## 4 Analytical Model
### Energy Efficiency
Let \(s\) be the number of active STAs, being \(s=1\) (apart from the AP). Being \(\rho_{r}\) the power consumed in the reception state, \(E_{r}^{T_{sl}>0}\) is given by  
\[E_{r}^{T_{sl}>0}=\left(\left(T_{RTS}+T_{CTS}\right)n+\alpha\left(2T_{DATA}+T_{ ACK}\right)s\right)\rho_{r}. \tag{8}\]  
To calculate the idle energy consumption \(E_{i}^{T_{sl}>0}\), it is assumed that \(n\) STAs and the AP consume energy to listen to the channel during a DIFS period. Both the source and destination STAs also consume energy to listen to the channel for the duration of \(1+2\alpha\) SIFS periods and \(1+2\alpha\) propagation delays. In addition, the \(n\!-\!s\) listening STAs consume energy to listen to the channel during the propagation delay following the RTS transmission. Thus, \(E_{i}^{T_{sl}>0}\) is derived as  
\[E_{i}^{T_{sl}>0}=\left(\left(T_{DIFS}+T_{SIFS}+2\delta\right)\left(n+1\right)+ \left(\left(1+2\alpha\right)T_{SIFS}+3\alpha\delta\right)\left(s+1\right) \right)\rho_{i}. \tag{9}\]  
In order to determine the switch energy consumption \(E_{sw}^{T_{sl}>0}\), it is considered that the \(n\!-\!s\) listening STAs consume energy during the transitions from idle to sleep \(T_{i\to sl}\) and from sleep to idle \(T_{sl\to i}\). Being \(\rho_{sl\to i}\) and \(\rho_{i\to sl}\) the power consumption for switching from sleep to idle and from idle to sleep, respectively, \(E_{sw}^{T_{sl}>0}\) is computed as  
\[E_{sw}^{T_{sl}>0}=\left(T_{i\to sl}\rho_{i\to sl}+T_{sl\to i}\rho_{sl\to i} \right)\left(n\!-\!s\right). \tag{10}\]  
To derive the sleep energy consumption \(E_{sl}^{T_{sl}>0}\), it is assumed that the \(n\!-\!s\) listening STAs consume energy to sleep during the microsleep period \(T_{sl}\). Hence, \(E_{sl}^{T_{sl}>0}\) is calculated as  
\[E_{sl}^{T_{sl}>0}=T_{sl}\left(n\!-\!s\right)\rho_{sl}, \tag{11}\]  
where \(\rho_{sl}\) is the power consumed in the sleep state. Based on the above, \(E_{s}^{T_{sl}>0}\) can be obtained as  
\[E_{\sigma}=\sigma\left(n+1\right)\rho_{i}, \tag{4}\]0  
Otherwise, if \(T_{sl}\) is equal to or shorter than zero (\(T_{sl}\leq 0\)), the listening STAs cannot sleep and remain awake consuming energy for listening to the whole data exchange. Let \(E_{s}^{T_{sl}\leq 0}\) be the energy consumed by all the STAs and the AP in a successful transmission when \(T_{sl}\leq 0\). The various energy consumption components of \(E_{s}^{T_{sl}\leq 0}\) for the TXOP PSM-based mechanism are computed as follows. The transmission energy consumption \(E_{t}^{T_{sl}\leq 0}\) is given by (7), since both the source and destination STAs consume energy to transmit their respective frames. To compute the reception energy consumption \(E_{r}^{T_{sl}\leq 0}\), it is considered that the source STA consumes energy to receive the CTS and \(\alpha\) ACK transmissions from the destination STA. In addition, the destination STA consumes energy to receive the RTS and \(\alpha\) data transmissions from the source STA. The \(n\!-\!1\) listening STAs consume energy to overhear the RTS, CTS, \(\alpha\) data, and \(\alpha\) ACK transmissions. This yields  
\[E_{\sigma}=\sigma\left(n+1\right)\rho_{i}, \tag{4}\]1  
To calculate the idle energy consumption \(E_{i}^{T_{sl}\leq 0}\), it is assumed that \(n\) STAs and the AP consume energy for being idle during a DIFS period, \(1+2\alpha\) SIFS periods, and \(2\left(1+\alpha\right)\) propagation delays interleaving the transmissions. Thus, \(E_{i}^{T_{sl}\leq 0}\) is expressed as  
\[E_{\sigma}=\sigma\left(n+1\right)\rho_{i}, \tag{4}\]2The listening STAs consume no energy for switching between idle and sleep states and for sleeping, since they remain awake during the whole data transfer. Therefore, \(E_{s}^{T_{sl}\leq 0}\) can be obtained as  
\[E_{\sigma}=\sigma\left(n+1\right)\rho_{i}, \tag{4}\]3  
Using \(T_{c}^{\prime}\), \(E_{c}^{\prime}\) can be written as  
\[E_{\sigma}=\sigma\left(n+1\right)\rho_{i}, \tag{4}\]4  
where \(n\) STAs and the AP consume energy for listening to the channel during an empty slot time \(\sigma\) following the end of a collision.