result provides a link between the buffer and delay violation probabilities. In the above formulation, $c$ is some positive constant, $q_{\max} = a d_{\max}$, and $a$ is the source arrival rate.

Now, the effective capacity for a given QoS exponent $\theta$ is obtained from $$\begin{gathered}
-\lim_{t\rightarrow\infty}\frac{1}{\theta
t}\log_e{\mathbb{E}\{e^{-\theta S[t]}\}} \stackrel{\text{def}}{=}
-\frac{\Lambda(-\theta)}{\theta}\end{gathered}$$ where $S[t] = \sum_{k=1}^{t}R[k]$ is the time-accumulated service process and $\{R[k], k=1,2,\ldots\}$ denote the discrete-time, stationary and ergodic stochastic service process. Note that in the model we consider, $R[k] = rT \text{ or } 0$ depending on the channel state being ON or OFF, respectively. In [@chang] and [@cschang Section 7.2, Example 7.2.7], it is shown that for such an ON-OFF model, we have $$\label{eq:bookresult}
\frac{\Lambda(\theta)}{\theta}=\frac{1}{\theta}\log_e\Big(\frac{1}{2}\Big(p_{11}+p_{22}e^{\theta
Tr}+\sqrt{(p_{11}+p_{22}e^{\theta Tr})^2+4(p_{11}+p_{22}-1)e^{\theta
Tr}} \Big)\Big).$$ Using the formulation in (\[eq:bookresult\]) and noting that $p_{11}+p_{22}=1$ in our model, we express the effective capacity normalized by the frame duration $T$ and bandwidth $B$, or equivalently spectral efficiency in bits/s/Hz, for a given statistical QoS constraint $\theta$, as $$\begin{aligned}
\hspace{-.2cm}{{\sf{R}}}_E({{\text{\footnotesize{SNR}}}},
\theta)=\frac{1}{TB}\max_{r\geq0}\Big\{-\frac{\Lambda(-\theta)}{\theta}\Big\}
&=\max_{r\geq0}\Big\{-\frac{1}{\theta
TB}\log_e\big(p_{11}+p_{22}e^{-\theta
Tr}\big)\Big\}\\
&=\max_{r\geq0}\Big\{-\frac{1}{\theta
TB}\log_e\big(1-P\{z>\alpha\}(1-e^{-\theta
Tr})\big)\Big\}\label{eq:specec}\\
&=-\frac{1}{\theta TB}\log_e\Big(1-P\{z>{\alpha_{\text{opt}}}\}\big(1-e^{-\theta
T{r_{\text{opt}}}}\big)\Big)\,\, \text{bits/s/Hz}\label{eq:spececopt}\end{aligned}$$ where ${r_{\text{opt}}}$ is the maximum fixed transmission rate that solves (\[eq:specec\]) and ${\alpha_{\text{opt}}}= (2^{\frac{{r_{\text{opt}}}}{B}}-1)/{{\text{\footnotesize{SNR}}}}$. Note that both ${\alpha_{\text{opt}}}$ and ${r_{\text{opt}}}$ are functions of ${{\text{\footnotesize{SNR}}}}$ and $\theta$.

The 