# An Effective Wide-Bandwidth Channel Access in Next-Generation WLAN-Based V2X Communications
## 4 Delay Analysis of All Back-Off AIFS Channel Access Scheme  
The performance of 20-MHz transmission can be measured with two metrics: 1. fairness with other devices transmitting a 10-MHz frame in a secondary channel, and 2. the throughput of frames transmitted using a 20-MHz channel. The fairness metric can be measured by the mean access delay of frames carried in a 10-MHz frame in the secondary 10-MHz band. The throughput metric can be measured by the mean access delay and frame transmission latency carried in the 20-MHz channel. For simplicity, in the performance analysis, there are several assumptions: full buffer traffic and one AC with the same AIFS number (AIFSN) and contention window value. The mathematical modeling of this paper is based on the Markov chain modeling in [20]. In addition, the transmitted frame size in each 10-MHz band is assumed to be the same in every transmission. Since the total latency is the sum of the access delay, frame transmission delay, and additional delay for retransmission, the average value of the latency can be expressed as  
\[L_{w}=d_{w}+T_{w}+\text{(mean retransmission delay)}, \tag{1}\]  
where \(L_{w}\) is the mean latency, \(d_{w}\) is the mean access delay, and \(T_{w}\) is the  
To obtain the access delay of frames carried in 20-MHz channels in the all back-off AIFS scheme, the transmission probability of each station should be calculated. Since the channel access scheme of all back-off AIFS includes an EDCA back-off operation with the modification of the channel state decision, the Markov-chain state transition diagram is the same as the Markov-chain diagram depicted in [20], with the assumption of the same AC. Based on the approach in [20], the transmission probability can be calculated as the sum of state probabilities with back-off counter 0, which is expressed as  
\[\tau_{w}=\sum_{i=0}^{m}b_{i,0}, \tag{2}\]  
where \(b_{i,0}\) is the state probability of the \(i\)th back-off procedure with back-off counter 0 of the Markov chain as defined in reference [20]. From the transmission probability \(\tau_{w}\), the probability of frame transmission in a 20-MHz channel by \(k\) stations in each slot, \(P_{w}\)(\(k\)), can be defined as  
\[P_{w}(k)=\left( \begin{array}{c}N_{w}\\ k\end{array} \right)\tau_{w}^{k}\cdot(1-\tau_{w})^{N_{w}-k}, \tag{3}\]  
where \(N_{w}\) is the number of stations with buffered data to transmit using 20-MHz frames. Since a collision occurs when two or more stations transmit frames in one slot, the collision probability of 20-MHz frame transmission in one slot \(P_{w}^{col}\) can be derived as  
\[P_{w}^{col}=1-P_{w}(0)-P_{w}(1). \tag{4}\]  
Similarly, the transmission probabilities transmitted in primary and secondary 10-MHz bands \(\tau_{p}\) and \(\tau_{s}\) can be calculated using the same scheme of Equation (2) from the model in reference [20]. The transmission probabilities of 10-MHz frames by \(k\) stations in each 10-MHz band in one slot \(P_{p}\)(\(k\)) and \(P_{s}\)(\(k\)) can be derived as  
\[P_{p}(k)=\left( \begin{array}{c}N_{p}\\ k\end{array} \right)\tau_{p}^{k}\cdot(1-\tau_{p})^{N_{p}-k}, \tag{5}\]  
\[P_{s}(k)=\left( \begin{array}{c}N_{s}\\ k\end{array} \right)\tau_{s}^{k}\cdot(1-\tau_{s})^{N_{s}-k}, \tag{6}\]  
where \(N_{p}\) and \(N_{w}\) denote the number of stations having buffered frames to transmit using 10-MHz frames in the primary and secondary channels, respectively. As in Equation (4), the collision probabilities of 10-MHz frame transmission in the primary channel and secondary channel can be obtained as \(P_{p}^{col}=1-P_{p}(0)-P_{p}(1)\) and \(P_{s}^{col}=1-P_{s}(0)-P_{s}(1)\), respectively. The mean access delay of frames transmitted in the 20-MHz channel, \(d_{w}\), can be converted to  
\[d_{w}=AIFS+E\text{ [length of a slot time for a 20 MHz frame]/}\tau_{w}, \tag{7}\]  
where \(AIFS=SIFS+AIFSN\cdot aSlotTime\). Assuming that \(T_{p}=T_{s}\), \(E\) [length of a slot time for a 20 MHz frame] is calculated as  
\[