## 3 The Model
### Characteristics of the Subscribed Customers  
For a customer subscribed to data plan \(i\), we introduce a vector \((U_{i},U_{i}^{f})\), where \(U_{i}\) is his/her total data usage (including full-speed data usage and restricted-speed data usage) in a period and \(U_{i}^{f}\) is his/her full-speed data usage in a period. In each period, the subscribed customer consumes data continuously throughout the entire period. Therefore, the vector \((U_{i},U_{i}^{f})\) is random at the beginning of the period and realized at the end of the period. We model two key characteristics of the subscribed customers' behavior. First, a customer's total data usage does not necessarily equal her/his data demand forecast, despite the fact that the customer chooses the data plan based on her/his data demand forecast. Generally, the customer's total data usage fluctuates around the volume of full-speed data in her/his data plan. This is reasonable because if the customer's total data usage is far below \(v_{i}\), then she/he should have subscribed to a \"low-priced\" data plan. If the customer's total data usage is far beyond \(v_{i}\), then she/he should have subscribed to a \"high-priced\" data plan. Therefore, we assume that the expected volume of the customer's total data usage equals the volume of full-speed data in her/his data plan; that is, \(\mathbb{E}[U_{i}]=v_{i}\). In addition, we assume that the total data usage of all customers subscribed to plan \(i\) are distributed independently and identically with a cumulative distribution function \(F_{i}(U_{i})\). Moreover, \(F_{i}(U_{i})\) is homogeneous over periods. Second, if a subscribed customer wants to consume more full-speed data than the allotted volume within his/her data plan, he/she purchases the supplementary data package repeatedly to maintain full speed throughout the entire period. This assumption is reasonable because the marginal price for any supplementary data package is constant. We can classify the subscribed customers of data plan \(i\) into three categories: \(A\), \(B\), and \(C\). For a customer of category \(A\), her/his total data usage does not exceed the allotted full-speed data \(v_{i}\). For a customer of category \(B\), the total data usage exceeds \(v_{i}\), but he/she does not purchase the supplementary data package. For a customer of category \(C\), the total data usage exceeds \(v_{i}\), and she/he purchases the supplementary data package \(i\) repeatedly. The customers of category \(A\) and category \(C\) use data at full speed throughout the entire period. However, the customers of category \(B\) use the restricted-speed data service, which leads to customer dissatisfaction. An overview of the three categories is given in Table 2.  
Figure 3: Decisions of the potential customers. Let \(S_{i,t}\) be the number of customers subscribed to data plan \(i\) at the beginning of period \(t\). Let \(S_{i,t}^{A}\), \(S_{i,t}^{B}\), and \(S_{i,t}^{C}\) be the number of customers in the \(A\), \(B\), and \(C\) categories, respectively. By definition, we have \(S_{i,t}=S_{i,t}^{A}+S_{i,t}^{B}+S_{i,t}^{C}\). In addition, we define \(w_{i}=S_{i,t}^{C}/(S_{i,t}^{B}+S_{i,t}^{C})\), which reflects the customers' willingness to purchase supplementary data packages. We assume that \(w_{i}\) is constant over periods. In period \(t\), the customers of both \(A\) and \(B\) categories pay \(b_{i}\). The service provider's revenue generated from the customers of category \(A\) and category \(B\) is:  
\[\sum_{i=1}^{m}(S_{i,t}^{A}+S_{i,t}^{B})\cdot b_{i}.\]  
To formulate the revenue generated from the customers of category \(C\), we need to further differentiate the data usage. We define \(k\) (\(k\in\mathbb{N}^{+}\)) as an integer that satisfies \(v_{i}+(k-1)\cdot v_{i}^{s}<U_{i}\leq v_{i}+k\cdot v_{i}^{s}\), where \(k\) denotes the number of supplementary data packages a category \(C\) customer with data usage \(U_{i}\) purchases in a single period.