# Optimal Task Offloading Strategy for Vehicular Networks in Mixed Coverage Scenarios
## 3 System Model and Problem Formalization
### Task Offloading Problem Formalization
#### 3.2.1 Time Delay Model  
Based on [PERSON]'s theorem, the data transmission rates \(r_{vr}\) and \(r_{vb}\) from vehicles to RSU or BS can be calculated as shown below:  
\[r_{vr}=B_{vr}log_{2}(1+\frac{P_{v}}{\sigma^{2}+N}), \tag{1}\]  
\[r_{vb}=B_{vb}log_{2}(1+\frac{P_{v}}{\sigma^{2}+N}), \tag{2}\]  
where \(\sigma\) represents white noise, \(N\) denotes interference among multiple transmissions, \(B_{vr}\) stands for the bandwidth allocated by the RSU to vehicles, and \(B_{vb}\) is the bandwidth allocated by the BS to vehicles.  
Since the delay of downlink transmission is much smaller than that of uplink transmission, we reasonably ignore the calculation result of downlink transmission delay. Therefore, we only focus on the computation delay and uplink transmission delay. When vehicles  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Notation** & **Descriptions** \\ \hline Parameters & \\ \(k\) & Computational complexity of the task \\ \(s\) & The data size of the task \\ \(\sigma^{2}\) & White noise \\ \(v_{i}\) & The current speed of the vehicle \\ \(N\) & Interference among multiple transmissions \\ \(r_{vr}\) & Transmission rate of tasks offloaded to the RSU \\ \(r_{vb}\) & Transmission rate of tasks offloaded to the BS \\ \(f_{r}\) & The CPU frequency for task processing at the RSU \\ \(f_{b}\) & The CPU frequency for task processing at the BS \\ \(f_{v}\) & The CPU frequency for task processing at the local area \\ \(B_{vr}\) & The bandwidth allocated by RSU to the vehicle \\ \(B_{vb}\) & The bandwidth allocated by BS to the vehicle \\ \(P_{v}\) & The transmission power of the vehicle \\ \(x_{i}\) & The current position of the vehicle \\ \(x_{r}\) & The coverage boundary position of the RSU where the vehicle is located \\ \(x_{b}\) & The coverage boundary position of the BS where the vehicle is located \\ Variables & \\ \(\alpha\) & The proportion of tasks allocated to RSU \(\alpha\in[0,1)\) \\ \(\theta\) & The proportion of tasks allocated to BS \(\theta\in[0,1)\) \\ \(\gamma\) & The proportion of tasks allocated to local \(\gamma\in(0,1]\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Mathematical notation.  
offload tasks to the RSU, the transmission delay \(T_{vr}^{t}\) from vehicles to the RSU and the computation delay \(T_{vr}^{c}\) at RSU can be expressed as shown below:  
\[T_{vr}^{t}=\frac{\alpha s}{r_{vr}}, \tag{3}\]  
\[T_{vr}^{c}=\frac{\alpha ks}{f_{r}}. \tag{4}\]  
Similarly, we obtain the computation delay and transmission delay models when a vehicle offloads a task to the BS.  
\[T_{vb}^{t}=\frac{\theta s}{r_{vb}}, \tag{5}\]  
\[T_{vb}^{c}=\frac{\theta ks}{f_{b}}. \tag{6}\]  
Therefore, the total delay of RSU and BS auxiliary processing tasks are shown as  
\[T_{vr}=T_{vr}^{t}+T_{vr}^{c}, \tag{7}\]  
\[T_{vb}=T_{vb}^{t}+T_{vb}^{c}. \tag{8}\]  
After task offloading, the size of the task that remains for local computation by the vehicle is \(\gamma s\), and the total delay for the vehicle to compute the remaining task locally is  
\[T_{v}=\frac{\gamma ks}{f_{v}}. \tag{9}\]  
According to the time delay models of vehicles to RSUs, vehicles to BSs and local processing, the total delay for the vehicle to compute tasks is  
\[T=\max\{T_{vr},T_{vb},T_{v}\}. \tag{10}\]