# Priority Pricing for Efficient Resource Usage of Mobile Internet Access
## 2 System Framework  
This study considers the price selection problem over a single-hop network link. The network offers different service levels to the users. Without loss of generality, let us call the highest priority level 1 and the lowest priority level \(I\). For service level \(i\), the service provider charges \(P_{i}\) per data unit. The service provider determines the price vector of \(I\) service levels, \(\{P_{i}\}\), with a certain objective, such as maximizing system profit or net value of the system. I assume that prices do not change once this vector is determined. However, multiple pricing policies can be implemented according to the various user and traffic scenarios using the suggested scheme, and wireless service providers may change the price policy accordingly. Let us call each user request to be transmitted a 'job.' The jobs are basically packets with QoS information. A user should negotiate with a base station to submit a job. He/She may decide not to submit the job if the cost is too expensive. When submitting a job, each user would essentially be requesting that the job be processed for the transmission. The submitted jobs are distributed to the priority queues by a classifier according to their types categorized by their QoS requirements. Here, the framework assumes that there are \(j\) types of QoS requirements, and each job is associated with QoS type \(j\). The scheduler chooses a job from the front of a non-empty priority queue according to its scheduling discipline and thrusts the chosen job into the sending queue. For example, a scheduler using a priority queuing (PQ) discipline serves the job with the highest priority first and uses all the bandwidths to send the jobs in the sending queue. By contrast, a scheduler using a weighted fair queuing (WFQ) discipline would allocate the bandwidths to each priority queue according to its weight [34]. Subsequently, the base station and the corresponding UE would prepare the radio resource to transmit the jobs. A scheduling model is shown in Figure 1.  
Although each job has a different size, performance requirements, and error characteristics, I consider that a job's completion delay is the main performance metric that determines its value. Thus, let us denote a job's utility function by \(U_{j}(T_{ij})\), where \(T_{ij}\) is the expected time to complete a type-\(j\) job when assigned to level \(i\). It is natural to assume that \(U_{j}(T_{ij})\) is a non-increasing function with respect to \(T_{ij}\). The proposed framework considers a non-preemptive strict priority scheduling (SPS) discipline, although the basic concept of this study is not tied to a particular scheduling discipline. Because no guarantee can be made on the exact completion time under SPS, this framework uses the expected completion time. The computation of \(T_{ij}\) depends on many factors, such as the job arrival process, job service process, and job size. For example, under  
Figure 1: PQ scheduling model. Poisson job arrival and exponential job length assumptions, \(T_{ij}\) with SPS can be computed using the \(M\)/\(M\)/\(I\) priority queuing model, as shown in Equation (1):  
\[T_{ij}\ =\ \frac{C_{j}}{\mu}+\frac{\frac{C^{2}\sum_{i=1}^{I}\lambda_{k}}{\mu^{2}} }{(1-\frac{C\sum_{k=1}^{I-1}\lambda_{k}}{\mu})(1-\frac{C\sum_{k=1}^{I}\lambda_ {k}}{\mu})} \tag{1}\]  
where \(\mu\) is the job service rate, \(\lambda_{k}\) is the job arrival rate for level \(k\), \(C_{j}\) is the average size of a type-\(j\) job, and \(C\) is the average size of all jobs. The mathematical expression \(\frac{C_{j}}{\mu}\) represents the job service time and \(\frac{C^{2}/\mu^{2}\cdot\sum_{k=1}^{I}\lambda_{k}}{(1-C/\mu\cdot\frac{\sum_{k= 1}^{I-1}\lambda_{k}}{\lambda_{k}})(1-C/\mu\cdot\frac{\sum_{k=1}^{I-1}\lambda_{k}} {\lambda_{k}})}\) represents the average waiting time of the job. Note that \(C\) and \(C_{j}\) are calculated from the empirical observations, although the job size has an exponential distribution. Readers may refer to [35] for the proof of Equation (1).