## 6 Conclusions  
In CRNs, considering communication failure and repair, to maximize system performance and meet the different transmission requirements of PUs, we propose a communication failure and repair mechanism with classified PUs to minimize the impact of communication failure on the transmission of PU packets with a higher priority. We divided PUs into two levels and set different preemption principles according to the system state. In addition, PU1 and PU2 packets can be transmitted at a low speed under the failure state of the system. We built an early-arrival discrete-time queueing model and solved the performance expressions of the system through the system's steady-state analysis. The curve characterization and performance analysis of some performance indexes were carried out by designing numerical experiments. When the communication failure rate increased, the blocking rates of the three types of packets increased, the throughput fell, and the loss rates of PU2 and SU packets increased. As the repair rate grew, the performance change trends were opposite to those caused by the rise in the communication failure rate. When PU1 packets or PU2 packets' arrival rate increased, the blocking rates of the three types of packets rose, and the loss rates of PU2 and SU packets grew. Meanwhile, the throughput of the corresponding type of packets increased, while the throughput of the other two types of packets decreased. By comparing the proposed mechanism with the traditional mechanism, we discovered that the proposed mechanism could more effectively meet the different transmission requirements of PUs and improve the performance of PU1 packets. In summary, the experimental results reflected the performance variation trends of the proposed communication failure and repair mechanism with classified PUs in different network environments and demonstrated its effectiveness.  
The limitations of this study and future work are summarized as follows:  
1. After the communication failure and repair mechanism with classified PUs is presented, a discrete-time queueing model is established and analyzed. However, due to the space limitation and the possible cumbersome statements, we only list the matrix and submatrices of the one-step state transition probability of the system and do not analyze the model in detail.
2. In the comparison experiments, we only compare the performance between the proposed communication failure and repair mechanism with classified PUs and that with unclassified PUs. It can be seen from the performance comparison diagrams that the proposed mechanism can effectively improve the performance of PU1 packets. However, we do not compare the proposed mechanism with existing methods, such as adaptive modulation and coding (AMC) or hybrid automatic repeat request (HARQ). In future work, we will further compare the proposed mechanism with existing techniques and methods to highlight the strengths and weaknesses of the mechanism proposed in this paper.
* In this paper, we deal with PUs with fixed positions. In future work, we will extend the study of spectrum allocation strategies in the context of communication failure and repair by taking mobile users into account.  
Conceptualization, Y.Z. and Q.L.; Data curation, S.Y. and Z.Y.; Formal analysis, Y.Z. and Q.L.; Investigation, Q.L. and S.Y.; Methodology, Q.L. and Z.Y.; Software, S.Y.; Supervision, Y.Z.; Validation, S.Y. and Z.Y.; Visualization, Q.L.; Writing--original draft, Y.Z. and Q.L.; Writing--review and editing, Y.Z., Q.L. and S.Y. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Fundamental Research Funds for the Central Universities (grant number N2323024), China.  
Not applicable.  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
The authors declare no conflicts of interest.