limits, and are differentiable for $\mu\in\mathbb{R}$, when the moment generating functions $\mathbb{E}_{u}\left[\exp\left(\mu U(t)\right)\right]$ and $\mathbb{E}_{p}\left[\exp\left(\mu P(t)\right)\right]$ exist for $t>0$, respectively. $U(t)$ is the cumulative harvested energy as defined before, and $P(t)\triangleq\sum_{i=1}^{t}p(i)$. Noting that (\[eq:arrival\_demand\_balance\]) holds for any stationary and ergodic energy arrival and demand processes with finite mean and variance, we provide the following two cases as examples in order to gain more insights.

### Constant energy demand

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

\[pro:mu\_zero\_and\_infinity\] When the energy decay 