## 3 Models and Preliminaries
### Main Idea  
We then work out our strategy for dispatching task allocation after model definition and develop the corresponding performance evaluation method. First, some assumptions are given as follows regarding the dispatching model:  
* It ignores the time gap between arrival and departure of a bicycle at time \(t\), and assumes that the departure time of a bicycle at a travel station is unified at the beginning of time \(t\), and then reach the destination at the end of time \(t\), during which the bicycle is independent of all the stations.
* It assumes that, during manual dispatching, the bicycle leaves the travel point at the beginning of time \(t\), and then arrives at the destination at the next moment.
* It assumes that the revenue and cost of the operator only come from three parts: the charge for bicycle riding, the cost for manual dispatching, and the cost for user incentive dispatching.
* It assumes that the dispatching of shared bicycles is only carried out within the research region, and will not involve places outside the research region.  
The main idea of this paper is to use a reasonable dispatching task allocation strategy to rebalance as many stations as possible. In order to achieve the above goal, we introduce the concept of overall station imbalance \(IMB\), which indicates the gap of bicycle's supply and demand within all stations and the smaller value represents the more balanced bicycle distribution in IoSB. The specific definition is as Formula (1):  
\[IMB(t)=\frac{1}{N}\sum_{i=1}^{N}|imb_{i}(t)| \tag{1}\]  
For optimizing the \(IMB\), the most important one is the prediction of the demand at the next moment. Trip record is a piece of data that records the starting point, ending point, time and other information of a trip. In our model, a quad Items is utilized to represent each trip record. The truth is that the trip record is a type of time series, which is an important branch of statistics. In this paper, we decided to use the ARMA model to predict the demand at the station for its prediction efficiency and the stability of time series. In the ARMA model, time series \(Y_{t}\) is a linear function of random error term of its current and early period and its early period value, which can be expressed as:  
\[Y_{t}=\phi_{1}Y_{t-1}+\phi_{2}Y_{t-2}+...+\phi_{p}Y_{t-p}+u_{t}-\theta_{1}u_{t -1}-...-\theta_{q}u_{t-q} \tag{2}\]  
Finally, in order to evaluate the performance of the final dispatching model, two indicators are defined, which are total revenue for operators \(B\) and Users' satisfaction \(\mathbf{\alpha}\). The indicator \(B\) is defined as Formula (3), which represents the operator's net income:  
\[B=c_{1}\cdot cost_{t}-c_{2}\cdot cost_{m}-c_{3}\cdot cost_{u} \tag{3}\]  
where \(c_{1}\) is total number of cycling orders within the observation period \([0,T-1]\); \(c_{2}\) is the total number of manual dispatching; \(c_{3}\) is the total number of user incentive dispatching. \(cost_{t}\) is the revenue of operator for single trip; \(cost_{m}\) is the total cost of manual dispatching; \(cost_{u}\) is the total cost of user incentive dispatching.  
Users' satisfaction \(\mathbf{\alpha}\) is used to describe the extent which cycling needs are met in IoSB. In our work, we calculate the users' satisfaction \(\mathbf{\alpha}\) via the value of the percent of successful rental in all trip records. When there is no bicycle at the station where the lease order occurs, the trip order is deemed to be failed.