rate, $\mu$, goes to zero, the minimum constant energy demand goes to the average energy arrival level, $\mathbb{E}_{u}\left[u(i)\right]$. In particular,$$p^{\star}=\lim_{\mu\to0}\lim_{t\to\infty}\frac{1}{t\mu}\ln\mathbb{E}_{u}\left[\exp\left(\mu U(t)\right)\right]=\mathbb{E}_{u}\left[u(i)\right].$$Furthermore, when the energy decay rate, $\mu$, goes to infinity, the minimum constant energy demand goes to the maximum energy arrival level, $\max\{u(i)\}$. In particular, $$p^{\star}=\lim_{\mu\to\infty}\lim_{t\to\infty}\frac{1}{t\mu}\ln\mathbb{E}_{u}\left[\exp\left(\mu U(t)\right)\right]=\max\{u(i)\}.$$

*Proof*: See Appendix \[app:proposition\_mu\_zero\_and\_infinity\].

### Constant energy arrival

Given an energy demand process, let us now assume that the energy arrival at the battery is constant[^9] over time. Then, we can easily find the following relation for a given energy decay rate, $\mu$: $u^{\star}=-\lim_{t\to\infty}\frac{1}{t\mu}\ln\mathbb{E}_{p}\left[\exp\left(-\mu P(t)\right)\right]$, where $u^{\star}$ is the maximum constant energy arrival such that the steady-state energy overflow probability is $\Pr\{e\geq e_{\text{th}}\}\approx\exp\left(-\mu e_{\text{th}}\right)$ for given $e_{\text{th}}$. In other words, targeting an energy decay rate, $\mu$, we should have the constant energy arrival less than or equal to $u^{\star}$ such that we can keep the steady-state energy overflow probability less than or equal to $\exp\big(-\mu e_{\text{th}}\big)$. Moreover, it is important to note that while having the energy arrival below $u^{\star}$ decreases the energy overflow probability, it causes an increase in the energy outage probability.

\[pro:mu\_zero\_and\_infinity\_2\] When the energy decay rate, $\mu$, goes to zero, the maximum constant energy arrival approaches the average energy 