# A Hybrid Handover Scheme for Vehicular VLC/RF Communication Networks
## 3 Hybrid Handover Scheme
### The MDP Setup
\begin{cases}\frac{ \lambda}{\lambda+\beta_{1}+\gamma_{-1}+\mathfrak{a}_{21}+H(\text{Off},\text{ On},2)}&i^{\prime}=(\text{Off},\text{On},2,b+1,|p(i)|)\\ \frac{\beta_{1}}{\lambda+\beta_{1}+\gamma_{-1}+\mathfrak{a}_{21}+H(\text{Off}, \text{On},2)}&i^{\prime}=(\text{Off},\text{Off},2,b,|p(i)|)\\ \frac{\beta_{2}}{\lambda+\beta_{1}+\gamma_{-1}+\mathfrak{a}_{21}+H(\text{Off}, \text{On},2)}&i^{\prime}=(\text{Off},\text{Off},1,b,|p(i)|)\\ \frac{\beta_{2}}{\lambda+\beta_{1}+\gamma_{-1}+\mathfrak{a}_{21}+H(\text{Off}, \text{On},2)}&i^{\prime}=(\text{Off},\text{On},2,b-1,|p(i)|)\end{cases} , \tag{5}\]  
where \(\mu_{(\text{Off},\text{On},2)}=(1-|p(i)|)\mu_{0}-\frac{1}{2}p(i)(1-p(i))\mu_{1}\). The transition rate for \((s,s^{\prime},h)=(\text{Off},\text{Off},2)\) is given by  
\[\Pr_{i\to i^{\prime}}\{(\text{Off},\text{Off},2),p(i)\}= \begin{cases}\frac{ \lambda}{\lambda+\beta_{1}+\beta_{-1}+\mathfrak{a}_{21}+H(\text{Off},\text{Off },2)}&i^{\prime}=(\text{Off},\text{Off},2,b+1,|p(i)|)\\ \frac{\lambda+\beta_{1}+\gamma_{-1}+\mathfrak{a}_{21}+H(\text{Off},\text{Off },2)}{\lambda+\beta_{1}+\gamma_{-1}+\mathfrak{a}_{21}+H(\text{Off},\text{Off })}&i^{\prime}=(\text{Off},\text{Off},2,b,|p(i)|)\\ \frac{\beta_{2}}{\lambda+\beta_{1}+\gamma_{-1}+\mathfrak{a}_{21}+H(\text{Off },\text{Off })}&i^{\prime}=(\text{Off},\text{Off},1,b,|p(i)|)\\ \frac{\beta_{2}}{\lambda+\beta_{1}+\gamma_{-1}+\mathfrak{a}_{21}+H(\text{Off },\text{Off })}&i^{\prime}=(\text{Off},\text{Off},1,b,|p(i)|)\\ \frac{\beta_{2}}{\lambda+\beta_{1}+\gamma_{-1}+\mathfrak{a}_{21}+H(\text{Off },\text{Off })}&i^{\prime}=(\text{Off},\text{Off},2,b-1,|p(i)|)\end{cases} , \tag{6}\]  
where \(\mu_{(\text{Off},\text{Off},2)}=(1-|p(i)|)\mu_{0}\) because packets can only be transmitted by RF links for states \((s,s^{\prime})=(\text{Off},\text{Off})\). Similarly, the transition rate for \((s,s^{\prime},h)=(\text{Off},\text{Off},1)\) is written as  
\[\Pr_{i\to i^{\prime}}\{(\text{Off},\text{Off},1),p(i)\}= \begin{cases}\frac{ \lambda}{\lambda+\beta_{1}+\beta_{1}+\beta_{-1}+\mu_{(\text{Off},1)}}&i^{ \prime}=(\text{Off},\text{Off},1,b+1,|p(i)|)\\ \frac{\beta_{1}}{\lambda+\beta_{1}+\mathfrak{a}_{21}+\mu_{(\text{Off},\text{Off },1)}}&i^{\prime}=(\text{On},\text{Off},1,b,|p(i)|)\\ \frac{\beta_{1}}{\lambda+\beta_{1}+\mathfrak{a}_{21}+\mu_{(\text{Off},\text{Off },1)}}&i^{\prime}=(\text{Off},\text{On},2,b,|p(i)|)\\ \frac{\beta_{1}}{\lambda+\beta_{1}+\mathfrak{a}_{21}+H(\text{Off},\text{Off },1)}&i^{\prime}=(\text{Off},\text{Off},1,b-1,|p(i)|)\end{cases} , \tag{7}\]  
where \(\mu_{(\text{Off},\text{Off},1)}=(1-|p(i)|)\mu_{0}\). There is no packet departure for \(b=0\) and no packet arrival for \(b=B\), which leads to tiny differences in the transition rates for \(b=0\) and \(b=B\). For example, the transition rate of \((\text{Off},\text{Off},2)\) for \(b=0\) is  
\[\Pr_{i\to i^{\prime}}\{(\text{Off},\text{Off},2),p(i)\}= \begin{cases}\frac{ \lambda}{\lambda+\beta_{1}+\beta_{-1}+\mathfrak{a}_{21}}&i^{\prime}=(\text{Off},\text{Off},2,1,|p(i)|)\\ \frac{\beta_{1}}{\lambda+\beta_{1}+\gamma_{-1}+\mathfrak{a}_{21}}&i^{\prime}=( \text{On},\text{Off},2,0,|p(i)|)\\ \frac{\beta_{2}}{\lambda+\beta_{1}+\gamma_{-1}+\mathfrak{a}_{21}}&i^{\prime}=( \text{Off},\text{Off},1,0,|p(i)|)\end{cases} . \tag{8}\]  
For \(b=B\), the transition rate for \((\text{Off},\text{Off},2)\) can be written as  
\[\Pr_{i\to i^{\prime}}\{(\text{Off},\text{Off},2),p(i)\}= \begin{cases}\frac{ \beta_{1}}{\beta_{1}+\gamma_{-1}+\mathfrak{a}_{21}+H(\text{Off},\text{Off },2)}&i^{\prime}=(\text{On},\text{Off},2,B,|p(i)|)\\ \frac{\beta_{1}}{\beta_{1}+\gamma_{-1}+\mathfrak{a}_{21}+H(\text{Off},\text{Off },2)}&i^{\prime}=(\text{Off},\text{On},2,B,|p(i)|)\\ \frac{\beta_{21}}{\beta_{1}+\gamma_{-1}+\mathfrak{a}_{21}+H(\text{Off},\text{Off },2)}&i^{\prime}=(\text{Off},\text{Off},1,B,|p(i)|)\\ \frac{H(\text{Off},\text{Off},2)}{\beta_{1}+\beta_{-1}+\mathfrak{a}_{21}+H(\text{Off },\text{Off},2)}&i^{\prime}=(\text{Off},\text{Off},2,B-1,|p(i)|)\end{cases}