# Priority Pricing for Efficient Resource Usage of Mobile Internet Access
## 5 Pricing Optimality Analysis
### Example 1  
This example considers a network with two service levels (\(I=2\)). There are 10 types of jobs (\(J=10\)), all of which have linear utility functions with a common maximum value \(\beta\) and a different delay cost \(\alpha_{j}\) for each job type (i.e., \(U_{j}(T)\;=\;\beta-\alpha_{j}T\)). \(\beta\) is 10, and \(\alpha_{1}=1.0\), \(\alpha_{2}=1.2\),..., \(\alpha_{10}=3.0\). The total job arrival rate \(\lambda\) was set to 5.0, and the job service rate \(\mu\) was set to 6.0. The size of the job is set to 1.0, for all job types (\(C_{j}=1.0\)), and the base service price \(P_{B}\) is set to 1.0. These parameters are designed to roughly mimic 5G mobile Internet access when users concurrently download big files from a server at maximum speed. In this setting, each job is accessing a server, and users are charged by the number of jobs they submit. According to empirical measurements in Korea [40], the average download speed of 5G mobile networks is approximately 700 Mbps and the average capacity of a base station is approximately 4.5 Gbps. Therefore, setting \(\mu\) to 6.0 corresponds to the total capacity of 4.2 Gbps, which is about the capacity of a base station. Figure 8 illustrates the results of search space pruning in this setting. The dotted area indicates the remaining search space after pruning, and the four lines indicate the boundary of each region defined by the four pruning rules. Rule 1 prunes out \(P_{1}\) larger than \(\min[U_{j}\big{(}C_{j}/\mu\big{)}/C_{j}]\), which is \(U_{10}(1/6)\;=\;10-3/6\). As a result, the \(P_{1}\) upper bound defined by Rule 1 is 9.5. Rule 2 ensures that the pruning of \(P_{2}\) is larger than \(P_{1}\). Rule 3 prunes \(P_{1}\) up to \(\min\Big{[}U_{j}\Big{(}T_{1j}^{*}\Big{)}/C_{j}\Big{]}\). The utility function \(U_{j}\Big{(}T_{1j}^{*}\Big{)}\) is the smallest when \(j=10\). As a result, the lower bound of \(P_{1}\) defined by Rule 3 is 7.0. Rule 4 computes the lower bound of \(P_{2}\) for a given \(P_{1}\). For example, when \(P_{1}\) is 9.0, \(\min\Big{[}U_{j}\Big{(}T_{2j}^{*}\Big{)}/C_{j}\Big{]}\) is 7.0 and \(\min\Big{[}U_{j}\Big{(}T_{2j}^{*}\Big{)}/C_{j}-U_{j}\big{(}C_{j}/\mu\big{)}/C_ {j}+P_{1}\Big{]}\) is (7.0 \(-\) 9.5 + 9.0). Therefore, the corresponding lower bound of \(P_{2}\) is 6.5, when \(P_{1}\) is 9.0. The effectiveness of pruning is determined by the diversity of the utility functions. In general, the wider the distribution of utility functions, the larger the remaining region after pruning. Figure 9 compares the size of the remaining search space after pruning for two different \(\alpha_{j}\) ranges. Except the \(\alpha_{j}\) range, the setting is the same as that in Figure 8.  
Now, let us examine how the system behaves in the pruned zone and in the remaining zone. Assume that the system objective is maximizing the system profit, so that the system objective function is \((P_{1}\lambda_{1}+P_{2}\lambda_{2}-P_{B}\mu)\). SAM is used to obtain the job arrival rate vector for a given price vector. Let us set the frequency of \(\{\lambda_{i}\}\) updates to 10 times per second. According to Rule 3, all \(P_{1}\)'s smaller than 7.0 belong to the pruned zone and those larger than 7.0 belong to the remaining zone. Figure 10 illustrates the system behavior of the former case, and Figure 11 illustrates that of the latter case. Figure 10 illustrates that for the price vectors belonging to the pruned zone, the system always converges on either a scenario that all jobs choose level 1 or a scenario that all jobs choose level 2. For instance, in the case of \(P_{1}\) = 7.0, if \(P_{2}\) is smaller than 5.0, all jobs choose level 2. If \(P_{2}\) is larger than 5.0, all jobs choose level 1. The system objective value (i.e., the system profit) in the case that all jobs choose level 1 is \((P_{1}\lambda_{1}-P_{B}\mu)\). The profit in the case that all jobs choose level 2 is \((P_{2}\lambda_{2}-P_{B}\mu)\). The highest system objective value for the pruned zone was (7.0 \(\times\) 5.0 \(-\) 1.0 \(\times\) 6.0) = 29.0. Figure 8: Search space pruning in example 1.