system with a first in-first out policy, assuming it exists, has a characteristic decay rate, i.e., the decay rate of the tail probability of the queue [@chang1994effective]. When the queueing capacity is infinite, we obtain a simple exponential expression for the queue overflow probability (i.e., the probability that the queue is greater than a threshold), which is a function of the desired threshold and the characteristic decay rate. In practical systems, when the queueing capacity is large, this exponential expression approximates the queue overflow probability very closely. Likewise, for the EH transmitter model introduced in Section \[sec:system\_model\], we assume that the battery size is infinite, i.e., $e_{\max}=\infty$, and define the decay rate of the tail distribution of the energy in the battery as follows.

\[def:definition\_decay\_rate\] Given a stationary and ergodic energy arrival process, $u(i)$, and a stationary and ergodic energy demand process, $p(i)$, under the stability condition[^6], i.e., $\mathbb{E}_{u}\left[u(i)\right]<\mathbb{E}_{p}\left[p(i)\right]$, where $\mathbb{E}\left[\cdot\right]$ is the expected value operator, the *energy decay rate* of the battery is defined as $$\label{decay_rate}
\mu\triangleq-\lim_{e_{\text{th}}\to\infty}\frac{\ln\Pr\{e\geq e_{\text{th}}\}}{e_{\text{th}}},$$ where $e_{\text{th}}$ is the desired energy level, and random variable $e$ corresponds to the steady-state distribution of the energy level in the battery.

The definition of $\mu$ in (\[decay\_rate\]) suggests an 