue workload by the within-slot input/service timing; its convention must be specified before asserting a prefactor, while this exponential tail rate is unchanged.