# Priority Pricing for Efficient Resource Usage of Mobile Internet Access
## 4 Convergence Analysis
### Case III  
In Case III, the equilibrium model used in Case II was extended to deal with multilevel systems with multiple job types [23]. In the extended model, the system objective is to maximize the system net value for a given system capacity \(\mu\). The objective function is expressed as \(\sum\limits_{j\in J}V_{j}\Big{(}\lambda_{j}^{*}\Big{)}-\sum\limits_{j\in J}v _{j}L_{j}\Big{(}\Big{\{}\lambda_{j}^{*}\Big{\}}\Big{)}\), where \(\lambda_{j}^{*}\) is the arrival rate of type-\(j\) jobs, \(V_{j}\Big{(}\lambda_{j}^{*}\Big{)}\) is the value function of type-\(j\) jobs, \(v_{j}\) is the delay cost of type-\(j\) jobs, \(\Big{\{}\lambda_{j}^{*}\Big{\}}\) denotes a \(j\)-dimensional arrival rate vector of type-\(j\) jobs, and \(L_{j}\Big{(}\Big{\{}\lambda_{j}^{*}\Big{\}}\Big{)}\) is the mean number of type-\(j\) jobs within the system in a steady state. Note the difference between the arrival rate of type-\(j\) jobs \(\lambda_{j}^{*}\) and the job arrival rate at level \(i\,\lambda_{i}\). For example, in a system with two service levels (\(I=2\)) with two job types (\(J=2\)), the first-order conditions for maximizing the system objective are as follows:  
\[V_{1}^{\prime}(\lambda_{1}^{*}\ )\ =\ v_{1}(\partial L_{1}/\partial\lambda_{1}^{*})+v _{2}(\partial L_{2}/\partial\lambda_{1}^{*}), \tag{13}\]  
\[V_{2}^{\prime}(\lambda_{2}^{*}\ )\ =\ v_{1}(\partial L_{1}/\partial\lambda_{2}^{*})+v _{2}(\partial L_{2}/\partial\lambda_{2}^{*}). \tag{14}\]  
From these equations, we can compute the optimal job arrival rates of type-\(1\) and type-\(2\) jobs, \(\lambda_{1}^{*}\) and \(\lambda_{2}^{*}\), for the given value functions, \(V_{1}\big{(}\lambda_{1}^{*}\ \big{)}\) and \(V_{2}\big{(}\lambda_{2}^{*}\ \big{)}\). Once the optimal \(\Big{\{}\lambda_{j}^{*}\Big{\}}\) is obtained, the optimal prices, \(P_{1}\) and \(P_{2}\), can be computed from the equilibrium conditions of \(V_{1}^{\prime}\big{(}\lambda_{1}^{*}\ )\ =\ P_{1}+v_{1}W_{1}\) and \(V_{2}^{\prime}\big{(}\lambda_{2}^{*}\ )\ =\ P_{2}+v_{2}W_{2}\), where \(W_{j}\) is the expected completion delay for a type-\(j\) job. At this point, the utility function becomes \(U_{j}(T)\ =\ P_{j}\ +\ v_{j}(W_{j}-T)\). I assume that \(v_{1}\) is greater than \(v_{2}\) and \(C_{1}=C_{2}=1\). Then, \begin{table}
\begin{tabular}{c c c c} \hline \hline \((P_{B},K,\,v_{1})\) & **Equilibrium Model** & **SAM (F = 2)** & **SAM (F = 1)** \\ \hline \((1.5,\,10,\,1)\) & \(P_{1}=1.5,\,\,\mu=39.0\) & \(\lambda_{1}=34.2\) & \(\lambda_{1}=[38.6,\,19.3]\) & \(\lambda_{1}=[38.6,\,0.0]\) \\ \((1.5,\,10,\,3)\) & \(P_{1}=1.5,\,\,\mu=35.0\) & \(\lambda_{1}=27.6\) & \(\lambda_{1}=[34.7,\,17.3]\) & \(\lambda_{1}=[34.7,\,0.0]\) \\ \((2.0,\,20,\,1)\) & \(P_{1}=2.0,\,\,\mu=92.9\) & \(\lambda_{1}=86.4\) & \(\lambda_{1}=[92.0,\,46.0]\) & \(\lambda_{1}=[92.0,\,0.0]\) \\ \((2.0,\,20,\,3)\) & \(P_{1}=2.0,\,\,\mu=87.8\) & \(\lambda_{1}=77.0\) & \(\lambda_{1}=[86.9,\,43.5]\) & \(\lambda_{1}=[86.9,\,0.0]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Small capacity networks with infrequent arrival rate broadcast (Case II). \begin{table}
\begin{tabular}{c c c c} \hline \hline \((P_{B},K,\,v_{1})\) & **Equilibrium Model** & **SAM (F = 100)** \\ \hline \((0.5,\,10,\,1)\) & \(P_{1}=0.5,\,\mu=371.7\) & \(\lambda_{1}=345.4\) & \(\lambda_{1}=[345.3]\) \\ \((0.5,\,10,\,3)\) & \(P_{1}=0.5,\,\mu=351.0\) & \(\lambda_{1}=308.0\) & \(\lambda_{1}=[305.1,\,308.6]\) \\ \((1.0,\,20,\,1)\) & \(P_{1}=1.0,\,\mu=380.0\) & \(\lambda_{1}=361.0\) & \(\lambda_{1}=[360.6]\) \\ \((1.0,\,20,\,3)\) & \(P_{1}=1.0,\,\mu=365.4\) & \(\lambda_{1}=333.7\) & \(\lambda_{1}=[332.2]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Large capacity networks with frequent arrival rate broadcast (Case II). owing to the incentive-compatibility [23], type-1 jobs will always prefer level 1 over level 2, whereas type-2 jobs always prefer level 2 over level 1. In other words, \(\lambda_{1}^{*}~{}=~{}\lambda_{1}\) and \(\lambda_{2}^{*}~{}=~{}\lambda_{2}\).