$${n_i}=\left \lceil {\log(\sqrt[B_i]{\text{P}_{\text o}^{\text{req}}})}\bigg/{\log(1- \frac{D_0}{\frac{1}{\sqrt{P_i}}D_1+\lambda_{\text a}\pi+\frac{ \mathcal N\gamma_{\text{th}}}{P_i\Omega   \alpha}})}\right \rceil.\label{rni}$$ Also, the constraint on $n_i$ is translated to a constraint on $P_i$ as: $$P_i\ge P_{\min} \buildrel \Delta \over =\big(\frac{-{D_1}\text{+}\sqrt{{D_1}^2\text{-}4\frac{\mathcal N\gamma_{\text{th}}}{\Omega \pi}(\lambda_{\text a}\pi\text{-}\frac{D_0}{1\text{-}\sqrt[n_{\max}B_i]{\text{P}_{\text o}^{\text{req}}}})}}{2(\lambda_{\text a}\pi\text{-}\frac{D_0}{1-\sqrt[n_{\max}B_i]{\text{P}_{\text o}^{\text {req}}}})}\bigg)^2.$$ Then, the optimization problem in reduces to a simple search over $ P_{\min} \le \mathcal{P}_i\le P_{\max} $ for minimization of : $$\label{ecprp}
 {\hat \beta_i E_\text{c}+   \hat \beta_in_i (\eta P_{i}+ P_{\text c}) \tau_i},$$ in which $n_i$ has been found as a function of $P_i$ in , $\hat \beta_i$ has been found as a function of $\text{P}_{\text s} (i)$ and $n_i$ in , and $\text{P}_{\text s} (i)$ has been found as a function of $P_i$ in . This operation control optimization problem is investigated numerically in the next section (Fig. \[oo\]).

  [*Parameters* ]{}                                   [*Value*]{}
  --------------------------------------------------- --------------------------------------------------------------------
  Service area                                        $20\times 20 \text{ Km}^2$
  Pathloss                                            $133+38.3\log (\frac{x}{1000})$
  Thermal noise power                                 $-174$ dBm/Hz
  Distribution of devices                             PCP$\big(\lambda_i\times$1e-6,$200$, Eq. with $\sigma$=100)
  Packet arrival of each device                       Poisson distributed with average reporting period ($T_i$) of 300 s
  Packet transmission time ($\tau_i$)                 100 ms
  Signal BW                                           10 KHz
  $E_0,P_{\text c} , E_{\text{st}}=0.5 E_{\text c}$   1000 J, 10 mW, 0.1 J
  $P_{\text r}$, $P_{\text a}$                        0.5 W, 1.5 W
  $\gamma_{\text{th}}$, $|\varpi|$, $\eta$            1,1,0.5
  