# Distributed Triggered Access for BSM Dissemination in 802.11 bd V2V Networks
## 3 Numerical Analysis
Once a TF is transmitted successfully, all nearby VSTAs within \(L_{pd}\) are supposed to transmit CTS frame simultaneously followed by the triggered access TB PPDU transmission of scheduled VSTAs. As long as the TF is not collided, the whole DTA transmission sequence will take place, so only the duration of the TF affects the vulnerable period. If we let \(P_{tr}\) be the probability that a VSTA successfully transmits a TF in an arbitrary time slot, we have  
\[ \begin{split} P_{tr}&=\tau\sum_{k=0}^{\infty}(1- \tau)^{k}P_{v}(k,N_{cd}-1)\left[\sum_{k=0}^{\infty}(1-\tau)^{k}P_{v}(k,N_{int}) \right]^{T_{val}^{ff}/T_{\omega}}\\ &=\tau e^{-N_{pd}}e^{-\tau(N_{cd}+N_{int}\cdot T_{val}^{ff}/T_{ \omega})-1},\end{split}  \tag{12}\]  
where \(T_{val}^{ff}\) denotes the vulnerable period of a TF that can be given by  
\[T_{val}^{ff}=2T_{tf} \tag{13}\]  
where \(T_{tf}\) is the duration of the PPDU containing the TF. Then, the probability that the tagged STA successfully participate in a triggered access transmission, \(P_{ta}\), can be given by  
\[P_{ta}=\frac{M-1}{N_{tr}}\binom{N_{tr}}{1}P_{tr} \tag{14}\]  
Now, we calculate the average virtual slot, \(T_{\omega}\). Based on the procedure of the proposed scheme, there are three types of possible virtual slots: (1) empty slots where all VSTAs are in backoff procedure or idle, (2) DTA successful slot and (3) MU-RTS collision slot with the duration \(T_{\omega}^{(1)}\), \(T_{\omega}^{(2)}\) and \(T_{\omega}^{(3)}\), respectively, and we have  
\[ \begin{cases}T_{\omega}^{(1)}=\sigma\\ T_{\omega}^{(2)}=T_{data}=T_{tf}+T_{sifs}+T_{cts}+T_{sifs}+T_{tb}+T_{aifs}\\ T_{\omega}^{(3)}=T_{tf}+T_{aifs}\end{cases}  \tag{15}\]  
where \(T_{aifs}\), \(T_{sifs}\), \(T_{cts}\) and \(T_{tb}\) denotes the duration of AIFS, SIFS, the PPDU carrying a CTS frame and the TB PPDU containing multi-vehicle BSM frames, respectively. Letting \(P_{\omega}^{(1)}\), \(P_{\omega}^{(2)}\) and \(P_{\omega}^{(3)}\) be the probabilities corresponding to the above slots, respectively, we have  
\[ \begin{cases}P_{\omega}^{(1)}=&1-\sum_{k=0}^{\infty}(1-\tau)^{N_{cd}}P_{v}(k,N _{cd})\\ P_{\omega}^{(2)}=&N_{cd}\tau\sum_{k=0}^{\infty}(1-\tau)^{N_{cd}}P_{v}(k,N_{cd}- 1)\\ P_{\omega}^{(3)}=&1-P_{\omega}^{(1)}-P_{\omega}^{(2)}\end{cases}  \tag{16}\]  
To simplify the analysis, in this paper, we neglect transmission events occurring outside of the interference range. To calculate the service rate of the proposed scheme, \(\mu\), we model the transmission procedure of the proposed scheme as an M/G/1 queue with unlimited queue and apply the Probability Generating Function (PGF) approach. As treated in [24], the service time of a packet can be modeled as a non-negative random variable denoted by \(S\), with the unit of the smallest time unit, \(\sigma\). We first provide the PGF of active BSM transmission procedure. Whenever a VSTA has an active BSM a backoff procedure is initiated. Therefore, the generalized state transition diagram of the service time of the BSM that has become the head of the queue can be modeled as depicted in Figure 6.  
In Figure 6, \(Y(z)\), \(X(z)\) and \(H(z)\) denote the PGF of the transmission sequence duration of a triggering VSTA, the PGF of the transmission duration of MU-RTS collision case and the PGF of the time that the backoff counter decrements by one, respectively. By the definition of the model, \(Y(z)\) and \(X(z)\) can be given by  
\[ \begin{cases}Y(z)=\frac{P_{tr}}{\tau}z^{\lfloor\frac{T_{\omega}^{(2)}}{\sigma }\rfloor}\\ X(z)=\frac{(1-P_{tr})}{\tau}z^{\lfloor\frac{T_{\omega}^{(3)}}{\sigma}\rfloor} \end{cases}  \tag{17}\]  
Using (14), (15) and (16), \(H(z)\) can be given by  
\[H(z)=P_{\omega}^{(1)}z+(P_{\omega}^{(2)}-P_{ta})z^{\lfloor\frac{T_{\omega}^{(2 )}}{\sigma}\rfloor}+P_{\omega}^{(3)}z^{\lfloor\frac{T_{\omega}^{(3)}}{\sigma}\rfloor} \tag{18}\]  
There are two main differences between the state transition model of the proposed scheme and the general EDCA model provided in [26; 27].