# Priority Pricing for Efficient Resource Usage of Mobile Internet Access
## 4 Convergence Analysis
### Case II  
The \"net-value maximization\" model [22] has the same system setting as in Case I, except for the objective function. That is, \(I=1\), \(J=1\), \(C=1\), \(V^{\prime}(\lambda^{*})\ =\ K/\sqrt{\lambda^{*}}\), and \(U_{j}(T)\ =\ \frac{K}{\sqrt{\lambda_{j}^{*}}}-v_{j}T\). Here, the system objective is to maximize the system net value, which can be expressed as \(V(\lambda^{*})-v_{1}L-\mu P_{B}\), where \(L\) is the mean number of jobs in the system in a steady state. In the M/M/1 model, \(L\ =\ f(\rho)\ =\ \lambda^{*}/(\mu-\lambda^{*})\). The procedure used to compute the Nash equilibrium is identical to that of Case I. The first-order conditions for maximizing the objective function are as follows.  
\[V^{\prime}(\lambda^{*})\ =\ v_{1}(\partial L/\partial\lambda), \tag{11}\]  
\[P_{B}\ =\ -v_{1}(\partial L/\partial\mu). \tag{12}\]  
By applying the assumed \(V^{\prime}(\lambda^{*})\), we obtain two equations, \(K^{2}\ =\ v_{1}P_{B}f^{\prime}(\rho)\) from Equation (11), and \(\mu\ =\ K^{2}\rho/P_{B}{}^{2}\) from Equation (12), respectively. As in Case I, \(\mu\) and \(\lambda_{1}\) are computed from these equations, and the optimal \(P_{1}\) is determined by applying \(\mu\) and \(\lambda_{1}\) to the equilibrium condition.  
The comparison results between the equilibrium method and the SAM are summarized in Tables 3 and 4, which both show a similar trend to Case I. With frequent arrival rate broadcasts in large-capacity networks, the system often converges to a steady state close to the equilibrium state (see Table 3). By contrast, Table 4 shows that the equilibrium  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \((P_{B},K,v_{1})\) & **Equilibrium Model** & **SAM (F = 100)** \\ \hline (0.5, 10, 1) & \(P_{1}=1.08\), \(\mu=85.9\) & \(\lambda_{1}=73.7\) & \(\lambda_{1}=[73.8,74.7]\) \\ (0.5, 10, 3) & \(P_{1}=1.16\), \(\mu=75.5\) & \(\lambda_{1}=57.0\) & \(\lambda_{1}=[56.6,57.4]\) \\ (1.0, 20, 1) & \(P_{1}=2.11\), \(\mu=90.0\) & \(\lambda_{1}=81.0\) & \(\lambda_{1}=[81.0,81.8]\) \\ (1.0, 20, 3) & \(P_{1}=2.21\), \(\mu=82.7\) & \(\lambda_{1}=68.4\) & \(\lambda_{1}=[67.8,68.6]\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Large capacity networks with frequent arrival rate broadcast (Case I).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \((P_{B},K,v_{1})\) & **Equilibrium Model** & **SAM (F = 2)** & **SAM (F = 1)** \\ \hline (1.5, 10, 1) & \(P_{1}=3.49\), \(\mu=8.4\) & \(\lambda_{1}=6.3\) & \(\lambda_{1}=[8.3,4.2]\) & \(\lambda_{1}=[8.3,0.0]\) \\ (1.5, 10, 3) & \(P_{1}=4.11\), \(\mu=6.4\) & \(\lambda_{1}=3.7\) & \(\lambda_{1}=[6.3,3.2]\) & \(\lambda_{1}=[6.3,0.0]\) \\ (2.0, 20, 1) & \(P_{1}=4.33\), \(\mu=21.5\) & \(\lambda_{1}=18.4\) & \(\lambda_{1}=[21.3,10.6]\) & \(\lambda_{1}=[21.3,0.0]\) \\ (2.0, 20, 3) & \(P_{1}=4.65\), \(\mu=18.9\) & \(\lambda_{1}=14.3\) & \(\lambda_{1}=[18.7,9.4]\) & \(\lambda_{1}=[18.7,0.0]\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Small capacity networks with infrequent arrival rate broadcast (Case I).  
method is not accurate when applied to small-capacity networks with infrequent arrival rate broadcasts.  
Note that the SAM results do not exactly match the equilibrium method results, even when the system converges to a steady state. This is not only because of the differences in the modeling assumptions (infinitesimal jobs and accurate knowledge of system status) but also because of the numerical round-off effects. For instance, \(\{\lambda_{i}\}\) broadcast values are truncated at five decimal places.  
I conclude from Case I and Case II that whether the system converges to an equilibrium state is independent of the system objective. In both cases, the users determine system behavior by making decisions based on their self-interest, which is oblivious to the system objective. If the conditions for equilibrium convergence are not met, the system can oscillate regardless of the system objective. Section 5 shows the evaluation of the price optimality computed using the equilibrium method for this case.