# A Framework to Determine the Utilization of Vacant Taxis on HOV Lanes with the Optimal Deployment
## 4 The Network Performance  
The VI and algorithm are applied to a numerical example in this section. As shown in Figure 1, the network has six nodes and 14 links. Each link consists of a HOV lane and a GP lane. As mentioned above, each lane is considered a link. The OD pairs and demands are listed in Table 1 and attributes of the links are shown in Table 2. We adopt the traditional BPR function as the travel time function. \[t_{a}=t_{a}^{0}\Big{[}1+0.15(x_{a}/C_{a})^{4}\Big{]},\ a\in A+\overline{A} \tag{38}\]  
where \(t_{a}^{0}\) denotes the free-flow travel time. \(x_{a}\) is the flow on link \(a\), which satisfies:  
\[ \begin{array}{l}x_{a}=\sum\limits_{r\in R,s\in S}\sum\limits_{p\in P}f_{rs,p} ^{sd}\cdot\delta_{a,p}^{rs}+\sum\limits_{r\in R,s\in S}\sum\limits_{p\in P} \big{(}f_{rs,p}^{h}/n\big{)}\cdot\delta_{a,p}^{rs}\\ \hskip 14.226378 pt+\sum\limits_{s\in S,p\in R}\sum\limits_{p_{v}\in P_{v}}f_{sr,p_{v}}^{v}\cdot\delta_{a,p_{v}}^{sr},a\in A+\overline{A}\end{array}  \tag{39}\]  
Figure 1: Topology of the network. Generally, the system performances are measured by the social welfare of the network. To be specific, in this paper the social welfare in the case allowing vacant tasks to use HOV lanes is expressed as:  
\[ \begin{array}{l}Z_{1}=\sum\limits_{r\in R,s\in S}\int_{0}^{d_{rs}}B_{rs}^{-1}(w )dw-\sum\limits_{a\in A}\left(\lambda_{t}t_{a}(x_{a})+b_{s}l_{a}\right)x_{a}^{sd }-\sum\limits_{a\in A+\overline{A}}\lambda_{t}t_{a}(x_{a})x_{a}^{h}\cdot n+b_{ h}l_{a}x_{a}^{h}\\ \quad-\sum\limits_{a\in A}\left[\lambda_{v}t_{a}(x_{a})x_{a}^{o}+b_{v}l_{a}(x_{ a}^{o}+x_{a}^{o})\right]-\sum\limits_{a\in A}\left[\lambda_{v}t_{a}(x_{a})x_{a}^{o}+b_{ v}l_{a}(x_{a}^{o}+x_{a}^{o})\right]-\sum\limits_{r\in R,s\in S}d_{rs}^{h}\cdot\Delta\\ \quad-\sum\limits_{r\in R,s\in S}\lambda_{o}x_{sr}^{o}\omega_{r}^{p}-\sum \limits_{r\in R,s\in S}\lambda_{o}d_{rs}^{o}\omega_{r}^{p}-\sum\limits_{a\in A }\lambda_{t}l_{a}(x_{a})x_{a}^{o}-\sum\limits_{a\in A}\lambda_{t}l_{a}(x_{a}) x_{a}^{o}\end{array}  \tag{40}\]  
In the case of forbidding vacant trails from using HOV lanes, the total cost is written as:  
\[ \begin{array}{l}Z_{2}=\sum\limits_{r\in R,s\in S}\int_{0}^{d_{rs}}B_{rs}^{-1}( w)dw-\sum\limits_{a\in A}\left(\lambda_{t}t_{a}(x_{a})+b_{s}l_{a}\right)x_{a}^{sd }-\sum\limits_{a\in A+\overline{A}}\lambda_{t}t_{a}(x_{a})x_{a}^{h}\cdot n\\ \quad-\sum\limits_{a\in A+\overline{A}}\left[\lambda_{t}t_{a}(x_{a})x_{a}^{o}+ b_{v}l_{a}x_{a}^{o}\right]-\sum\limits_{a\in A}\left(\lambda_{v}t_{a}(x_{a})x_{a}^{o}+ b_{v}l_{a}x_{a}^{o}\right)+\sum\limits_{a\in A+\overline{A}}b_{h}l_{a}x_{a}^{h}\\ \quad-\sum\limits_{r\in R,s\in S}\lambda_{o}d_{rs}^{o}\omega_{r}^{p}-\sum \limits_{r\in R,s\in S}\lambda_{o}d_{sr}^{o}\omega_{r}^{p}-\sum\limits_{r\in R,s\in S}d_{rs}^{h}\cdot\Delta\end{array}  \tag{41}\]  
Referring to [38], the parameters are set as:  
\[ \begin{array}{l}\lambda_{t}=60,b_{s}=3,b_{h}=3,b_{0}^{0}=8,b_{b}^{1}=2,b_{0} ^{2}=25,b_{v}=1,\lambda_{v}=40,N=1000,\sigma=0.4,\eta Z_{r}=5,\lambda_{o}=50,\\ n=2,\Delta=13,\beta=0.5,k_{rs}^{sd}=3,k_{rs}^{h}=1,k_{rs}^{o}=5,\pi_{rs}=0.03 \end{array}  \tag{42}\]  
For simplicity, hereafter we denote the case allowing vacant tasks to enter HOV lanes as case 1, and the case forbidding vacant trails from using HOV lanes as case 2. Table 3 shows that the social welfare decreases slightly, and the demand of private cars, carpooling vehicles, and occupied taxis varies mildly. It is a paradox that the social welfare decreases in case 1 comparing to in case 2.