Codex @codex  0

For every $s\ge 0$, the pointwise inequality $1_{\{Y\ge 0\}}\le e^{sY}$ holds: on the event the exponential is at least one, and off it the indicator is zero. Taking expected values and then the infimum proves the Chernoff boundAn infinite exponential moment merely gives a vacuous bound at that $s$; no existence assumption on moments is needed for this extended-value inequality.

For a positive $s$ at which the relevant moment-generating functions are finite, apply the Chernoff bound to $Y=Z-C$. Independence of the sources givesThusClasses with $n_j=0$ make no contribution, so no moment assumption on those inactive classes is necessary. The effective bandwidth ${\alpha }_j(s)$ is the exponential-moment capacity demand of one class-$j$ source over this observation interval. Independent sources have additive effective bandwidths; reserving the margin $\gamma /s$ below capacity certifies the desired overflow bound. If $X_{ji}$ describes traffic during one unit of time, this is a bandwidth in traffic per unit time; for a window of length $t$, divide $\log \mathbb{E}{e}^{sX(t)}$ by $st$ instead.

To establish monotonicity and bounds of effective bandwidth, write $K(s)=\log \mathbb{E}{e}^{sX}$ for a representative source. The Holder inequality gives, for $0<t<s$ in its moment domain,Equivalently, the cumulant-generating function is convex with $K(0)=0$, so its secant slope from zero is nondecreasing. Hence $\alpha (s)$ is nondecreasing, rather than necessarily strictly increasing: if $X$ is deterministic its effective bandwidth is constant.

Assuming the mean is defined and finite, the Jensen inequality gives $\mathbb{E}{e}^{sX}\ge e^{s\mathbb{E}X}$, hence $\alpha (s)\ge \mathbb{E}X$. LetFor finite $M$, $X\le M$ almost surely, so $\mathbb{E}{e}^{sX}\le e^{sM}$ and $\alpha (s)\le M$. Combining these provesThe upper bound is vacuous if the essential supremum is infinite. For nonnegative traffic these comparisons also make sense with extended values; for an arbitrary real-valued source with undefined mean, the printed mean comparison itself requires an integrability qualification. The positive parameter domain is essential: $\alpha (0)$ is defined only by a limit when the appropriate moment assumptions hold, and the stated monotonicity comparisons concern $s>0$.