# Distributed Triggered Access for BSM Dissemination in 802.11 bd V2V Networks
## 3 Numerical Analysis
As there is no change on the contention window size of the transmitter, the EDCA procedure with the fixed contention windows size, \(W\), can be modeled as one-dimensional Markov chain model as depicted in Figure 5[38]. In Figure 5, \(P_{arr}\), \(P_{emp}\) and \(P_{b}\) denote the probability of a packet arrival from the higher layer during a virtual time slot [39], \(T_{\omega}\), the probability that no active BSM exists in the queue and the backoff blocking probability [40], respectively. Denoting the rate of generating a BSM to \(\lambda\), \(P_{arr}\) can be given by  
\[P_{arr}=1-e^{-\lambda T_{\omega}} \tag{3}\]  
The backoff blocking probability, \(P_{b}\), is the probability that the backoff counter of the tagged VSTA is frozen by other VSTAs occupying the channel or by other ACs. In this paper, the target BSM message is supposed to use the highest AC, the impact of internal contention within a STA is not considered. By the definition of \(P_{b}\), we have  
\[P_{b}=1-\sum_{k=0}^{\infty}(1-\tau)^{N_{ed}-1}P_{b}(k,N_{ed}-1) \tag{4}\]  
Figure 5: Markov chain model for the backoff procedure. In (4), \(\tau\) is the transmission probability that a VSTA transmits a frame in a randomly chosen time slot [38], and can be obtained by solving the Markov chain model of the proposed scheme in Figure 5.  
According to the stationary distribution of the Markov model in Figure 5, we have  
\[ \begin{cases}b_{k}=\frac{W-k+1}{W+1}b_{0}\\ b_{Idle}=P_{emp}\cdot b_{0}+b_{Idle}(1-P_{arr})\end{cases}  \tag{5}\]  
\[b_{0}=\left[\frac{W}{2(1-P_{b})}+\frac{P_{emp}+P_{arr}}{P_{arr}}\right]^{-1} \tag{6}\]  
Unlike conventional EDCA transmissions, in DTA, a VSTA may transmit an active BSM with non-zero backoff counter by the triggered access from other VSTAs. In this case, the backoff procedure is resumed after the triggered access, but the VSTA does not transmit a BSM when its counter reaches to zero unless there exists an active BSM in the buffer. In the proposed scheme, therefore, an EDCA transmission event occurs when one of the following conditions is met: (1) the VSTA has more than one active BSM, (2) the VSTA has one active BSM and no triggered access happens during the backoff procedure, (3) the only active BSM is transmitted by an triggered access during the backoff procedure, and another active BSM arrives before the counter reaches to zero. In this paper, we consider the probability that multiple triggered access scheduling events or packet arrival events occur during one backoff procedure is negligible. According to [41] the steady-state probability that \(k\) packets exist in the system, \(P_{q}^{(k)}\) is  
\[P_{q}(k)=(1-\rho)\rho^{k}, \tag{7}\]  
where \(\rho\) is the busy probability of the system which can be derived by  
\[\rho=\lambda/\mu \tag{8}\]  
where \(\mu\) is the average service rate of the proposed scheme. Thus, denoting the probability that a VSTA has at least one active BSM to transmit when the counter reaches to zero as \(P_{a}\), we have  
\[ \begin{split} P_{a}=&\frac{1-p_{q}(0)-p_{q}(1)}{1-p _{q}(0)}+\frac{p_{q}(1)}{1-p_{q}(0)}\frac{1}{W+1}\\ &\cdot\sum_{k=1}^{W}\left[(1-p_{ta})^{k}+\sum_{j=1}^{k}p_{ta}(1-p _{ta})^{j-1}\left(1-e^{-\lambda(k-j+1)T_{a}}\right)\right],\end{split}  \tag{9}\]  
and we can obtain \(P_{emp}\) and \(\tau\) as  
\[P_{emp}=1-P_{a}, \tag{10}\]  
and  
\[\tau=P_{a}b_{0} \tag{11}\]  
In (9), \(P_{ta}\), is the probability that a VSTA transmits a BSM by triggered access in an arbitrary virtual time slot. A VSTA transmits a BSM via triggered access when it successfully receives a TF containing the scheduling information of the VSTA which means that the received TF is transmitted within \(R_{tr}\) and the VSTA is included in \(M\), the number of RUs for triggering, out of \(N_{tr}\) VSTAs. An EDCA transmission is regarded successful if only one VSTA transmits the Trigger frame within the ED range at the time slot, and no transmission occurs from the VSTAs in the hidden range during the vulnerable period [26].