## 7 Conclusions  
The results obtained in this study made it possible to draw the following conclusions:  
1. Based on the spatiotemporal linkage (SPL) of the mining and backfilling operations for single supports, the operations were divided into four processes involving cylinder actions for single support: support motion, support adjusting, dumping, and tamping. Theoretical formulas for calculating the actuation time of single support in the three solid backfilling technologies were derived and further validated via a case study.
2. The SPL of different mining and backfilling processes in the whole face for intelligent parallel operations was studied. The intelligent solid backfilling operation for the whole face was subdivided into four distinct stages. We further presented the theoretical formula for estimating the time of cyclic operation for the whole face in the three mining technologies.
3. The influence of key factors on the time of cyclic operation of three backfilling techniques was analyzed. It is concluded that the time of cyclic operation is positively correlated with the time gap between hydraulic cylinder actions and the number of tampings; it is hyperbolically correlated with the number of supports in parallel operation and the intelligence index. Moreover, reducing the number of tampings, increasing the number of supports in parallel operation, and the intelligence index can significantly improve the backfilling efficiency.
4. A case study was performed for the #11233 filling working face of Xingdong Mine, China, operated by Jizhong Energy Resources Co., Ltd. Keeping other parameters unchanged, by adopting the optimized method of grain diameter grading of backfilling materials to reduce the number of tampings to 3 times and increasing the number of parallel supports to 4, with the intelligence index being 1, the backfilling efficiency can be increased by 39%. The improvement of backfilling efficiency is the most significant. It has great significance for improving production capacity, reducing staff, and improving efficiency and safety of the backfilling working face.  
Conceptualization and methodology, T.Z., G.Z. and Q.Z.; software and validation, Y.H., H.L. and J.C.; resources and data curation, K.Y., H.L. and Y.W.; writing--original draft preparation, T.Z.; writing review and editing, Q.Z., G.Z. and K.Y.; supervision and funding acquisition, Q.Z. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the National Natural Science Foundation of China (Grant No.52174134), the Fundamental Research Funds for the Central Universities (Project No. 2021 GJZPY12), Funded by the Graduate Innovation Program of China University of Mining and Technology (Project No.2023 WJLCRCZL036), Funded by the Postgraduate Research and Practice Innovation Program of Jiangsu Province (Project No. KYCX23_2807).  
Not applicable.  
The data presented in this study are available upon request from the corresponding author. The data are not publicly available. The key data used in this study represent field measurement statistics of a ZC5160/30/50-type solid backfilling mining hydraulic support from the Xingdong Coal Mine, and the related data are displayed in the text in Figure 5. The data contain detailed technical parameters of this type of support, and obtaining them again would infringe upon the rights and interests of Xingdong Mine and the factory that manufactures the hydraulic supports. In addition, the other data are all derived from the previously mentioned statistics and theoretical formulas.  
Thanks to Henan Guoxing Technology Co., Ltd., and the Xingdong coal mine for the parameters of the backfilling hydraulic support.  
[PERSON] was employed by the Anhui RONDS Technology Co., Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.