## 3 Methodology  
### Markovian Queueing Networks  
#### 3.1.1 Assumptions and Notations  
The dockless bike-sharing system is modeled by a closed Markovian queueing network with \(N\in\mathbb{N}^{*}\) connected nodes and a total of \(K\in\mathbb{N}^{*}\) bikes. The bikes are distributed among the nodes. \(\mathcal{N}=\{1,2,\cdots,N\}\) and \(\mathcal{K}=\{1,2,\cdots,K\}\) are the sets of \(N\) nodes and \(K\) bikes, separately. Customers arrive at node \(i\in N\) to pick up bikes to ride and return bikes at node \(j\in N\). Let \(p_{ij}\) be the transition probability from node \(i\in N\) to node \(j\in N\). All of the nodes are fully connected because of the customers' bike trips. The transition probabilities can reflect the information of customers' destinations as well as geographical environment. If the route between two nodes is long and steep, there will be a low probability of riding bikes between these two nodes. High demand leads to a large value of the arrival rate. The main assumptions are made as follows:  
1. Customers arrive at a node one by one for picking up bikes rather than arriving in groups.
2. The inter-arrival times of customers are exponentially distributed (i.e., the number of customer arrivals within a unit time interval is Poisson distributed) with an arrival rate \(\lambda_{i}\) at node \(i\), and the arrivals at each node are completely independent.
3. All of the probabilities are the same for each route from a start node to a destination node (which can also be the start node itself), which means \(p_{ij}=\frac{1}{N}\) for every pair of \(i\) and \(j\).
4. The time spent on picking up or dropping off bikes is negligible, which means customers do not have to wait in lines at any node in the network, and the trip time is not considered as an independent parameter in the theoretical models as our focus is the steady states in the long run, but it is included in the simulation model.
5. The number of bikes is evenly distributed at each node at the beginning of the operation, and the total number of bikes in the system is fixed.
6. The capacity of each node is large enough to accommodate \(K\) bikes.
7. If a node has no bike during a period of time, customers will still arrive randomly and leave immediately, and the service for these customers will be regarded as the lost demand.  
The notations are summarized in Table 1.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Notation** & **Units** & **Definitions** \\ \hline \(N\) & [nodes] & Total number of nodes in a system \\ \hline \(K\) & [bikes] & Total number of bikes in a system (\(K=\epsilon N\)) \\ \hline \(\epsilon\) & [bikes] & The initial number of bikes at each node in the system \\ \hline \(\lambda_{i}\) & [people/min] & Arrival rate at node \(i\in N\) \\ \hline \(\lambda\) & [people/min] & A certain value of the arrival rate \\ \hline \(\alpha_{(m)}\) & [people/min] & \
\begin{tabular}{c} Transition rate of returning one bike to node \(i\) by a customer riding from \\ one of the other nodes except node \(i\) (\(m=0,1,\cdots,K-1\)) \\ \end{tabular} \\ \hline \(\beta_{(m+1)}\) & [people/min] &
\begin{tabular}{c} Transition rate of renting one bike from node \(i\) to one of the other nodes \\ except node \(i\) (\(m=0,1,\cdots,K-1\)) \\ \end{tabular} \\ \hline \(\alpha\) & [people/min] & A certain value of the transition rate \\ \hline \(\beta\) & [people/min] & A certain value of the transition rate \\ \hline \(p_{l}\) & - & Probability of having \(l\) bikes at node \(i\) with no relocation (\(l=0,1,\cdots,K\)) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Overview of notations.