# Priority Pricing for Efficient Resource Usage of Mobile Internet Access
## 4 Convergence Analysis
### Case I  
In the profit center pricing structure model [22], one type of job (\(J=1\)) with a unit size (\(C=1\)) exists in a system with a single service level (\(I=1\)). Thus, all jobs share the same utility function and choose to join or not join the system. Because a single service level exists, the effective job arrival rate \(\lambda^{*}\) is equal to the job arrival rate at level 1, \(\lambda_{1}\). The system objective function, which is the expected profit per time unit, is equal to \(\lambda^{*}P_{1}-\mu P_{B}\), where \(P_{B}\) is the base service price of the system. An iso-elastic marginal value function, \(V^{\prime}(\lambda^{*})\ =\ K/\sqrt{\lambda^{*}}\), was assumed, and that makes the utility function \(U_{j}(T)\ =\ \frac{K}{\sqrt{\lambda_{j}^{*}}}-v_{j}T\). The equilibrium model simultaneously finds the optimal price (\(P_{1}\)) and optimal network capacity (\(\mu\)) by computing the conditions for a Nash equilibrium. A Nash equilibrium is satisfied when jobs are indifferent to joining or not joining the system. That is, the condition of \(V^{\prime}(\lambda^{*})-v_{1}T_{11}\ =\ P_{1}\) is satisfied at the equilibrium state. In the equilibrium state, the system objective function becomes \(\lambda^{*}[V^{\prime}(\lambda^{*})-v_{1}T_{11}]-\mu P_{B}\). The first-order conditions for maximizing the objective function are as follows. \[V^{\prime}(^{*})+V^{\prime}(^{*})\ =\ (v_{1}/\mu)\ f^{\prime}(\lambda^{*}/\mu), \tag{9}\]  
\[P_{B}\ =\ v_{1}\Big{(}\lambda^{*}/\mu^{2}\Big{)}\ f^{\prime}(\lambda^{*}/\mu), \ \text{where}\ f(\rho)\ =\ \rho/(1-\rho)\ \text{and}\ \rho\ =\ \lambda^{*}/\mu, \tag{10}\]  
Two equations are generated when applying the assumed equation \(V^{\prime}(\lambda^{*})=K/\sqrt{\lambda^{*}}\), which are \(K^{2}\ =\ 4v_{1}P_{B}f^{\prime}(\rho)\) from Equation (9), and \(\mu\ =\ K^{2}\rho/\big{(}4P_{B}{}^{2}\big{)}\) from Equation (10), respectively. The optimal system utilization rate, \(\rho\), was computed from the former and the optimal \(\mu\) from the latter. The optimal job arrival rate, \(\lambda^{*}\), is derived from \(\rho\) and \(\mu\) (\(\rho\ =\ \lambda^{*}/\mu\) in Equation (10)). The optimal \(P_{1}\) is derived from the equilibrium condition by applying the \(T_{11}\) value computed from Equation (2). Readers may refer to [22] for more details. This case compares the job arrival rates from this equilibrium model with those obtained from the SAM while varying the system settings. I tested two groups of system settings: one with high-capacity networks (i.e., large \(\mu\)) and frequent \(\{\lambda_{i}\}\) updates (100 times per second, i.e., \(F=100\)), and the other with low-capacity networks (i.e., small \(\mu\)) and infrequent \(\{\lambda_{i}\}\) updates (once or twice per second, i.e., \(F=1\) or \(F=2\)). The comparison results are summarized in Tables 1 and 2. The equilibrium model takes \(P_{B}\), \(K\), and \(v_{1}\) as input parameters and generates the optimal price, \(P_{1}\), the optimal capacity, \(\mu\), and the optimal job arrival rate, \(\lambda_{1}\), at the equilibrium state. Using the same \(P_{B}\), \(K\), and \(v_{1}\), the SAM takes \(P_{1}\) and \(\mu\) computed by the equilibrium model as inputs. The total job arrival rate \(\lambda\) was set to \(0.99\mu\). The key comparison target is the job arrival rate, \(\lambda_{1}\), which is always a single value when computed by the equilibrium model and indicates convergence in an equilibrium state. By contrast, the SAM may generate a set of \(\lambda_{1}\) values in the potential oscillation loop. For example, \(\lambda_{1}=[73.8,74.7]\) indicates that the system oscillates between two states: one with \(\lambda_{1}=73.8\), and the other with \(\lambda_{1}=74.7\). As expected, Table 1 shows that the results of the equilibrium method are far off the with those of the SAM in large-capacity networks that have frequent arrival rate updates.