# Analysis and Design of Functional Device for Vehicular Cloud Computing
## 4 Performance of Two-Hop Relay-Assisted Vehicular Networks
### \(S\to R_{i}\to D\) Link  
The capacity of the \(S\to R_{i}\to D\) link can be inferred using the Shannon theory and in combination with Equation (9). The capacity of this link is denoted as \(C_{AF}\),  
\[C_{AF}=\frac{1}{2}\log_{2}\left(1+\frac{\gamma_{1}\gamma_{2}}{\gamma_{1}+ \gamma_{2}+1}\right). \tag{18}\]  
To ensure QoS, the SNR threshold \(\gamma_{th}\) is defined, and it essentially depends on the modulation type. In this case, the outage probability of the link \((S\to R_{i}\to D)\) is defined as the probability with an instantaneous SNR, \(\gamma\), that falls below a protection threshold, namely [34]  
\[P_{out}=P\left(\gamma<\gamma_{th}\right). \tag{19}\]  
Here, \(\gamma_{th}\) is given by  
\[\gamma_{th}=2^{2R}-1. \tag{20}\]  
Substituting Equation (14) into Equation (13) leads to the approximate expression in Equation (13) for the case in Equation (10), which is given by \[P_{out} =P\left(\left(\frac{1}{\frac{1}{\gamma_{1}}+\frac{1}{\gamma_{2}}} \right)^{-1}<2^{2R}-1\right) \tag{21}\] \[=1-\frac{2\gamma_{th}}{\sqrt{\gamma_{1}\gamma_{2}}}K_{1}\left( \frac{2\gamma_{th}}{\sqrt{\gamma_{1}\gamma_{2}}}\right)e^{\left(\frac{-\gamma_ {th}}{\gamma_{1}}+\frac{-\gamma_{th}}{\gamma_{2}}\right)},\]  
where \(K\left(\bullet\right)\) is a first-order modified Bessel function of the second kind. Therefore, from the above context and Equation (8), we see that the distribution of the approximated expression of in Equation (15) is given as follows [35]:  
\[P_{out}\approx 1-\frac{2\left(2^{2R}-1\right)}{\sqrt{\gamma_{1}\gamma_{2}}}K_{1 }\left(\frac{2\left(2^{2R}-1\right)}{\sqrt{\gamma_{1}\gamma_{2}}}\right)e^{ \left(\frac{-\left(2^{2R}-1\right)}{\gamma_{1}}+\frac{-\left(2^{2R}-1\right) }{\gamma_{2}}\right)}. \tag{22}\]  
This complex operation can be handled at VC for reducing communication loads on a vehicle. Although the introduction of VC technology has increased the computing power for the vehicular networks, complex calculations will increase the time consumption of a VC computing process [36]. This will increase the safety messages transmission delay. Therefore, further computational simplification is necessary.  
When \(K_{1}\left(x\right)\approx 1/x,x\to 0\)[37], \(K_{1}\left(\frac{2\gamma_{th}}{\sqrt{\gamma_{1}\gamma_{2}}}\right)\approx \frac{\sqrt{\gamma_{1}\gamma_{2}}}{2\gamma_{th}}\) at a higher SNR. Hence, we can utilize the property to re-calculate the outage probability in Equation (22) as  
\[P_{out}\approx 1-e^{\left(\frac{-\left(2^{2R}-1\right)}{\gamma_{1}}+\frac{- \left(2^{2R}-1\right)}{\gamma_{2}}\right)}. \tag{23}\]  
By utilizing the first-order [PERSON] formula approximation about the exponential function, we can approximate the expression in Equation (23) further as  
\[P_{out}\approx\left(\frac{1}{\gamma_{1}}+\frac{1}{\gamma_{2}}\right)\left(2^{ 2R}-1\right). \tag{24}\]  
Equation (24) for the outage probability, in addition to being an approximate formula for the choice of destination SNR in Equation (10), is also a simple lower bound for AF relay-assisted vehicular networks. To benefit the device deployment, the process also reduces the computational complexity of the equipment.