# Distributed Triggered Access for BSM Dissemination in 802.11 bd V2V Networks
## 3 Numerical Analysis
First, the traffic can transition to end state from any state due to the triggered access, Second, the state-in probability from the start point is no longer uniformly distributed, because of the residual backoff counter after a triggered access. In Figure 6, \(R(z)\) denotes the PGF of the transmission sequence duration of a triggered VSTA, and can be obtained as  
\[R(z)=P_{ta}z^{\lfloor\frac{T_{\omega}^{(2)}}{\sigma}\rfloor} \tag{19}\]  
If an active BSM in service is transmitted via triggered access during the backoff procedure and another active BSM exists in the queue, the next serving BSM can use the residual backoff counter so that the service time can be reduced. Therefore, whether a BSM initiates a backoff procedure or reuses the  
Figure 6: The generalized state transition diagram for the proposed scheme. previous backoff procedure depends on the number of buffered BSM. Using the steady-state probability (7), we can calculate the initial state-in probability for state \(k\), \(P_{in}^{(k)}\), as follows  
\[P_{in}^{(k)}=\frac{p_{q}(1)}{1-p_{q}(0)}\frac{1}{W+1}+\frac{1-p_{q}(1)}{1-p_{q}( 0)}\left(\frac{P_{rib0}^{(0)}}{W+1}+P_{rib0}^{(k)}\right) \tag{20}\]  
In (20), \(P_{rib0}^{(k)}\) is the probability that the residual backoff counter of the previous active BSM is \(k\) which can be given by  
\[P_{rib0}^{(k)}=\frac{1}{W+1}\sum_{j=0}^{W+1-k}P_{ta}(1-P_{ta})^{j} \tag{21}\]  
Applying [PERSON]'s formula, there exists \(W+1\) loops in the proposed transition model. Since all loops in the model touch each other at least once, the determinant of the transition model, \(\Delta\) can be given by  
\[\Delta=1-\frac{X(z)}{W+1}\sum_{k=0}^{W}H(z)^{k} \tag{22}\]  
For state \(k\), there exists \(k+1\) paths that exits the system via state \(k\), and each path equally touches the right most \(W-k+1\) loops sharing the same cofactor value of the determinant, \(\Delta_{k}\). If we let \(\Delta_{k}\) be the cofactor value for the paths exiting the system via state \(k\), we have  
\[ \begin{cases}\Delta_{0}=1\\ \Delta_{1}=1-\frac{X(z)}{W+1}\\ \Delta_{2}=1-\frac{X(z)}{W+1}(1+H(z))\\ \Delta_{3}=1-\frac{X(z)}{W+1}(1+H(z)+H(z)^{2})\\ \vdots\\ \Delta_{W}=1-\frac{X(z)}{W+1}\sum_{k=0}^{W}H(z)^{k-1}\end{cases}  \tag{23}\]  
If we denote the sum of gains of all paths with respect to \(\Delta_{k}\) by \(G_{k}\),  
\[ \begin{cases}G_{0}=Y(z)\sum_{k=0}^{W}P_{in}^{(k)}H(z)^{W-k}\\ G_{1}=R(z)\sum_{k=1}^{W}P_{in}^{(k)}H(z)^{W-k}\\ G_{2}=R(z)\sum_{k=2}^{W}P_{in}^{(k)}H(z)^{W-k}\\ \vdots\\ G_{W-1}=R(z)\sum_{k=W-1}^{W}P_{in}^{(k)}H(z)^{W-k}\\ G_{W}=R(z)P_{in}^{(W)}\end{cases}  \tag{24}\]As a result, the PGF of the service time, \(S(z)\), can be given by  
\[S(z)=\frac{\sum\limits_{k=0}^{W}G_{k}\Delta_{k}}{\Delta} \tag{25}\]  
Finally, using the property of PGF, the average service time, \(\overline{S}\), can be derived as  
\[\overline{S}=\frac{1}{\mu}=S^{\prime}(z)\bigg{|}_{z=1} \tag{26}\]  
According to the Pollaczek-Khintchine mean value formula [41] for M/G/1 queue, the average queuing delay \(\sqrt{Q}\), can be derived by  
\[\overline{Q}=\frac{\lambda\left[P_{S}{}^{\prime\prime}(1)+P_{S}{}^{\prime}(1) \right]}{2\left[1-\lambda/\mu\right]} \tag{27}\]  
Thus, using (25)-(27), the average transmission delay \(\overline{D}\) can be obtained by  
\[\overline{D}=\overline{Q}+\overline{S} \tag{28}\]