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

*Proof*: See Appendix \[app:proposition\_mu\_zero\_and\_infinity\_2\].

Proposition \[pro:mu\_zero\_and\_infinity\] suggests that when the constant energy demand is less than or equal to the average value of the energy arrival process, energy overflows are inevitable. It also shows that, as long as the energy demand is greater than or equal to the maximum possible energy arrival in any time frame, there is no energy overflow in the battery. Likewise, according to Proposition \[pro:mu\_zero\_and\_infinity\_2\], when the value of a constant energy arrival process is greater than or equal to the average value of the energy demand process, the energy overflows are imminent, and when the constant energy arrival is less than or equal to the minimum energy demand, there is no energy overflow in the battery. Although the statements in Propositions \[pro:mu\_zero\_and\_infinity\] and \[pro:mu\_zero\_and\_infinity\_2\] are rather intuitive, they are confirmed by the characterization in (\[eq:arrival\_demand\_balance\]), where we express the probability that the energy level in the battery is above a defined threshold as an exponential function of the energy decay rate and the defined threshold. In other 