# An Analytical Model for Interference Alignment in Broadcast Assisted VANETs
## 3 Model of Broadcast Assisted Transmission Based IA Application in VANETs
### Optimization Objective  
It is obvious that the safety related information should be transmitted punctually. During each scheduling frame, we have \(K\) time slots and \(TU(t)\) is the state of time slot \(t\). If there is no vehicle initiates communication at time slot \(t\), \(TU(t)\) equals 0, then we get  
\[\left\{\begin{array}{l}TU(t)=1,(\sum\limits_{i=1:N}\delta\,i\,\tau(t)\geq 1 )\\ TU(t)=0,(\sum\limits_{i=1:N}\delta\,i\,\tau(t)\!=0)\end{array}\right.,\ (1\leq t\leq K) \tag{26}\]  
Fewer time slots are occupied, more efficient of safety related information propagation is. Therefore, we take time slot taken-up ratio denoted as \(PTU\) to be the optimization objective, then we get  
\[PTU=\sum_{t=1:K}TU(t)/K \tag{27}\]  
Combined with other constraints, we have the optimization problem \(OPT-PTU\):  
\begin{tabular}{l l l} \hline \hline
**OPT:** & **Min** & \(PTU\) \\  & \(s.t.\) & \((1)\)-\((3)\),\((5)\),\((7)\)-\((27)\) \\ \hline \hline \end{tabular}  
Among all the constraints, Formula (1)-(2) regulate half-duplex mode of vehicle terminals; (5)-(7) give constraints on receiving state and communication links; (8)-(10) introduce constraints on V2V broadcast communication; (11) presets initialization values to reduce searching range of feasible solutions; (14)-(18) regulate V2I and IA constraints for base station. (19)-(25) impose constraints on communicate route and data-driven assisted transmission. Among them, \(\lambda_{i,j}(t),il_{i,j}(t),al_{i,j}(t)\) and \(rot_{i,j}(t)\) are positive integer variables. \(PTU\) is non-negative variable. \(hcount\) can be adjusted according to delay requirement of VANETs.  
Due to type of variable \(TU(t)\) and constraint (18) which is not linear, the prime optimization problem is Mixed Integer Nonlinear Programming (MINLP). In order to solve this problem, we have to reform some constraints into much neater forms. The next two lemmas will help us to reduce complexity of \(OPT-PTU\) problem. Proofs of these two lemmas are postponed to Appendix A.  
**Lemma 1**.: _Constraint (26) with integer variable \(TU(t)\) is identical with the form listed in (28)_  
\[\left\{\begin{array}{l}TU(t)\leq 1-1/N+\sum\limits_{i=1:N}\delta\, \overset{i}{T}(t)/N\\ TU(t)\geq\sum\limits_{i=1:N}\delta\,\overset{i}{T}(t)/N\end{array}\right. \tag{28}\]  
**Lemma 2**.: _Nonlinear constraint (16) combined with (17), (18) can be reformed in to the following linear constraint (29)-(31)_  
\[il_{i,j}(t)\geq\delta\,\overset{j}{T}(t)+\delta\,\overset{i}{R}(t)-\lambda\, \,\overset{i}{,j}(t))-1,(IL(i,j)\
eq 0) \tag{29}\]  
\[il_{i,j}(t)\leq\delta\,\overset{i}{R}(t),(IL(i,j)\
eq 0) \tag{30}\]  
\[\left\{\begin{array}{l}il_{i,j}(t)\leq 1-\lambda\,\,\overset{i}{,j}(t)\\ 0\leq il_{i,j}(t)\end{array}\right.,(IL(i,j)\
eq 0) \tag{31}\]  
With these two lemmas, prime optimization problem is identical with the following  
\begin{tabular}{c c c} \hline \hline
**OPT:** & **Min** & _PTU_ \\  & \(s.t.\) & \((1)\)-\((3)\),\((5)\),\((7)\)-\((15)\),\((19)\)-\((25)\),\((27)\)-\((31)\) \\ \hline \hline \end{tabular}  
\(OPT-PTU\) is reformed into Mixed Integer Linear Programming that can be solved more efficiently compared with prime one.