## 4 Analytical Model
### Energy Efficiency  
The energy efficiency under network saturation \(\eta\), hereafter referred to as saturation energy efficiency, is defined herein as the ratio of the average number of payload bits successfully transmitted in a slot time to the average amount of energy consumed in a slot time. The saturation energy efficiency is derived following a similar rationale to that used in the derivation of the saturation throughput in [21; 24]. Using (1) and the derivation above to compute \(S\), \(\eta\) can be expressed as  
\[\eta=\frac{\beta P_{tr}P_{s}E[P]^{\prime}}{\left(1-P_{tr}\right)E_{\sigma}+P_{ tr}P_{s}E_{s}^{\prime}+P_{tr}\left(1-P_{s}\right)E_{c}^{\prime}}, \tag{3}\]  
where \(E_{\sigma}\) is the energy consumed during an empty slot time. In addition, \(E_{s}^{\prime}\) and \(E_{c}^{\prime}\) are the average energy consumed during a successful transmission and a collision, respectively. Since \(n\) STAs and the AP consume energy for being idle during an empty slot time \(\sigma\), \(E_{\sigma}\) is given by  
\[E_{\sigma}=\sigma\left(n+1\right)\rho_{i}, \tag{4}\]  
where \(\rho_{i}\) is the power consumption corresponding to the idle state. Using \(T_{s}^{\prime}\), \(E_{s}^{\prime}\) can be derived as  
\[E_{s}^{{}^{\prime}}=\frac{E_{s}}{1-B_{0}}+\sigma\left(n+1\right)\rho_{i}, \tag{5}\]  
where \(E_{s}\) is given by the sum of various energy consumption components, being \(E_{t}\), \(E_{r}\), \(E_{i}\), \(E_{sw}\), and \(E_{sl}\) the energy consumed in the transmission, reception, idle, switch, and sleep states, respectively. The expression of \(E_{s}\) depends on whether the listening STAs are able to sleep during a successful transmission or not. To determine this, the microsleep period during a successful transmission \(T_{sl}\) is computed as the time difference between the total transmission duration available for sleeping and the transition time between awake and sleep states. More specifically, when the TXOP PSM-based mechanism is employed this duration comprises all that occurs following the RTS transmission plus the propagation delay. As shown in Figure 1, this duration includes the CTS transmission, \(\alpha\) data transmissions, \(\alpha\) ACK transmissions, \(1+2\alpha\) SIFS periods, and \(1+2\alpha\) propagation delays. Being \(T_{i\to sl}\) and \(T_{sl\to i}\) the time required to switch from idle (i.e., awake) to sleep and from sleep to idle, respectively, \(T_{sl}\) is obtained as  
\[T_{sl}=T_{CTS}+\alpha\left(2T_{DATA}+T_{ACK}\right)+\left(1+2\alpha\right) \left(T_{SIFS}+\delta\right)-\left(T_{i\to sl}+T_{sl\to i}\right). \tag{6}\]  
If \(T_{sl}\) is longer than zero (\(T_{sl}>0\)), the listening STAs are able to sleep during a successful transmission. For the TXOP PSM-based mechanism, this scenario is illustrated in Figure 1.  
Let \(E_{s}^{T_{sl}>0}\) be the energy consumed by all the STAs and the AP in a successful transmission when \(T_{sl}>0\). The various energy consumption components of \(E_{s}^{T_{sl}>0}\) are obtained as follows. In order to derive the transmission energy consumption \(E_{t}^{T_{sl}>0}\) it is assumed that the source STA consumes energy to perform the RTS and \(\alpha\) data transmissions to the destination STA. In addition, the destination STA consumes energy to perform the CTS and \(\alpha\) ACK transmissions to the source STA. Being \(\rho_{t}\) the power consumption corresponding to the transmission state, \(E_{t}^{T_{sl}>0}\) is expressed as  
\[E_{t}^{T_{sl}>0}=\left(T_{RTS}+T_{CTS}+\alpha\left(T_{DATA}+T_{ACK}\right) \right)\rho_{t}. \tag{7}\]  
To compute the reception energy consumption \(E_{t}^{T_{sl}>0}\), it is considered that the source STA consumes energy to receive the CTS and \(\alpha\) ACK transmissions from the destination STA. Then, the destination STA consumes energy to receive the RTS and \(\alpha\) data transmissions from the source STA. In addition, the rest of STAs (i.e., \(n\!-\!1\)) consume energy to overhear the RTS and CTS transmissions only, as they can sleep for the remainder of the data transmission.