## 7 Conclusions  
For implementing and operating the satellite-terrestrial integrated network, network slicing is adopted as a solution to meet the differentiated service requirements and orchestrate the underlying network infrastructures in a unified manner. To improve resource utilization efficiency and motivate the participation of different service providers, slice admission control is further introduced. However, admission control for multi-domain network slicing becomes much more complex when multi-sided participants are involved and there is no perfect solution to date.  
In this paper, a multi-sided auction approach is proposed for modeling multi-domain network slicing admission control as a mechanism design problem. Theoretical analysis shows that the proposed ascending-price auction mechanism is individually rational, strongly budget balanced, and obviously truthful. Numerical experiments based on both real-world and simulated data demonstrate that the ascending-price auction mechanism is asymptotically optimal with the increase in users compared with the optimal trade mechanism.  
Some potential future research directions are also considered. The first consideration is a more complex market structures, for example, more than two alternative routes are available, or a single domain may be used within multiple routes. From the user's perspective, more choices bring more opportunities for using the network slicing service with a lower price. From the service provider's perspective, both a deeper entangled cooperation relationship among the domains in the same route and a harsh competition relationship among the domains in the different routes exist. More numerical experiments are expected to be designed to evaluate the ascending-price auction mechanism in more complex and realistic settings.  
Another consideration is the extension from the bandwidth requirements to more requirements, e.g., transmission delay and energy consumption constraints. Similar constraints have been considered in relevant multi-domain tasks, e.g., network slicing orchestration with impatient tenants [58] and energy-efficient service function chaining orchestration [57]. It is worth further consideration to incorporate these factors in the ascending-price auction mechanism, while maintaining the desirable properties.  
In addition, some statistical analysis and analysis of variance can be conducted to demonstrate the statistical significance of the performance for the proposed ascending-price auction mechanism in future numerical experiments.  
Conceptualization, W.J. and Y.Z.; methodology, W.J. and Y.Z.; software, W.J. and Y.Z.; validation, W.J. and Y.Z.; formal analysis, W.J. and Y.Z.; investigation, W.J. and Y.Z.; resources, W.J. and Y.Z.; data curation, W.J. and Y.Z.; writing--original draft preparation, W.J. and Y.Z.; writing--review and editing, W.J., Y.Z. and X.X.; visualization, W.J. and Y.Z.; supervision, Y.Z. and X.X.; project administration, Y.Z. and X.X.; funding acquisition, Y.Z. and X.X. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the Science and Technology Project of the State Grid Corporation of China (5400-202255158A-1-1-ZN).  
Not applicable.  
The authors declare no conflict of interest.