$\boldsymbol{\pi}=M\boldsymbol{\pi}$ and $\sum_{i=0}^{\infty}\pi_{i}=1$, where $[\cdot]^{T}$ is the transpose operator. We can notice that the steady-state probability $\pi_{0}$ gives us the energy outage probability, which is $\pi_{0}=\frac{1}{1+\sum_{m=1}^{\infty}\prod_{i=1}^{m}q_{i}}$. Having $\pi_{0}$ and $M$, we can easily provide the other steady-state probabilities as follows: $\pi_{k}=\pi_{k-1}q_{k}=\pi_{0}\prod_{i=1}^{k}q_{i}=\frac{\prod_{i=1}^{k}q_{i}}{1+\sum_{m=1}^{\infty}\prod_{i=1}^{m}q_{i}}$ for $k\in\{1,2,\cdots\}$. Recalling that both $u(i)$ and $p(i)$ are stationary and ergodic, and that the stability condition given in Definition \[def:definition\_decay\_rate\] is satisfied, we can show that $\pi_{0}\geq\pi_{1}\geq\pi_{2}\geq\cdots\geq\pi_{m}\geq\cdots$. Furthermore, using (\[eq:state\_transition\_probability\_2\_add\]) and assuming that consecutive energy arrivals and consecutive energy demands are independent and identically distributed (i.i.d.), we can employ an inductive method and realize that $q_{1}\leq q_{2}\leq q_{3}\leq\cdots\leq q_{m}\leq\cdots$, and find an upper bound on the outage probability as follows: $$\begin{aligned}
\pi_{0}&=\frac{1}{1+\sum_{m=1}^{\alpha-1}\prod_{i=1}^{m}q_{i}+\sum_{m=\alpha}^{\infty}\prod_{i=1}^{m}q_{i}}=\frac{1}{1+\sum_{m=1}^{\alpha-1}\prod_{i=1}^{m}q_{i}+\prod_{i=1}^{\alpha}q_{i}\left[1+\sum_{m=\alpha+1}^{\infty}\prod_{i=\alpha+1}^{m}q_{i}\right]}\nonumber\\
&\leq\frac{1}{1+\sum_{m=1}^{\alpha-1}\prod_{i=1}^{m}q_{i}+\frac{\prod_{i=1}^{\alpha}q_{i}}{1-q_{\alpha+1}}}\label{eq:upper_bound_pi_0}\end{aligned}$$ for $\alpha\in\{1,2,\cdots\}$. The upper bound in (\[eq:upper\_bound\_pi\_0\]) comes from the fact that $\sum_{m=\alpha+1}^{\infty}\prod_{i=\alpha+1}^{m}q_{i}\geq\sum_{m=\alpha+1}^{\infty}\prod_{i=\alpha+1}^{m}q_{\alpha+1}$. With increasing $\alpha$, the bound becomes tighter. Performing the sum in the denominator in (\[eq:upper\_bound\_pi\_0\]) up to a finite number provides us the energy outage probability closely.

Transmission Throughput {#sec:transmission_throughput}
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In this section, we concentrate on the performance levels of the aforementioned EH wireless communication system. Therefore, we invoke the average data service rate in the wireless channel and the effective capacity in the data link layer as performance measures. The 