## 5 Numerical Evaluation
### Sensitivity Analysis  
To obtain a sense of which parameters affect the service provider's revenue most, we conduct a sensitivity analysis. In the following, we list the two most significant results and then derive some managerial suggestions based on our analysis. Figure 6 shows how the SP's revenue changes with network capacity. For both APAO and DPC, the network capacity \(\tau\) takes values over [2.05,2.35], and the other parameters are set the same as in the base case. As network capacity changes, the SP's revenue changes under both DPC and APAO, but the revenue under DPC is more robust than that under APAO. This is because network congestion occurs only under APAO. The service provider using DPC can take the limited network capacity into consideration and change the control policy accordingly. In addition, we note that the revenue under APAO is not necessarily continuous in network capacity. This can be explained by the shifts in network congestion from one period to another. Take the jump at \(\tau=2.13\) for instance. When \(\tau\) takes values over \([2.05,2.13)\), network congestion continues to occur in Period 2. Within the interval \([2.05,2.13)\), the small change in network capacity is not enough to shift network congestion. However, when \(\tau\) exceeds 2.13 (and \(<\)2.20), network congestion shifts from Period 2 to Period 3. Note again that network congestion pulls down the SP's total market share and, in turn, decreases revenue. Therefore, the later network congestion occurs, the less the loss in revenue is. Analogously, as \(\tau\) exceeds 2.20 (and \(<\)2.25), network congestion shifts from Period 3 to Period 4, etc. As a result, the curve for APAO jumps from one segment to another and is not continuous in network capacity. Figure 5: Trends of the market share under APAO and DPC. Figure 7 shows how the SP's revenue is affected by potential customers' willingness to join the network. For both APAO and DPC, the willingness \(\lambda\) takes values over [5.8%,11%], and the other parameters are set the same as in the base case. Again, DPC provides more robust revenues than APAO. As one might expect, revenues should be monotonically increasing in the willingness \(\lambda\) under both APAO and DPC. To our surprise, this occurs only for the SP using DPC. In Figure 6, we observe the non-monotonicity of the curve of APAO: revenue is increasing in each segment of the curve, but decreasing from one segment to another. As \(\lambda\) increases, additional potential customers join the network in each period. When \(\lambda\) takes values over [5.8%,6.2%), revenue increases in \(\lambda\) because network congestion continues to occur in Period 6. However, when \(\lambda\) exceeds 6.2% (and \(<\)6.6%), network congestion does not occur in Period 6, but shifts from Period 6 to Period 5. The earlier network congestion occurs, the greater the loss in revenue is. Analogously, as \(\lambda\) exceeds 6.6% (and \(<\)7.1%), network congestion shifts from Period 5 to Period 4, etc. Therefore, the curve of APAO drops from one segment to another, which explains the non-monotonicity of the curve of APAO. This leads to an interesting finding: a service provider using DPC can always benefit from an increase in potential customers' willingness to join the network, but a service provider using APAO cannot. One practical implication is that the dynamic plan control method can help service providers increase their revenue. Specifically, based on the results of the sensitivity analysis, it is highly recommended that mobile service providers with small network capacity use DPC. Small network capacity makes the network more prone to congestion. By using DPC, the service provider allocates the limited network capacity in a more reasonable way and thus avoids network congestion as much as possible. service providers faced with a high level of potential customers' willingness (usually service providers who have a good reputation and use advertising) will benefit more by utilizing DPC than other service providers.