## 2 Related Work  
The IEEE TSN standards define two different shaping approaches, namely time-aware shaping (TAS) in IEEE 802.1 Qbv [5] for scheduled streams and credit-based shaping (CBS) for unscheduled flows in IEEE 802.1 Qav [4]. Although other shapers can implement the traffic schedule in TT-Ethernet, such as the burst-limiting shaper (BLS) [14; 15] and the peristaltic shaper (PS) [16], the TSN task group has standardized the TAS to shape TT streams. The TAS mechanism is more applicable than the others to ensure a bounded latency for control traffic [17]. As an alternative to the TAS technique, some researchers have proposed asynchronous time shaping (ATS) as it does not require the selected nodes to be fully synchronized. The ATS mechanism is presented in the ongoing IEEE P802.1 Qcr project. ATS-aware nodes can temporally reshape incoming frames in a different queuing system using an urgency-based scheduler (UBS). [PERSON] et al. [18; 19] and [PERSON] et al. [20; 21] proved that the ATS mechanism provides low and bounded worst-case latencies with high link use, low complexity, temporal reshaping, and complete synchronization independence. In [22], the authors concluded that the flow control strategy in the ATS method provides a better real-time performance guarantee for aperiodic traffics than the CBS in a heavy network load. However, after comparing TAS and ATS performances, [PERSON] et al. [23] confirmed that the TAS method with tight configurations achieves the related latency targets for both sporadic and periodic data streams. In contrast, the preferred use of the ATS approach is only in the case of light network loads. Several scheduling models have been implemented based on the TAS mechanism and have examined the latency performance for TT traffic under the complete synchronization assumption. The targeted TT latency deadlines have been guaranteed using simulation-based algorithms, e.g., in [24; 25], and analytically, e.g., in [16; 26]. Besides, many researchers have suggested improving the GCL design to increase the scalability and adaptability that fit critical time applications. Fixed-timing algorithms result in a fluctuated quality of service (QoS) for all traffic types, depending on the instantaneous loading condition and network size. Therefore, [PERSON] et al. [23] proposed an adaptive duration for scheduled and unscheduled transmission intervals in the GCL design. The window duration is adjusted based on each traffic class's latency threshold and the incoming flows' specifications. OMNet++ results confirmed that queuing delay increases with decreasing window duration for related traffic and vice versa. [PERSON] et al. [27] added a protection interval to transmit the emergency event flows in each transmission cycle rapidly. The simulation results ensured the required services for emergency traffic, with minimal impact on the scheduled flows. [PERSON] et al. [28] and [PERSON] et al. [29] addressed the essential design compromises between scheduling efficiency and physical resources affecting the required scalability, adaptability, and self-configuration of the Industrial Internet of Things (IIoT) system. The authors also suggested adapting the hyper-period length and raster size according to the incoming traffic. In [30], [PERSON] et al. proposed adjusting the GCL according to the network topology and loading conditions' availability. [PERSON] et al. [31] presented a scheduling design with online updating to suitably integrate incoming data. Several other pieces of the research have proposed joint routing and scheduling algorithms, as in [32; 33; 34]. Other interesting works have suggested using the software-defined network (SDN) protocol to manage TSN scheduling, as in [1; 35; 36]. However, all the above algorithms are simulation based, which cannot express all pessimistic transmission cases and, then, leads to unsafe latency evaluations [37]. As known, critical time applications have hard and soft real-time traffic.