## 5 Discussion  
By analyzing the results of Markovian queueing network, we can obtain the several interesting findings. Compared with the high-demand node, the number of bikes at the lower-demand node changes at a much slower rate. As a whole, we guess that the number of bikes at the higher-demand node is likely to be smaller than the number of bikes at each of the middle-demand nodes (if there are a relatively large number of middle-demand nodes) and is much smaller than the number of bikes at the lower-demand node. The bike distribution at the lower-demand node is not sensitive to the increase of the arrival rate at the higher-demand node, which means bikes at the higher-demand node are likely to be scattered at both the lower-demand node and some middle-demand nodes. It is suggested that the operators should pay more attention to the nodes with higher demands, where the number of bikes decreases rapidly, rather than the nodes with lower demands, where the number of bikes increases relatively slowly, although these all belong to two extremes of the nodes with different customer demand. When the total number of bikes in the system is fixed, the expected number of bikes at the low-demand node in the long run tends to decrease and its possible range becomes broader with the increase of the number of nodes and the decrease of the initial number of bikes at each node. It suggests that a large number of nodes in the network can reduce the influence of the unequal demands moderately and make the long-term bike distribution more balanced than a small number of nodes in the network. Meanwhile, a larger initial number of bikes at each node can lead to a more imbalanced distribution of shared bikes in the system without human intervention. Figure 7: Effect of revenue- and cost-related parameters and arrival rates. (**a**) The maximum profit after implementing the relocation strategy with respect to the optimal relocation frequency under the influence of \(c_{ini}\); (**b**) The maximum profit after implementing the relocation strategy with respect to the optimal relocation frequency under the influence of \(c_{re}\); (**c**) The maximum profit after implementing the relocation strategy with respect to the optimal relocation frequency under the influence of \(c_{p}\). (**d**) The maximum profit after implementing the relocation strategy with respect to the optimal relocation frequency under the influence of \(\frac{\lambda}{\lambda_{i}}\). The rebalance problem is solved by an operator-based rebalance strategy based on the proper utilization of probabilistic results provided by the theoretical models. The main objective is to maximize the total profit and get the optimal relocation frequency. It is found that the relationship between the total profit and the relocation frequency takes the shape of a concave curve, which makes it quick to find the global optimal point by taking derivative of the expression. There are various key parameters related to the optimization problem (i.e., the indexes which can measure the effectiveness of each relocation \(c\) and \(\mu\), the unit price \(c_{in}\), the unit cost of relocation \(c_{re}\), the unit cost of unmet demands \(c_{p}\) and the different ratios of arrival rates). It is common sense from a business perspective that operators prefer the solution which can bring more profits and requires a low frequency of relocating bikes. The numerical experiments show that small values of \(c\) and \(c_{p}\) and large values of \(\mu\) can fit the preference. A small value of \(c\) and a large value of \(\mu\) can reduce the probability of having zero bikes at the high-demand node, which means an effective relocation strategy can result in a low frequency of relocation and increase the total profit mainly by saving the total cost. When the value of \(c\) is small, the maximum profit is more sensitive to the change of the optimal relocation frequency, while, when the value of \(\mu\) is large, the maximum profit is less sensitive to the change of the optimal relocation frequency.