respect to the battery lifetime. Using the battery lifetime definition in , one may define the optimization problem for deriving the optimized operating point of type $i$ IoT devices as follows:

$$\begin{aligned}
\operatorname*{maximize}_{n_i, P_i } & \hspace{3mm}\mathbb L(i); \label{op2}\\
&\text{s.t.:} ~\text{P}_{\text o}(i)\le \text{P}_{\text o}^{\text{req}}(i), n_i\le n_{max}, P_i\le P_{\max}, \nonumber\end{aligned}$$

where $\text{P}_{\text o}^{\text{req}}(i)$ is the maximum tolerated outage probability for type $i$ IoT devices. The reliability constraint in could be rewritten as the minimum required success probability in communications as follows: $$\label{con}1-\sqrt[n_iB_i]{\text{P}_{\text o}^{\text{req}}}(i)\le \text{P}_{\text s}(i).$$ Furthermore, by using the $\text{P}_{\text s}(i)$ expression in theorem \[t3\], we have: $$\begin{aligned}
\text{P}_{\text s}(i)&= \int\nolimits_{0}^{\infty} X_0 \exp(\text{-}X_5r^{2})2rdr\nonumber\\
&=\frac{0.5\sqrt{\pi}{\lambda_{\text a}\pi}\exp\big(-{\hat\upsilon_{i,2}} \big)}{\sum_{k}\lambda_{k}\hat \upsilon_{k,2} 
  (\frac{P_k\gamma_{\text{th}}}{P_i\Omega   })^{0.5} \frac{\pi^{2}}{2} \text{csc}(\frac{\pi}{2})\text{+}\lambda_{\text a}\pi\text{+}\frac{ \mathcal N\gamma_{\text{th}}}{\Omega  P_i \alpha}},\label{den}\end{aligned}$$ in which, $\ell_{\max}=1$, $\delta=2$, and $\mathcal Q\approx1$ have been assumed for brevity of expressions. Also, $X_5$ is an auxiliary variable equal to the denominator of . The expression in could be rewritten as: $$\text{P}_{\text{s}}(i)=\frac{D_0}{\frac{1}{\sqrt{P_i}}D_1+\lambda_{\text a}\pi+\frac{ \mathcal N\gamma_{\text{th}}}{P_i\Omega   \alpha}},\label{rpi}$$ where the auxiliary variables $D_0$ and $D_1$ are defined as: $$\begin{aligned}
D_0&=0.5\sqrt{\pi}{\lambda_{\text a}\pi}\exp\big(-{\hat\upsilon_{i,2}} \big),\nonumber\\
 D_1&=\sum\nolimits_{k}\lambda_{k}\hat \upsilon_{k,2} 
  (\frac{P_k\gamma_{\text{th}}}{\Omega   })^{0.5} \frac{\pi^{2}}{2} \text{csc}(\frac{\pi}{2}).\nonumber
  \end{aligned}$$ Satisfying with equality, we have: $$\sqrt[n_iB_i]{\text{P}_{\text o}^{\text{req}}}(i)=1- \frac{D_0}{\frac{1}{\sqrt{P_i}}D_1+\lambda_{\text a}\pi+\frac{ \mathcal N\gamma_{\text{th}}}{P_i\Omega   \alpha}}.$$ By simplifying the expression, $n_i$ is derived as a function of $B_i$ as follows: 