[@nL]: $$\mathbb L(i)= \frac{E_{0}}{{E_{\text{st}}+\hat \beta_i E_\text{c}+   \hat\beta_in_i (\eta P_{i}+ P_{\text c}) \tau_i}}T_i,\label{lif}$$ where $E_\text{c}$ represents the average energy consumption in listening after each trial for ACK reception, and $\hat \beta_i$ represents the average number of trials and is derived as: $$\label{beta}\hat \beta_i=\sum\nolimits_{j=1}^{B_i}j\big[1\text{-}[1\text{-}\text{P}_{\text{s}}(i )]^{n_i}\big]\big[1\text{-}\text{P}_{\text s}(i )\big]^{n_i[j-1]},$$ where $\text{P}_{\text s}(i)$ have been derived in theorem \[t3\].

Optimized Operation Control
===========================

From the battery lifetime analysis in , one sees that battery lifetime of devices may decrease in $n_i$ and $P_i$ because of the potential increase in the energy consumption per reporting period. Furthermore, when reliability of communication is lower than a threshold, increase in $n_i$ and $ {P}_i$ may decrease the need for listening to the channel for ACK arrival and retransmissions, and hence, increasing $n_i$ and $ {P}_i$ may increase the battery lifetime. Taking this into account, one sees there should be an operation point beyond which, increase in $ {P}_i$ and/or $n_i$ offers a tradeoff between reliability and lifetime, and before it, increase in $ {P}_i$ and/or $n_i$ increases both reliability and durability of communications. This observation will be evaluated using simulation results in the next section. Here, we aim at finding the optimized operation point of the network with 