# An Analytical Model for Interference Alignment in Broadcast Assisted VANETs
## 3 Model of Broadcast Assisted Transmission Based IA Application in VANETs
### V2V and V2I Broadcast Model  
In our paper, all terminals adopt a half-duplex mode of operation to switch between receiving and transmitting state. \(\delta\,^{i}_{\ T}(t)\) is the sending state of vehicle \(i\) at time slot \(t\) and \(\delta\,^{i}_{\ R}(t)\) is receiving state. Then we get  
\[\left\{\begin{array}{l}0\leq\delta\,^{i}_{\ T}(t)\leq 1\\ 0\leq\delta\,^{i}_{\ R}(t)\leq 1\end{array}\right.,\ (1\leq i\leq N,1\leq t \leq K) \tag{1}\]  
\[\delta\,^{i}_{\ T}(t)+\delta\,^{i}_{\ R}(t)\leq 1,\ (1\leq i\leq N,1\leq t \leq K) \tag{2}\]  
Since all vehicles should broadcast their safety related information in one scheduling frame, which means they must make a broadcast task during. Thus, we have  
\[\left\{\begin{array}{l}\sum\limits_{t=1:K}\delta\,^{i}_{\ T}(t)\geq 1\\ \sum\limits_{t=1:K,j=1:N}\lambda\,_{ij}(t)\geq 1,(L(i,j)=1)\end{array} \right.,\ (i\in Src) \tag{3}\]  
For V2V communication, IA cannot be applied, so the number of valid session and interference received by a vehicle cannot exceed the number of its antennas [8]. When a vehicle broadcasts to surrounding vehicles, number of receivers is equal to the number of vehicles in receiving state. Thus, the following equation should be met:  
\[\left\{\begin{array}{l}\sum\limits_{j=1:N}\lambda\,_{ij}(t)\leq\sum\limits _{j=1:N}L(i,j)\cdot\delta\,^{j}_{\ R}(t),(\delta\,^{i}_{\ T}(t)=1)\\ \sum\limits_{j=1:N}\lambda\,_{ij}(t)=0,(\delta^{i}_{\ T}(t)=0)\end{array}\right. \tag{4}\]  
Constraints in (4) can be further combined as (5):  
\[\begin{array}{l}\sum\limits_{j=1:N}\lambda\,^{t}_{\ i,j}\leq\delta\,^{i}_{ \ T}(t)\cdot\sum\limits_{j=1:N}\left[L(i,j)\cdot\delta\,^{j}_{\ R}(t)\right]\\ \Rightarrow\left\{\begin{array}{l}\sum\limits_{j=1:N}\lambda\,^{t}_{\ i,j}\leq\sum\limits_{j=1:N}\left[L(i,j)\cdot\delta\,^{j}_{\ R}(t)\right]\\ \sum\limits_{j=1:N}\lambda\,^{t}_{\ i,j}\leq N_{a}(i)\cdot\delta\,^{i}_{\ T}(t )\end{array}\right.\end{array} \tag{5}\]  
Since vehicle \(i\) may receive interference from its surrounding vehicles at the same time, the intended information might not be decoded. Therefore, at the same time, the amount of vehicle data received by vehicle \(i\) cannot exceed its number of antennas, then we get  
\[\left\{\begin{array}{l}\sum\limits_{1\leq j\leq N}^{i\
eq j}\lambda\,_{j,i} (t)\leq N_{a}(i),(\delta\,^{i}_{\ R}(t)=1)\\ \sum\limits_{j=1:N}^{i\
eq j}\lambda\,_{j,i}(t)=0,(\delta\,^{i}_{\ R}(t)=0) \end{array}\right.,(i\
eq St) \tag{6}\]which can be combined as  
\[\sum\limits_{1\leq j\leq N}^{i\
eq j}\lambda_{\ j,i}(t)\leq\delta_{\ R}^{\ i}(t) \cdot N_{a}(i),\ (i\
eq St) \tag{7}\]  
When a vehicle is in sending or silent state, and it cannot receive sessions from neighboring vehicles, then the following constraint should be met:  
\[\left\{\begin{array}{l}\lambda_{\ i,j}(t)\leq N_{a}(i)\cdot\delta_{\ T}^{\ i}(t)\\ \delta_{\ T}^{\ i}(t)\leq\sum\limits_{t=1:K,j=1:N}\lambda_{\ i,j}(t)\\ \end{array},(L(i,j)=1)\right. \tag{8}\]  
If a terminal switches to receiving mode and there is no vehicle sends message to it, it will only receive interference or nothing. Thus, we get  
\[\left\{\begin{array}{l}\sum\limits_{j=1:N}\lambda_{\ i,j}(t)\leq\sum\limits _{j=1:N}\delta_{\ R}^{\ j}(t)\\ \delta_{\ R}^{\ j}(t)\leq\sum\limits_{j=1:N}\lambda_{\ i,j}(t)\\ \end{array},(i\
eq St,L(i,j)=1)\right. \tag{9}\]  
For a vehicle in receiving state, the number of interference a vehicle can receive is no exceed the number of its antennas. Then we get  
\[il_{i,j}(t)\leq\delta_{\ T}^{\ i}(t)\cdot N_{a}(i),\ (j\
eq St) \tag{10}\]  
Some constraints can be set default value in advance based on practical situation. When the distance between vehicles exceeds communication range, \(\lambda_{i,j}(t)\) equals 0. Similarly, when vehicles out of interference range with each others, there is no interference. Then, we have the following equations  
\[\left\{\begin{array}{l}\lambda_{\ i,j}(t)\leq N_{a}(i)\cdot L(i,j)\\ il_{i,j}(t)\leq N_{a}(i)\cdot IL(i,j)\\ al_{i,j}(t)\leq N_{a}(i)\cdot IL(i,j)\\ \end{array}\right. \tag{11}\]