## 3 Methodology
### Markovian Queueing Networks
#### 3.1.3 A Markovian Queueing Network with Higher Demands and Lower Demands  
In reality, a bike-sharing system may include multiple nodes with complex and diverse demands from customers. To make the models more related to a real-world network, the Markovian queueing network model is extended in the paper to analyze the system under the influence of unequal demands, which include the higher demands and the lower demands. The reduced \(N\)-node network of the extension model is shown in Figure 3. Node \(i\) refers to a node with higher demands, and node \(j\) refers to a node with lower demands than the nodes (whose arrival rates are the same and represented by \(\lambda\)) inside a virtual node (i.e., \(\lambda_{j}\leq\lambda\leq\lambda_{i}\)). With the existence of a virtual node and two single nodes, a three-tuple state space \(\{(K,0,0),\cdots,(0,K,0),\cdots,(0,0,K)\}\)is used to represent the changing states of the whole system. Thanks to the fixed total number of bikes, a two-tuple state space \(\{(K,0),(K-1,1),\ldots,(1,K-1),(0,K)\}\) performs a similar function, which represents the number of  
Figure 3: A reduced \(N\)-node network with the same probabilities and unequal demands. bikes at nodes \(i\) and \(j\), separately (see Figure 4a,b). Each transition of the states means transferring one bike at a time. The six types of transition rates are defined as follows:  
\[\beta_{i(m_{1}+1,m_{2})}=\beta_{i}=\lambda_{i}\frac{N-2}{N}, \forall m_{1},m_{2} \tag{8}\] \[\beta_{j(m_{1},m_{2}+1)}=\beta_{j}=\lambda_{j}\frac{N-2}{N}, \forall m_{1},m_{2}\] (9) \[\alpha_{j}\beta_{i(m_{1}+1,m_{2})}=\frac{\lambda_{i}}{N}, \forall m_{1},m_{2}\] (10) \[\alpha_{i}\beta_{j(m_{1},m_{2}+1)}=\frac{\lambda_{j}}{N}, \forall m_{1},m_{2}\] (11) \[\alpha_{i(m_{1},m_{2})}=\frac{N-2}{N}\frac{\lambda(K-m_{1}-m_{2} )}{K-m_{1}-m_{2}+N-3}\] (12) \[\alpha_{j(m_{1},m_{2})}=\frac{N-2}{N}\frac{\lambda(K-m_{1}-m_{2} )}{K-m_{1}-m_{2}+N-3} \tag{13}\]  
Figure 4: The detailed state transition diagram. strategy we tend to adopt is to relocate the extra bikes from the relatively low-demand nodes (which have been aggregated into a virtual node in the network model with one high-demand node) to the high-demand node (which is represented as node \(i\) in the network model with one high-demand node) to reach the initial number of bikes at each node. The relocation strategy can be implemented as many times as needed during normal operation of the bike-sharing system, and the relocation frequency can reflect the number of times of relocation. In this paper, we assume that a more effective relocation strategy is able to reduce the value of \(p0\) more rapidly, which can be described as \(p^{{}^{\prime}}_{0}\propto cr^{-\mu}p_{0}(c>0,\mu>0)\). \(p^{{}^{\prime}}_{0}\) represents the steady-state probability of having zero bikes at node \(i\) under the influence of relocation. The formulas and descriptions are based on the Markovian queueing network model with higher demands. When operators implement the relocation strategy more frequently, \(p^{{}^{\prime}}_{0}\) decreases from a basic value of \(p_{0}\). To determine the relationship between the probabilities without relocation and the probabilities with relocation, we have \(p^{\prime}_{0}=cr^{-\mu}p_{0}\). When \(p^{{}^{\prime}}_{0}\) equals \(p_{0}\), \(cr^{-\mu}=1\), where the value of \(r\) represents no relocation is implemented in the system. \(r>c^{\frac{1}{\mu}}\) means the operators start to relocate bikes. Considering that the sum of all the probabilities equals 1, the other probabilities are assumed to increase proportionately to satisfy the constraint, which are given by  
\[p^{\prime}_{l}=Bp_{l},\quad\forall l=1,\cdots,K \tag{24}\]  
For \(\sum_{l=0}^{K}p^{\prime}_{l}=1\), we have  
Figure 3:0 (25) Figure 3:1  
The total profit after implementing the relocation strategy can be calculated by the following formula:  
Figure 3:2 Figure 3:3  
\(Y(r)\) means the total profit with the corresponding relocation frequency \(r\).