approximation for the energy overflow probability in the steady-state given a large battery size[^7], i.e., $\Pr\{e\geq e_{\text{th}}\}\approx \exp\big(-\mu e_{\text{th}}\big)$. In particular, the energy decay rate characterizes the exponential decay rate of the tail distribution of the energy level in the battery, and the exponential approximation holds when we have stationary and ergodic energy arrival and demand processes [@chang1995effective_conf]. With a large battery size and a target overflow probability, we can use the energy decay rate, $\mu$, as a tool to identify the probability that the energy level in the battery is above a defined threshold. The energy decay rate primarily depends on the energy arrival and demand processes. Large $\mu$ refers to an energy demand process that consumes the stored energy rapidly, while smaller $\mu$ means a moderate energy demand process.

For an infinite-size battery the instantaneous energy level is given by $e(i)=[e(i-1)+u(i)-p(i)]^{+}$. Moreover, because energy in our model is used for data transmission only, we further consider a work-conserving energy demand process[^8] in the following analysis. Hence, noting that $u(i)$ and $p(i)$ are independent of each other, we have a unique $\mu^{\star}$ that satisfies [@chang_book Remark 9.1.2] $$\begin{aligned}
\label{eq:arrival_demand_balance}
\Lambda_{u}(\mu^{\star})+\Lambda_{p}(-\mu^{\star})=0,\end{aligned}$$ where $\Lambda_{u}(\mu)\triangleq\lim_{t\to\infty}\frac{1}{t}\ln\mathbb{E}_{u}\left[\exp\left(\mu U(t)\right)\right]$ and $\Lambda_{p}(\mu)\triangleq\lim_{t\to\infty}\frac{1}{t}\ln\mathbb{E}_{p}\left[\exp\left(\mu P(t)\right)\right]$ are the Gärtner-Ellis 