# Priority Pricing for Efficient Resource Usage of Mobile Internet Access
## 2 System Framework
The expected job completion time is computed assuming that the queues associated with each service level are in a steady state. This assumption is appropriate for reasonably long broadcast intervals. Broadcast intervals are discussed in the last paragraph of this section, and update times longer than the time required to adjust to the new steady state are often sensible. The user may adopt a formula that considers transient states. A user selects the service level of his jobs based solely on self-interest, and they are oblivious to the system objective. A user may decide not to submit a job, but he/she cannot change his/her decision once the job has been submitted. I assume that job arrival at each service level follows the Poisson process. The job arrival rate at service level \(i\) is denoted by \(\lambda_{i}\), and \(\{\lambda_{i}\}\) is the job arrival rate vector. For notational convenience, \(\lambda_{0}\) denotes the arrival rate of jobs that were not submitted. When the total job arrival rate is \(\lambda\), the sum of the job arrival rates at all levels is equal to \(\lambda\) (i.e., \(\sum\limits_{i\ =\ 0}^{I}\lambda_{i}\ =\ \lambda\)). The network serves jobs at a rate of \(\mu\), which is essentially determined by the capacity of the wireless network. The capacity of a wireless network is often variable, as the wireless channel condition fluctuates, and the link throughput depends on the channel condition. In such cases, existing methods for estimating wireless link capacity may be used [36; 37]. To achieve system stability, the effective job arrival rate \((\lambda-\lambda_{0})\) must be smaller than \(\mu\). I assume that the system status is periodically broadcast and that users make decisions based on the latest broadcast \(\{\lambda_{i}\}\) instead of the actual (but unknown) job arrival rates between the periodic updates. This assumption is reasonable for wireless networks, because all existing wireless access technologies use some form of periodic system broadcast channels (e.g., beacons). Keeping an accurate track of \(\{\lambda_{i}\}\) for each user by sniffing all of the network traffic is usually infeasible in wireless networks, and for each user to ask the network for the current \(\{\lambda_{i}\}\) whenever he has a job to send is undesirable for efficiency reasons. Because this study assumes that each user's decision is based solely on self-interest, a user with a type-\(j\) job is assumed to select the service level \(i\) that maximizes his utility value minus the associated cost, regardless of the system objective. The following expression captures the decision-making process of a user with a type-\(j\) job:  
\[Argmax_{i}\ [U_{j}(T_{ij})-P_{i}C_{j}]. \tag{2}\]  
If the utility value minus the cost is negative for all levels, the user will not submit the job. The user's decision depends on \(T_{ij}\) (in Equation (2)), and \(T_{ij}\) depends on the job arrival rates at higher levels (in Equation (1)). The job arrival rates, in turn, depend on the decisions of other users. This interdependency raises the possibility of system oscillation. Note that while pricing is static, the decisions made by users may be dynamic.