## 3 The Model  
A mobile service provider (SP) employs connection-speed-restriction pricing and offers \(m\) different data plans. The information about the data plans is pre-announced to the customers. We consider a market composed of a large number of customers. The customer population, denoted by \(N\), is deterministic. All data plans give customers unlimited data. The differences between the data plans are their prices and allotted volumes of full-speed data. We denote \(b_{i}\) as the price per period of data plan \(i\) and \(v_{i}\) as the allotted volume of full-speed data in data plan \(i\). To use the data service, a customer needs to subscribe to a data plan first. Connection-speed-restriction pricing implies that for a customer subscribed to data plan \(i\), his/her individual data speed will be restricted after he/she uses up the allotted volume of full-speed data \(v_{i}\) in his/her data plan. If the customer wants to re-obtain full-speed data connection, he/she needs to buy the supplementary data package or wait until the beginning of the next period. The supplementary data package (SDP) provides the customers with additional full-speed data. In addition, the purchase of the supplementary data package can be performed repeatedly as desired. According to the SP's arrangement, different data plans may offer different supplementary data packages. A customer can only buy the supplementary data package that is attached to his/her data plan. In this paper, we consider the situation in which each data plan offers only one type of supplementary data package. For the ease of exposition, we address the supplementary data package attached to data plan \(i\) as \"supplementary data package \(i\)\". We denote \(b_{i}^{s}\) as the price per purchase of supplementary data package \(i\) and \(v_{i}^{s}\) as the additional volume of full-speed data within supplementary data package \(i\). For example, if a customer subscribes to data plan \(i\) and wants to consume data volume \(v_{i}+3v_{i}^{s}\) at full speed in one period, then she/he needs to buy supplementary data package \(i\) three times, and her/his total cost in this period is \(b_{i}+3b_{i}^{s}\). An overview of the data plan parameters is given in Table 1, and the mechanism of connection-speed-restriction pricing is illustrated in Figure 2.  
For the ease of discussion, we sort all data plans by price; that is, \(b_{1}<\cdots<b_{i}<\cdots<b_{m}\). Then, the allotted volumes of full-speed data within all data plans satisfy \(v_{1}<\cdots<v_{i}<\cdots<v_{m}\). In addition, we make the following assumptions. **Assumption**. : _For all data plans and supplementary data packages, we have:_  
1. \(\dfrac{b_{1}}{v_{1}}\geq\cdots\geq\dfrac{b_{i}}{v_{i}}\geq\cdots\geq\dfrac{b_{ m}}{v_{m}}\)_,_
2. \(\dfrac{b_{1}^{s}}{v_{1}^{s}}\geq\cdots\geq\dfrac{b_{i}^{s}}{v_{i}^{s}}\geq \cdots\geq\dfrac{b_{m}^{s}}{v_{m}^{s}}\)_,_ \begin{table}
\begin{tabular}{c c c} \hline \hline Data plan \(i\) & \(v_{i}\) & = The allotted volume of full-speed data within data plan \(i\) \\ \hline \multirow{3}{*}{
\begin{tabular}{c} Supplementary data package \(i\) \\ \end{tabular} } & \(b_{i}\) & = The price per period of data plan \(i\) \\ \cline{2-3}  & \(v_{i}^{s}\) & = The additional volume of full-speed data gained by purchasing supplementary data package \(i\) \\ \cline{2-3}  & \(b_{i}^{s}\) & = The price per purchase of supplementary data package \(i\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the data plan and supplementary data package (SDP). Figure 2: Mechanism of connection-speed-restriction pricing. 3. \(b_{i+1}<b_{i}+\left\lceil\frac{v_{i+1}-v_{i}}{v_{i}^{s}}\right\rceil\cdot b_{i}^{s},\quad\forall i=1,2,\cdots,m-1\). These three assumptions are mild and easily satisfied in real practice. Assumptions 1 and 2 are straightforward: a high-priced data plan (and its corresponding supplementary data package) means a low unit price of full-speed data.