= P\{z > \alpha\}$. Similarly, the other transition probabilities become $$\begin{aligned}
p_{11}&=p_{21}=P\{z \le \alpha\}=\int_0^\alpha p_z(z)dz\\
p_{22}&=p_{12}=P\{z >  \alpha\}=\int_\alpha^\infty p_z(z)dz\end{aligned}$$ where $p_z$ is the probability density function of $z$. We finally note that $rT$ bits are successfully transmitted and received in the ON state, while the effective transmission rate in the OFF state is zero.

Preliminaries – Effective Capacity and Spectral Efficiency-Bit Energy Tradeoff
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In [@dapeng], Wu and Negi defined the effective capacity as the maximum constant arrival rate[^5] that a given service process can support in order to guarantee a statistical QoS requirement specified by the QoS exponent $\theta$. If we define $Q$ as the stationary queue length, then $\theta$ is the decay rate of the tail distribution of the queue length $Q$: $$\lim_{q \to \infty} \frac{\log P(Q \ge q)}{q} = -\theta.
$$ Therefore, for large $q_{\max}$, we have the following approximation for the buffer violation probability: $P(Q \ge q_{\max}) \approx
e^{-\theta q_{max}}$. Hence, while larger $\theta$ corresponds to more strict QoS constraints, smaller $\theta$ implies looser QoS guarantees. Moreover, if $D$ denotes the steady-state delay experienced in the buffer, then it is shown in [@liu-isit] that $P\{D \ge d_{\max}\} \le c \sqrt{P\{Q\ge q_{\max}\}}$ for constant arrival rates. This 