}{4\sigma^2\sqrt{\xi}}  )\sin(\frac{||{\bf z}||^2 }{4\sigma^2\sqrt{\xi}}  )-\nonumber\\
&\hspace{1.7cm}\text{si}(\frac{||{\bf z}||^2 }{4\sigma^2\sqrt{\xi}}  )\cos( \frac{||{\bf z}||^2 }{4\sigma^2\sqrt{\xi}} )\bigg],\nonumber\end{aligned}$$ where $\text{si}(\cdot)$ and $\text{ci}(\cdot)$ are well-known sine and cosine integrals, as follows: $$\text{si}(x)=-\int\nolimits_{x}^{\infty}\frac{\text{sin} (t)}{t}dt,\hspace{2mm} \text{ci}(x)=-\int\nolimits_{x}^{\infty}\frac{\text{cos}(t)}{t}dt.$$

The proof is given in corollary 3.4 of the extended version [@opd].

\[r1\] Analysis of $\text{H}\big({\bf z},\text{f}^*({\bf x}), \xi)$ shows that it can be well approximated by $1$ for $\frac{\sqrt\xi||{\bf z}||^2   }{4  \sigma^2  }\gg1$. For theorem \[t1\] in which $\xi=Q_j\gamma_{\text {th}}/{\Omega}$, $ \text{H}\big({\bf z},\text{f}^*({\bf x}), \xi)\approx 0$ for $j=1$ because $Q_1=\mathcal Q\approx 0$; and $\text{H}\big({\bf z},\text{f}^*({\bf x}), \xi)\approx 1$ for $j=2$ when $z\gg z_0 \buildrel \Delta \over =\frac{2\sigma \sqrt[4]\Omega}{\sqrt[4]{\gamma_{\text{th}}}}$ because $Q_2=1$.

From theorem \[t1\], one sees that probability of success, $\text{p}_{\text s}(i,{\bf z})$, is a function of $||\bf z||$ rather than phase of $\bf z$. Then, hereafter we use $\text{p}(i,z)$ to denote probability of success for communication distance of $z$.

Until now, we have derived the probability of success for a given communication distance to an AP. In the following, we investigate success probability where multiple APs might be able to decode a packet, i.e. the coverage areas of neighboring APs are overlapping. Regarding the fact that theorem \[t1\] provides probability of success as a function of communication distance, given the distribution 