## 5 Discussion
Specifically, the use of cross-domain resource sharing, as demonstrated in the Markov chain analysis, results in a marked increase in channel availability, surpassing the reliability benchmarks reported in previous works [35]. This advancement is particularly evident in scenarios with limited redundant resources, where our strategies demonstrate superior performance in maintaining high availability under failure conditions. At the same time, the cross-domain redundancy approach optimizes resource utilization since the BLACK radios provide additional redundancy only when needed, rather than being dedicated RED backups. The flexibility afforded by the ability to utilize radios across domains has tangible advantages over a siloed redundancy strategy. The cost savings from reduced hardware needs while maintaining high redundancy are significant. However, the discussion must also consider potential limitations and challenges involved in implementing this cross-domain redundancy. The model does not yet account for specific RED selection logic for BLACK reserves. Moreover, the instant cross-domain switchover assumptions require validation against scenarios with encrypted session migrations. Analyzing associated switching device vulnerabilities could strengthen trust arguments. This study contributes to the current body of knowledge by offering a theoretical model for enhancing communication reliability within the specialized context of air traffic management. However, it also provides practical considerations for the design and implementation of these systems. The proposed model and findings emphasize the critical role of dynamic redundancy and cross-domain resource utilization in strengthening the reliability of ATM communication systems, particularly those operating within the RED domain. Future work should focus on refining the Markov model to include considerations for the strategy of selecting m radio stations from the BLACK domain and exploring the impact of non-instantaneous switching devices. Additionally, further empirical validation of the model in operational settings would be beneficial to substantiate the theoretical findings and to fine-tune the model for practical applications in air traffic control systems worldwide. While the application of Markov chain models has provided valuable insights into the reliability of dual-priority communication systems, it is important to acknowledge the limitations of this approach, particularly in relation to state space explosion. The strategies we employed, such as state aggregation and model simplifications, are effective for managing the complexity, but there are scenarios where the number of states may still become large, making exact solutions computationally challenging. Future research could explore alternative modeling techniques, which may offer a more scalable approach to analyzing systems with extensive state spaces. Additionally, further investigation into hybrid models that combine Markov chains with other methods could provide a balanced solution to the state space explosion problem. Future research could explore the application of alternative modeling formalisms, such as stochastic reward networks (SRNs) [36; 37] and multidimensional universal generating functions (MUGFs) [38; 39; 40], which have demonstrated efficacy in characterizing modern and complex network architectures. SRNs could provide a more nuanced analysis of the reliability and performance of dual-priority communication systems, particularly in environments like 5G networks, where system states and rewards can vary significantly. MUGFs, on the other hand, could offer a scalable solution for analyzing the reliability of multi-tenant service function chains, potentially overcoming the limitations posed by state space explosion in traditional Markov models. These approaches represent promising directions for further enhancing the robustness and applicability of reliability models in complex communication systems.