## 2 Background and Related Work  
The IEEE TSN standards released IEEE 802.1 Qbv standard [21] to support the time-triggered mechanism with low bounded latency and jitter guarantees. Before network configuration and operation, a scheduler collects network topo information and timing parameters of time-triggered flows as input and generates a valid scheduling table. Then, it sends down scheduling tables to each switch, and switches periodically perform scheduling table entries to achieve deterministic communication and latency guarantees. Transmission scheduling in TSN networks to impact real-time performance is a well-researched problem [22; 23; 24; 25; 26; 27]. The resource allocation mechanism in TSN can be considered from two aspects in general. The first subproblem is the routing algorithm, i.e., how to select a path from source to destination for each flow. The second subproblem is the scheduling method, i.e., when to send which frame to be sent or forwarded. Concerning the routing subproblem, the optimal objective is to minimize the number of hops for routing traffic traditionally [28]. However, [PERSON] et al. proved that the shortest path algorithm is not an excellent routing algorithm for TSN because this approach may result in nonschedulable solutions. Hence, many heuristic routing algorithms are proposed [8; 9; 10; 11; 29] to determine the route paths with different objective optimization. The work in [30] proposes a K-shortest path heuristic and a metaheuristic randomized adaptive search procedure to solve the joint routing and scheduling problem for time-triggered traffic. Some existing approaches use the Steiner Tree as the default route [24; 25]. In the schedule synthesis, a routing and scheduling procedure is devised to improve the schedule success rate. The scheduling procedure is based on the formerly fixed routing, and thus the execution time of the schedule is sharply reduced. To solve the scheduling problem in TSN, several prior works used mathematical programming Satisfiability Modulo Theories (SMTs) based on solver (\(Z3\)) to compute schedules for TTEthernet, profiNET, and IEEE 802.1 Qbv [31; 32; 33; 34]. Additionally, the results in [12] show that the computational complexity increase exponentially when the number of flows increases linearly. [PERSON] et al. [13] observed that prior works separate the flow routing subproblem from the flow scheduling subproblem. The typical solution is to solve the routing subproblem first and then compute the schedule based on fixed routing results. They formulated routing and scheduling jointly as an ILP problem using a Gurobi optimizer to solve the problem. [PERSON] et al. [15] studied how to improve schedulability and proposed an ILP-based algorithm with a maximum scheduled traffic load (MSTL) metric to determine the routing of each TT flow. [PERSON] and [PERSON] [20] address a special problem named no-wait packet scheduling, where the frames are not allowed to wait in the switches. The switch forwards the frame to the next node as soon as it receives the frame. This method avoids the frames jitter by assuming that all flows have the same period, and thus, the problem is restricted to one frame per flow. [PERSON], [PERSON] et al. [35] proposed a joint algorithm of appropriate message fragmentation and no-wait scheduling to improve the schedule success rate up to 50% further than previous algorithms. [PERSON] et al. [27] propose an approach that schedules the transmission of time-triggered flows only on the hosts. Moreover, authors of [17; 36; 37] observed the effect of different routing results on scheduling and recommended joint routing and scheduling algorithms to use proper network resources and to target load balancing. [PERSON] et al. [36] considered frame period as a factor of the degree of conflict measured and incrementally added to the schedule; this approach is based on assuming densely connected typologies such that the possible paths between any two nodes outnumber those required for messages exchanged between these nodes.