# Deadline-Aware Scheduling for Transmitted RSU Packets in Cooperative Vehicle-Infrastructure Systems
## 3 System Model
### Prediction Model of Vehicle Communication Quantity  
In the research scenario of this paper, the connection between vehicles and RSU is not continuous. Therefore, we must determine how many vehicles establish communication links with N RSU. When a vehicle enters the RSU coverage area, its communication state is defined as ON. When the vehicle drives out of the RSU coverage area, its communication state is defined as OFF [19]. In this paper, we analyzed the free flow traffic model to predict the number of vehicles establishing communication links with RSU on the road as \(\kappa(t)\). We assumed that the probability of the vehicle and establishing communication links obey the discontinuous Bernoulli distribution (IBP) with parameter \((\lambda,\alpha,\beta)\)[20]. The vehicle's travel time in the ON/OFF state shows a geometric distribution, and the expected values are \(\alpha\) and \(\beta\), respectively.  
The cumulative distribution function of vehicle residence time \(t_{r}\) on this road section can be expressed as  
\[F_{R}(t_{r})=1-F_{V}\bigg{(}\frac{L}{t_{r}}\bigg{)}=1-\frac{1+erf\left(\left( \frac{L}{t_{r}}-\overline{V}\right)/\sigma\sqrt{2}\right)}{erf\left(\left(V_{ \max}-\overline{V}\right)/\sigma\sqrt{2}\right)-erf\left(\left(V_{\min}- \overline{V}\right)/\sigma\sqrt{2}\right)} \tag{7}\]  
\(F_{V}(L/t_{r})\) is the probability distribution function corresponding to the truncated probability density function in the previous section.  
The arrival of vehicles obeys the Poisson distribution with parameter \(\lambda\); therefore, the probability expression of k vehicles arriving in the road section within the interval of \((0,t)\) is as follows:  
\[A_{k}(t)=\frac{(\lambda t)^{k}e^{-(\lambda t)}}{k!} \tag{8}\]  
The t-slot, \(1-F_{R}(t-t_{i})\), represents the probability of any vehicle arriving in the \(t_{i}\)-slot. The vehicle's arrival obeys the Poisson distribution. Therefore, the vehicle arrival time distribution depends on whether k vehicles arrive within the time interval of \((0,t)\). This is the same as the uniform distribution of k points within \((0,t)\). The probability expression for the existence of any k vehicles on the road section within the t-slot is as follows:  
\[p_{k}(t)=\int_{0}^{t}[1-F_{R}(t-t_{i})]\frac{dt_{i}}{t}=\frac{1}{t}\int_{0}^{t} {[1-F_{R}(t_{i})]dt_{i}} \tag{9}\]  
The probability of the vehicle leaving the road section in the t-slot during the time interval of \((0,t)\) is as follows:  
\[1-p_{k}(t)=\frac{1}{t}\int_{0}^{t}{F_{R}(t_{i})dt_{i}} \tag{10}\]  
Depending on whether k vehicles arrive within the time interval of \((0,t)\), the probability of y vehicles driving on the road section within the t-slot is as follows:  
\[P_{y|k}(t)=\left\{\begin{matrix}C_{y}^{k}[p_{k}(t)]^{y}[1-p_{k}(t)]^{k-y}&,y \leq k\\ 0&,y>k\end{matrix}\right. \tag{11}\]  
The probability of establishing a communication link between a vehicle and \(RSU_{n}\) follows a discontinuous Bernoulli distribution with a parameter of \((\lambda,\alpha,\beta)\). Therefore, the probability of y vehicles driving on the road section in t-slot and r vehicles with RSU coverage is as follows:  
\[P\{\kappa(t)=r\}=\frac{\alpha}{\alpha+\beta}\sum\limits_{k=y}^{\infty}C_{y}^{ k}[p_{k}(t)]^{y}[1-p_{k}(t)]^{k-y}\cdot\frac{(\lambda t)^{k}e^{-(\lambda t)}}{k!}=\frac{\alpha[\lambda t\cdot p_{k}(t)]^{y}e^{-(\lambda t)\cdot p_{k}(t)}}{( \alpha+\beta)y!} \tag{12}\]