process of APs, the expected communication distance to the neighboring APs, and hence, probability of success in data transmission could be derived. In PPP deployment of APs with density $\lambda_{\text{a}}$, the PDF of distance from a random point to the $\ell$th nearest AP, denoted by $d_{\ell}$ is given by [@dis]: $$\text{P}_{d_{\ell}}(r)=\exp(-\lambda_{\text a}\pi r^2) {2(\lambda_{\text a}\pi r^2)^{\ell}}/[{r({\ell}-1)!}].$$ Then, one can derive the average probability of success in packet transmission from a random point for type-$i$ as: $$\text{P}_\text{s}(i)=  1-\prod\nolimits_{{\ell}=1}^{\ell_{\max}} \int\nolimits_{0}^{\infty} \big(1-\text{p}_\text{s}(i,r)\big) ~ \text{P}_{d_{\ell}}(r) dr.\label{cov}$$

\[t3\] For $\text{f}(x)$ given in , and $\text{g}({\bf z})=\alpha||{\bf z}||^{-4}$, we have: $$\text{P}_\text{s}(i)\approx  1-\prod\nolimits_{{\ell}=1}^{\ell_{\max}} \big[1-\frac{X_0}{\sqrt{{X_1}^{\ell-1}}} \exp(\frac{{X_2}^2}{4{X_1}^2} ) \mathcal G(X_3,\ell)\big],$$ $$\begin{aligned}
&\text{where } X_0=\frac{(\lambda_{\text a}\pi)^\ell}{(\ell-1)!}\exp\big(-{\hat\upsilon_{i,2}} \big), X_1=\frac{ \mathcal N\gamma_{\text{th}}}{\Omega  P_i \alpha},\nonumber\\ 
  & X_2=\sum\limits_{j,k} \lambda_k\hat \upsilon_{k,j} 
  (\frac{\gamma_{\text{th}}Q_j P_k}{\Omega  P_i})^{0.5} \frac{\pi^{2}}{2} \text{csc}(\frac{\pi}{2})+\lambda_{\text a}\pi, X_3=\frac{X_2}{2\sqrt{X_1}}.\nonumber
   \end{aligned}$$ Also, $\mathcal G(X_3,\ell)=\int\nolimits_{\frac{{X_2}^2}{2X_1}}^{\infty}  (z\text{-}X_3)^{(\ell-1)}\exp(-z^2)dz,$ and could be derived for any $\ell$ in the form of error function, e.g. for $\ell_{\max}=2$: $$\begin{aligned}
&\mathcal G(X_3,1)=-(\sqrt{\pi}(\text{erf}(X_3) - 1))/2,\nonumber\\
&\mathcal G(X_3,2)=\exp(-X_3^2)/2 + (X_3 \sqrt{\pi}(\text{erf}(X_3) - 1))/2.\nonumber\end{aligned}$$

The proof is given in theorem 3.5 of the extended version [@opd].

### Reliability of IoT Communication {#rels}

Now, we have the required tools to investigate reliability of IoT communications. Once a type-$i$ device has a packet to transmit, it transmits $n_k$ replicas of 