## 4 Analytical Model
### Throughput  
The saturation throughput \(S\) is defined in [21] as the ratio of the average amount of payload bits successfully transmitted in a slot time to the average length of a slot time. \(S\) can be expressed as [21; 24]:  
\[S=\frac{P_{tr}P_{s}E[P]^{\prime}}{\left(1-P_{tr}\right)\sigma+P_{tr}P_{s}T_{s} ^{\prime}+P_{tr}\left(1-P_{s}\right)T_{c}^{\prime}}. \tag{1}\]  
With probability \(P_{tr}=1-\left(1-\tau\right)^{n+1}\) there is at least one transmission in a slot time, given \(n\) STAs and the AP contending on the channel. Similarly, with probability \(P_{s}=\left(n+1\right)\tau\left(1-\tau\right)^{n}/P_{tr}\) an attempt to transmit a frame is successful. In addition, \(E[P]^{\prime}=E[P]/1-B_{0}\) is the average amount of payload bits transmitted in a successful transmission, where \(B_{0}=1/W\) is the probability that a successfully transmitting STA can get access to the channel in the first slot time immediately after a DIFS period and \(W=CW_{min}+1\). The probability that a slot time is empty is given by \(1-P_{tr}\), and \(\sigma\) is the duration of an empty slot time. \(P_{tr}P_{s}\), is the probability that a successful transmission occurs in a slot time, being \(T_{s}^{\prime}=\left(T_{s}/1-B_{0}\right)+\sigma\) the average duration of a successful transmission. With probability \(P_{tr}(1-P_{s})\), a collision occurs in a slot time, and \(T_{c}^{\prime}=T_{c}+\sigma\) is the average duration of a collision.  
It is worth noting that the throughput Expression (1) is obtained without specifying the access mechanism employed. In order to compute the throughput for a given DCF access mechanism it is necessary to determine the values of \(T_{s}\) and \(T_{c}\).  
\(T_{s}\) represents the duration of a successful transmission. For the TXOP PSM-based mechanism, this duration includes a DIFS period, an RTS transmission, a CTS transmission, \(\alpha\) data transmissions, \(\alpha\) ACK transmissions, \(1+2\alpha\) SIFS periods, and \(2(1+\alpha)\) propagation delays, as shown in Figure 1. Therefore, \(T_{s}\) is given by  
\[T_{s}=T_{RTS}+T_{CTS}+\alpha\left(T_{DATA}+T_{ACK}\right)+T_{ DIFS}+\left(1+2\alpha\right)T_{SIFS}+2\left(1+\alpha\right)\delta, \tag{2}\]  
where \(T_{DIFS}\) and \(T_{SIFS}\) are the duration of the DIFS and SIFS periods, respectively, and \(\delta\) is the propagation delay. \(T_{RTS}\), \(T_{CTS}\), \(T_{DATA}\), and \(T_{ACK}\) are the times required to transmit the RTS, CTS, data, and ACK frames, respectively. In addition, \(\alpha\) is the number of data frames that the source STA is allowed to transmit to the destination STA in a successful channel access attempt.  
\(T_{c}\) represents the duration of a collision. Either if DCF or the TXOP PSM-based mechanism is employed, this duration comprises the RTS transmission plus the propagation delay and an EIFS period, thus \(T_{c}=T_{RTS}+\delta+T_{EIFS}\).  
In order to account for burst transmissions, \(\beta\) is defined as the number of data frames transmitted in a successful channel access attempt. Hence, the numerator in (1) is rewritten as \(\beta P_{tr}P_{s}E[P]\)Since each successful channel access attempt using the TXOP PSM-based mechanism includes the transmission of \(\alpha\) data frames, \(\beta=\alpha\).