## 6 Conclusions  
In this article, the flow routing and scheduling problems in time-sensitive networking were studied. A novel method based on the FC step is proposed, which covered a finer-grained classification and the no-wait scheduling of TT flows by ILP techniques. The proposed MPFRS-FC technique covered multidisjoint path routing based on flow period correlation and links pruning to improve the scheduling performance and reduce the execution time of the algorithm. Given the characteristics of different network structures, no disjoint paths can occur in routing. An optimal scheduling method called APCRS-FC was further proposed to add an adaptive period compensation factor to make it possible for flows to increase their schedulability in the network. We designed twelve experiments, each of which used 100 test cases to evaluate the applicability scenarios and scalability of the proposed algorithm, and compared the performance with two baseline algorithms, namely, the SPRF-EtoED algorithm and LBF-EtoED algorithm. The results show the outperformance of the proposed algorithms in handling large traffic load, large differential traffic periods, large network scale, and varied network connectivity scenarios in terms of success rate and execution time. We also calculate the overall end-to-end delays in each method, though both MPFRS-FC and APCRS-FC algorithms increase the end-to-end delay of partial flows, which is not higher than 15% on average. The main reason for this depends on the routing planning, since the scheduling methods are all based on the constraints of no-wait transmission. Partial flows planned by the MPFRS-FC algorithm choose the path that is not the shortest path to the destination node to improve the global scheduling success rate. Furthermore, the APCRS-FC algorithm makes it possible to transmit in the network by compensating the traffic period at the source, which results in some flows having a delay of compensation at the first hop. This latency is accumulated in each hop relative to the original flow. Optimization here depends on a tradeoff between the latency performance of partial flows and the success rate of traffic scheduling. The larger the compensated flow period the greater the impact on the end-to-end delay performance of that flow. It is noticed that MPFRS-FC and APCRS-FC can lower the sum of end-to-end delays in large-scale networks with high traffic loads. In our future work, we would extend the routing and scheduling approach to improve scheduling flexibility by introducing deep reinforcement learning. On the other hand, the issue of converged wired and wireless routing and scheduling in TSN is also an interesting direction. Figure 14: The sum of end-to-end delays of the SPRF-EtoED, LBF-EtoED, MPFRS-FC, and APCRS-FC for 100 test cases with 30 flows chosen from flow group 2, tested in different types of network. * (21) IEEE, 802.1 Qbv. IEEE Standard for Local and Metropolitan Area Networks--Bridges and Bridged Networks--Amendment 25: Enhancements for Scheduled Traffic. In IEEE Std 802.1 Qbv-2015, 18 March 2016, pp. 1-57. Available online: [[https://ieeexplore.ieee.org/Xplore/home.jsp](https://ieeexplore.ieee.org/Xplore/home.jsp)]([https://ieeexplore.ieee.org/Xplore/home.jsp](https://ieeexplore.ieee.org/Xplore/home.jsp)) (accessed on 5 April 2022). * (22) [PERSON]; [PERSON] Scheduling in time sensitive networks (TSN) for mixed-criticality industrial applications. In Proceedings of the 2018 14 th IEEE International Workshop on Factory Communication Systems (WFCS), Imperia, Italy, 13-15 June 2018; pp. 1-4. * (23) [PERSON]; [PERSON]; [PERSON]; [PERSON] Design optimisation of cyber-physical distributed systems using IEEE time-sensitive networks. _IET Cyber-Phys. Syst. Theory Appl._**2016**, \(1\), 86-94. [CrossRef]
* (24) [PERSON]; [PERSON] Combined task- and network-level scheduling for distributed time-triggered systems. _Real-Time Syst._**2016**, _52_, 161-200. [CrossRef]
* (25) [PERSON]; [PERSON]; [PERSON] IEEE 802.1 Qbv gate control list synthesis using array theory encoding.