## 5 Modelling the Transmission Time  
As previously highlighted, the network load imposed by a message stream \(S_{i}\) is constrained by the CSMA-CA parameters, such as channel access time and transmission time of a data frame. Due to the probabilistic behaviour of the CSMA-CA protocol, we consider the use of a set of communication models proposed by [PERSON] et al. [34] to estimate the frame transmission capacity within a superframe duration.  
Within the active duration of a superframe, the transmission time (\(T_{TXD}\)) for a single data frame can be modelled as follows:  
\[T_{TXD}=T_{BOT}+T_{PACKET}+T_{radio\_transition}+T_{ACK} \tag{13}\]  
where \(T_{BOT}\) is the total backoff time, \(T_{PACKET}\) is the packet transmission time, that is given by \(\frac{L_{PACKET}}{D}\) (where \(L_{PACKET}\) corresponds to packet frame length and \(D\) is the radio data rate). \(T_{radio\_transition}\) corresponds to time duration that the radio takes to switch between different operating modes, for example, from sleep to receive mode and from receive to transmit mode. \(T_{ACK}\) corresponds to the acknowledgement transmission time, that is given by \(\frac{L_{ACK}}{D}\) (where \(L_{ACK}\) corresponds to the ack frame length). Figure 8 illustrates the basic scheme for transmission of a single data frame. Note that the backoff period is aligned with the beacon interval and nodes must perform the backoff algorithm to transmit a data frame.  
Figure 7: Shortest Beacon interval.  
According to the CSMA-CA algorithm, a node needs to listen two CCAs before transmitting a packet (_CWinit_ parameter default is 2). For this reason, [PERSON] et al. [34] model the probability (\(P_{c}\)) of a node to perform two consecutive CCAs as follows:  
\[P_{c}=(1-q)^{2\times(N_{nodes}-1)}, \tag{14}\]  
where \(q\) corresponds to the probability that a node transmits a single message (with its ACK) during the CAP, which can be modelled as [34]:  
\[q=\frac{L_{PACKET}}{SD\times R} \tag{15}\]  
The CSMA-CA algorithm also defines a maximum number of backoffs (\(b\)), which correspond to the number of attempts that the backoff algorithm is repeated in case of unsuccessfully CCA evaluations. This value is defined by _macMaxBackoffPeriod_ MAC parameter and its default value is 4. Hence, the probability (\(P_{s}\)) of a node to perform a CCA with the maximum backoff number (\(b\)) can be modelled as:  
\[P_{s}=\sum_{NB=1}^{b}P_{c}\times(1-P_{c})^{NB-1} \tag{16}\]  
Then, the average backoff number (\(r\)) for each message can be modelled as [34]:  
\[r=(1-P_{s})\times b+\sum_{NB=1}^{b}NB\times P_{c}\times(1-P_{c})^{NB-1} \tag{17}\]  
Also, the average backoff time (\(T_{BO}\)) for each message can be modelled as function of BE [34]:  
\[T_{BO}(BE)=\frac{2^{BE}-1}{2}\times T_{BOL}, \tag{18}\]  
where \(T_{BOL}\) corresponds to the backoff period length and its value is defined by _aUnitBackoffPeriod_ MAC parameter (default value is 20 _symbols_).  
Therefore, the total backoff time can be modelled based on Equations (17) and (18) and considering that, for each backoff period, in average \(\frac{3}{2}\) CCA assessments are performed [34]. Thus, the total backoff time (\(T_{BOT}\)) is given by:  
\[T_{BOT}=\frac{3}{2}r(T_{CCA})+\sum_{NB=0}^{r-1}T_{BO}(min(macMinBE+NB,macMax BE)), \tag{19}\]  
where \(T_{CCA}\) corresponds to the CCA listening time.  
Within the context of this work, this set of analytical models is used to predict the number of messages transferred during a minimum superframe duration.  
Figure 8: Transmission time duration of a data frame.