## 3 Methodology
### Markovian Queueing Networks
#### 3.1.2. A Markovian Queueing Network with Higher Demands
\begin{table}
\begin{tabular}{c c c} \hline \hline \(\alpha_{i(m_{1},m_{2})}\) & [people/min] & Transition rate of renting one bike at the virtual node and returning it at \\  & & node \(i\) (\(m_{1},m_{2}=0,1,\cdots,K-1\)) \\ \hline
**Notation** & **Units** & **Definitions** \\ \hline \(\alpha_{j(m_{1},m_{2})}\) & [people/min] & Transition rate of renting one bike at the virtual node and returning it at \\  & & node \(j\) (\(m_{1},m_{2}=0,1,\cdots,K-1\)) \\ \hline \(\beta_{i(m_{1},m_{2})}\) & [people/min] & Transition rate of renting one bike at Node and returning it at the virtual \\  & & node \(i\) (\(m_{1},m_{2}=0,1,\cdots,K-1\)) \\ \hline \(\beta_{j(m_{1},m_{2})}\) & [people/min] & Transition rate of renting one bike at Node and returning it at the virtual \\  & & node \(j\) (\(m_{1},m_{2}=0,1,\cdots,K-1\)) \\ \hline \(\alpha_{i}\beta_{j(m_{1},m_{2})}\) & [people/min] & Transition rate of renting one bike at Node and returning it at node \(i\) \\  & & (\(m_{1},m_{2}=0,1,\cdots,K-1\)) \\ \hline \(\alpha_{j}\beta_{i(m_{1},m_{2})}\) & [people/min] & Transition rate of renting one bike at Node and returning it at node \(j\) \\  & & (\(m_{1},m_{2}=0,1,\cdots,K-1\)) \\ \hline \(p_{m_{1}}\) & - & Probability of having \(m_{1}\) bikes at node \(i\) (\(m_{1}=0,1,\cdots,K\)) \\ \hline \(p_{m_{2}}\) & - & Probability of having \(m_{2}\) bikes at node \(j\) (\(m_{2}=0,1,\cdots,K\)) \\ \hline \(p_{m_{1},m_{2}}\) & - & Probability of having \(m_{1}\) bikes at node \(i\) and having \(m_{2}\) bikes at node \(j\) with \\  & & no relocation (\(m_{1},m_{2}=0,1,\cdots,K\)) \\ \hline \(p_{l}^{{}^{\prime}}\) & - & Adjusted probability of having \(l\) bikes at node \(i\) with relocation \\  & & (\(l=0,1,\cdots,K\)) \\ \hline \(r\) & [times] & \multicolumn{2}{c}{Relocation frequency during operation time} \\ \hline \(c\) & - & Coefficient of variation in probability of having zero bikes at node \(i\) under the influence of relocation \\ \hline \(\mu\) & - & Index of measuring the influence of relocation on steady-state probabilities \\  & & \\ \hline \(B\) & - & Coefficient of variation in probability of having bikes at node \(i\) under the influence of relocation \\ \hline \(c_{\text{in}}\) & [RMB \(\cdot\)min/(bike\(\cdot\)person)] & Unit revenue per bike per person during operation time \\ \hline \(c_{\text{re}}\) & [RMB /time] & Unit cost of one-time relocation \\ \hline \(c_{p}\) & [RMB\(\cdot\)min/person] & Penalty per person for unmet demands during operation time \\ \hline \(Y(r)\) & [RMB] & Total profit with relocation \\ \hline \(r_{\text{max}}\) & [times] & Optimal relocation frequency which corresponds to the maximum total profit with relocation \\  & & profit with relocation \\ \hline \(Y(r_{\text{max}})\) & [RMB] & Maximum total profit with relocation by adopting the optimal relocation frequency \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._with one single node (i.e., node \(i\)) and a combination of \(N-1\) nodes as a virtual node is shown in Figure 1.  
In this subsection, we build network models based on the model reduction method and differentiate a node with higher demands from the other nodes (i.e., \(\lambda\leq\lambda_{i}\)). We assume that it is possible to have zero bikes at the nodes which are inside a virtual node during the state transition process of the dockless bike-sharing system (see Figure 2). The digits in circles represent the changing number of bikes at the high-demand node. The number of bikes at a virtual node is \(K\) minus the circled number. Therefore, each circled number can be described as a state of the system. \(\alpha_{(0)},\alpha_{(1)},\cdots,\alpha_{(K-1)}\) and \(\beta_{(1)},\beta_{(2)},\cdots,\beta_{(K)}\) are the state transition rates. \(\{\alpha_{(m)},m=0,1,\cdots,K-1\}\) means the rate of returning one bike, which is picked up at one of the nodes from the virtual node, at node \(i\). \(\{\beta_{(m+1)},m=0,1,\cdots,K-1\}\) means renting one bike at node \(i\) and returning it at one of the other nodes except node \(i\).