# An Effective Wide-Bandwidth Channel Access in Next-Generation WLAN-Based V2X Communications
## 4 Delay Analysis of All Back-Off AIFS Channel Access Scheme
\begin{array}{l}E[\text{length of a slot time for a 20 MHz frame}]=P_{p}(0)\cdot P_{s}(0)\cdot P_{w}(0)\cdot aSlotTime\\ \qquad+P_{p}(0)\cdot P_{s}(0)\cdot P_{w}(1)\cdot(T_{w}+AIFS+aSlotTime)\\ \qquad+P_{p}(1)\cdot P_{s}(0)\cdot P_{w}(0)\cdot m_{s}(T_{p}+AIFS+aSlotTime)\\ \qquad+P_{p}(0)\cdot P_{s}(1)\cdot P_{w}(0)\cdot m_{p}(T_{s}+AIFS+aSlotTime)\\ \qquad+P_{p}(1)\cdot P_{s}(1)\cdot P_{w}(0)\cdot(T_{p}+AIFS+aSlotTime)\\ \qquad+P_{p}^{\text{col}}\cdot P_{s}(0)\cdot m_{s}(T_{p}+EIFS+aSlotTime)\\ \qquad+P_{p}^{\text{col}}\cdot P_{s}^{\text{col}}\cdot P_{w}(0)\cdot(T_{p}+EIFS +aSlotTime)\\ \qquad+P_{p}^{\text{col}}\cdot P_{s}^{\text{col}}\cdot P_{w}(0)\cdot(T_{p}+ EIFS+aSlotTime)\\ \qquad+P_{p}(1)\cdot P_{s}^{\text{col}}\cdot P_{w}(0)\cdot(T_{p}+AIFS+aSlotTime+m _{p}(EIFS-AIFS))\\ \qquad+P_{p}^{\text{col}}\cdot P_{s}(1)\cdot P_{w}(0)\cdot(T_{p}+AIFS+aSlotTime +m_{s}(EIFS-AIFS))\\ \qquad+P_{p}(0)\cdot P_{s}(0)\cdot P_{w}^{\text{col}}\cdot(T_{w}+EIFS+aSlotTime )\\ \qquad+(1-P_{p}(0)\cdot P_{s}(0))\cdot(P_{w}(1)+P_{w}^{\text{col}})\cdot(\max( T_{w},T_{p})+EIFS+aSlotTime)\end{array}  \tag{8}\]  
where \(m_{s}(x)\) is the mean holding time for the secondary channel for duration \(x\), and \(m_{p}(x)\) is the mean holding time for the primary channel for duration \(x\). Since busy-channel states may occur in the primary channel and the secondary channel in a cascaded manner, the mean holding time for the secondary channel, \(m_{s}(x)\), can be obtained by a recursive function:  
\[ \begin{array}{l}m_{s}(x)=x+\sum_{i=1}^{[x/aSlotTime]}\left(P_{s}(0)\right)^ {i-1}P_{s}(1)m_{p}(T_{sch\_AIFS}-x+i\times aSlotTime)\\ \qquad+\sum_{i=1}^{[x/aSlotTime]}\left(P_{s}(0)\right)^{i-1}P_{s}^{\text{col}} m_{p}(T_{sch\_EIFS}-x+i\times aSlotTime)\end{array}  \tag{9}\]  
for \(x\leq T_{sch\_AIFS}\), and  
\[ \begin{array}{l}m_{s}(x)=x+\sum_{j=1}^{n_{s}(x)}\left(P_{s}(0)\right)^{j-1} P_{s}(1)(m_{p}(x-T_{sch\_AIFS}-j\times aSlotTime)-x+T_{sch\_AIFS}+j\times aSlotTime)\\ \qquad+\sum_{i=1}^{[(x-n_{s}(x)\times aSlotTime)/aSlotTime]}\left(P_{s}(0) \right)^{n_{s}(x)}\left(P_{s}(0)\right)^{i-1}P_{s}(1)m_{p}(T_{sch\_AIFS}-(x-n_ {s}(x)\times aSlotTime)+i\times aSlotTime)\\ \qquad+\sum_{i=1}^{[x/aSlotTime]}\left(P_{s}(0)\right)^{i-1}P_{s}^{\text{col}} m_{p}(T_{sch\_EIFS}-x+i\times aSlotTime),n_{s}(x)=\left[(x-T_{sch\_AIFS})/aSlotTime\right] \end{array}  \tag{10}\]  
where \(T_{sch\_AIFS}<x\leq T_{sch\_EIFS}\) and \(T_{sch\_AIFS}=T_{s}+AIFS+aSlotTime\) and \(T_{sch\_EIFS}=T_{s}+EIFS+aSlotTime\). The mean holding time for the primary channel, \(m_{p}(x)\), can be calculated in the same way as (9) and (10), with the substitution of \(P_{s}(k)\), \(P_{s}^{\text{col}}\), \(m_{p}(x)\), \(T_{sch\_AIFS}\), and \(T_{sch\_EIFS}\) for \(P_{p}(k)\), \(P_{p}^{\text{col}}\), \(m_{p}(x)\), \(T_{pch\_AIFS}\), and \(T_{pch\_EIFS}\) when \(T_{pch\_AIFS}=T_{p}+AIFS+aSlotTime\) and \(T_{pch\_EIFS}=T_{p}+EIFS+aSlotTime\). The mean access delay of frames carried in the secondary 10-MHz channel, \(dp\), can be converted to  
\[d_{p}=AIFS+E\text{ [length of a slot time in primary 10 MHz]/}\tau_{p}, \tag{11}\]  
where \(E\) [length of a slot time in primary 10 MHz] is calculated as  
\[ \begin{array}{l}E[\text{length of a slot time in primary 10 MHz}]=P_{p}(0)\cdot I_{wp}(0)\cdot aSlotTime\\ \qquad+P_{p}(1)\cdot I_{wp}(0)\cdot(T_{p}+AIFS+aSlotTime)\\ \qquad+P_{p}(0)\cdot I_{wp}(1)\cdot(T_{w}+AIFS+aSlotTime)\\ \qquad+P_{p}^{\text{col}}\cdot I_{wp}(0)\cdot(T_{p}+EIFS+aSlotTime)\\ \qquad+P_{p}(0)\cdot I_{wp}^{\text{col}}\cdot(T_{w}+EIFS+aSlotTime)\\ \qquad+\left\{(1-P_{p}(0))\cdot(1-I_{wp}(0))\right\}\cdot(\max(T_{p},T_{w})+ EIFS+aSlotTime)\end{array}  \tag{12}\]  
where \(I_{wp}(k)\) is the interference probability to the primary channel by the 20-MHz frame sent by \(k\) stations. When \(i_{up}\) denotes the interference probability to primary channel by a certain station transmitting a 20-MHz frame, \(I_{up}(k)\) can be derived as  
\[I_{up}(k)=\left( \begin{array}{c}N_{up}\\ k\end{array}