## 6 Numerical Experiments
### Impact of Domain Numbers  
In this part, the impact of different domain numbers is further evaluated in both the thin market (i.e., \(N=10\)) and the thick market (i.e., \(N=100\)). The motivation is to validate whether it would be more difficult to provide network slicing services involving more domains in the service chain route and whether the service shortcut route would become more favorable or not. Similarly to the previous part, to avoid an unfair comparison, the ask price of the service providers from the domains in the service chain route decreases accordingly when the domain number decreases to maintain a stable overall service cost.  
The evaluation of different domain numbers in the service chain route is shown in Figure 8. Both the ascending auction and the optimal trade mechanisms are hardly affected by the increasing domain numbers in the service chain route, as long as the service cost for using the two alternative routes remains at the same level. Intuitively, when more domains with a fixed ask price level are required to build the network slices, fewer users who are capable of affording the increasing service cost remain. Our result implies that the difficulty of providing multi-domain network slicing services does not rely on the domain number but the overall service cost along the service chain route.  
Figure 7: The comparison with the cases of different provider/user ratios (M/N): (**a**) the admission ratio (N = 10); (**b**) the average served bandwidth per user (N = 10); (**c**) the gain from trade per user (N = 10); (**d**) the admission ratio (N = 100); (**e**) the average served bandwidth per user (N = 100); and (**f**) the gain from trade per user (N = 100).  
Figure 9 is used to compare the user gain and service provider profit with different domain numbers in the service chain route. The gain from trade per use is decomposed into two parts, namely, the user gain and the service provider profit, in the results of the ascending auction mechanism. The user gain is defined as the gap between the user's value and the auction price. Similarly, the service provider profit is defined as the gap between the auction price and the service provider's value. For the second route, i.e., the service chain route, the sum of the service provider profit is used to compare with the profit of the first route. As shown in Figure 9, the service provider's profit increases with a larger domain number, and the increase part is filled with the decrease in the user gain. This implies that more gain from trade is transferred from the user side to the service provider side when it becomes more cumbersome to provide the network slicing service in the service chain route involving more domains. In other words, the service provider can earn more profit, even without increasing the overall service cost and impairing the overall performance in the ascending auction mechanism. Furthermore, this internal impact is more obvious in a thin market (i.e., Figure 9a), compared with a thick market (i.e., Figure 9b). In other words, in a thin market with only 10 users or service providers, the decrease in the user gain is more obvious with the increase in the domain number \(K\), compared to that in a thick market with 100 users or service providers.  
Figure 8: The comparison with the cases of different domain numbers in the service chain route: (**a**) the admission ratio (N = 10); (**b**) the average served bandwidth per user (N = 10); (**c**) the gain from trade per user (N = 10); (**d**) the admission ratio (N = 100); (**e**) the average served bandwidth per user (N = 100); and (**f**) the gain from trade per user (N = 100).  
Figure 9: The comparison with the cases of different domain numbers in the service chain route: (**a**) the user gain and the service provider profit (N = M = 10); and (**b**) the user gain and the service provider profit (N = M = 100).