# Priority Pricing for Efficient Resource Usage of Mobile Internet Access
## 5 Pricing Optimality Analysis
### Example 3  
This example compares the algorithm with another equilibrium method (Case II in Section 4). Case II uses a net-value maximization scheme with a single service level. As in Example 2, the comparison targets are the price vector and the expected system objective value. The comparison results are summarized in Tables 8 and 9. \(\mathcal{R}_{theory}\) is the system objective value (i.e., the system net value or social welfare) computed at the optimal equilibrium state as follows:  
\[\mathcal{R}_{theory}\ =\ V(\lambda^{*})-v_{1}L-\mu P_{B},\] \begin{table}
\begin{tabular}{c c c} \hline \hline \((\textbf{P}_{B},\textbf{K},\textbf{v}_{1},\textbf{\mu})\) & **Equilibrium Method** & **Heuristic Search** \\ \hline \((0.5,10,1,85.9)\) & \(P_{1}=1.08\), \(\mathcal{R}=36.9\), \(\mathcal{R}_{theory}=36.9\) & \(P_{1}=1.10\), \(\mathcal{R}=37.1\) \\ \((0.5,10,3,75.5)\) & \(P_{1}=1.16\), \(\mathcal{R}=28.1\), \(\mathcal{R}_{theory}=28.5\) & \(P_{1}=1.15\), \(\mathcal{R}=29.0\) \\ \((1.0,20,1,90.0)\) & \(P_{1}=2.11\), \(\mathcal{R}=81.0\), \(\mathcal{R}_{theory}=81.0\) & \(P_{1}=2.15\), \(\mathcal{R}=81.8\) \\ \((1.0,20,3,82.7)\) & \(P_{1}=2.21\), \(\mathcal{R}=67.4\), \(\mathcal{R}_{theory}=68.4\) & \(P_{1}=2.25\), \(\mathcal{R}=68.9\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Large capacity networks with frequent arrival rate broadcast (\(F=100\)). \begin{table}
\begin{tabular}{c c c} \hline \hline \((\textbf{P}_{B},\textbf{K},\textbf{v}_{1},\textbf{\mu})\) & **Equilibrium Method** & **Heuristic Search** \\ \hline \((1.5,10,1,8.4)\) & \(P_{1}=3.49\), \(\mathcal{R}=6.8\), \(\mathcal{R}_{theory}=9.5\) & \(P_{1}=4.15\), \(\mathcal{R}=10.47\) \\ \((1.5,10,3,6.4)\) & \(P_{1}=4.11\), \(\mathcal{R}=3.45\), \(\mathcal{R}_{theory}=5.5\) & \(P_{1}=5.50\), \(\mathcal{R}=7.86\) \\ \((2.0,20,1,21.5)\) & \(P_{1}=4.33\), \(\mathcal{R}=33.8\), \(\mathcal{R}_{theory}=36.9\) & \(P_{1}=4.60\), \(\mathcal{R}=38.6\) \\ \((2.0,20,3,18.9)\) & \(P_{1}=4.65\), \(\mathcal{R}=20.4\), \(\mathcal{R}_{theory}=28.5\) & \(P_{1}=5.55\), \(\mathcal{R}=31.6\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Small capacity networks with infrequent arrival rate broadcast (\(F=5\)). where \(L\) is the mean number of jobs, and \(\lambda^{*}\) is the job arrival rate in the equilibrium state. \(\mathcal{R}\) is the system objective value computed from the SAM result using the following equation:  
\[\mathcal{R} = \sum_{\{S\}}\bigl{(}W(S)\lambda^{*}(S)\bigl{[}V^{\prime}(\lambda^{* }(S))-v_{1}T(S)\bigr{]}\bigr{)}, \tag{17}\]  
where \(\{S\}\) denotes the set of recurring states in the SAM transition graph, \(W(S)\) is the stationary probability of state \(S\), \(\lambda^{*}(S)\) is the job arrival rate in state \(S\), \(V^{\prime}\) is the marginal value function, and \(T(S)\) is the average job completion time in state \(S\). Table 8 offers the same observations as those in Table 6. The equilibrium method accurately models the actual system behavior in large capacity networks with frequent arrival rate updates, and as a result, the outcome of the algorithm matches that of the equilibrium method. Table 9 is similar to Table 7, with a few differences. First, \(\mathcal{R}\) of the equilibrium method is lower than \(\mathcal{R}_{theory}\). Unlike in Example 2 (the profit-maximizing network), \(\mathcal{R}\) of the algorithm is also lower than \(\mathcal{R}_{theory}\). This is because the sub-optimal decisions of users, owing to inaccurate knowledge of the system status, do not increase the system objective value in the net-value-maximizing network. Nonetheless, the algorithm results in higher \(\mathcal{R}\) values than the equilibrium method and achieves a system objective closer to the theoretical maximum value \(\mathcal{R}_{theory}\). Second, the sensitivity to price is generally lower than that in the profit-maximizing network because the price that generates the highest \(\mathcal{R}\) is not a single point but rather a range of prices. This occurs because the system objective function is not directly affected by the price (see Equation (17)).