## 2 System Model
### Gate Control Constraints  
The generation of a valid Gate Control List (GCL) by any scheduling algorithm must ensure that all data streams, or their constituent sub-streams if applicable, meet a set of predefined constraints. These constraints are essential for guaranteeing deterministic transmission, meeting deadlines, and avoiding conflicts within the Time-Sensitive Network. The primary variable to be determined by the scheduling process is the transmission start time, \(\varphi_{i,j,\mu}\), for each sub-stream instance. The key constraints are as follows:  
1. Stream Transmission Order and Deadline (Edge Constraints): For each original data stream \(F_{i}\), divided into \(N_{i}\) sub-streams (\(N_{i}\geq 1\)), these sub-streams must be transmitted sequentially within each of its \(U_{i}\) cycle instances in the hyperperiod \(\mathcal{T}\). The transmission of the last sub-stream must complete before the stream's deadline \(D_{i}\) for that cycle instance. \begin{table}
\begin{tabular}{l l} \hline \hline
**Symbol** & **Definition** \\ \hline \(S\) & Set of schedulable data streams \\ \(F_{i}\) & An individual data stream \(i\) \\ \(L_{i}\) & Length of data stream \(F_{i}\) (e.g., in bytes) \\ \(t_{i}\) & Transmission time of data stream \(F_{i}\) on a given link \\ \(T_{i}\) & Period of data stream \(F_{i}\) (original stream’s period) \\ \(D_{i}\) & Deadline of data stream \(F_{i}\) (relative to its release in a cycle) \\ \(P_{i}\) & Transmission path (sequence of links/switches) for data stream \(F_{i}\) \\ \(PCP_{i}\) & Priority Code Point value assigned to data stream \(F_{i}\) \\ \(\mathcal{T}\) & Hyperperiod of the GCL schedule \\ \(A_{i}\) & Available time duration calculated for potential sub-flow division of stream \(F_{i}\) within its period \\ \(N_{i}\) & Number of sub-streams data stream \(F_{i}\) is divided into (if \(N_{i}=1\), no division) \\ \(a_{i,x}\) & Transmission time of the \(x\)-th sub-stream of data stream \(F_{i}\) \\ \(U_{i}\) & Number of cycle instances (occurrences) of stream \(F_{i}\) within one hyperperiod \(\mathcal{T}\) \\ \(\delta_{i}\) & Maximum allowed inter-cycle start time variation (jitter) for stream \(F_{i}\) \\ StreamSchedulingTime & Reserved time duration to account for TSN switch internal processing and scheduling overhead per hop \\ LatencyRequirements\({}_{i}\) & Maximum allowed end-to-end latency for stream \(F_{i}\) \\ JitterRequirements\({}_{i}\) & Maximum allowed end-to-end jitter for stream \(F_{i}\) (variation in latency) \\ \(\phi_{i,\mu}\) & Scheduled start time of transmission for an undivided data stream \(F_{i}\) in its \(\mu\)-th cycle instance within the hyperperiod \\ \(\varphi_{i,j,\mu}\) & Scheduled start time of transmission for the \(j\)-th sub-stream of data stream \(F_{i}\) in its \(\mu\)-th cycle instance within the hyperperiod \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Symbols and definitions used in this paper. \[(\mu-1)T_{i}<\varphi_{i,1,\mu}<\varphi_{i,1,\mu}+a_{i,1}<\varphi_{i,2,\mu}<\cdots< \varphi_{i,N_{i},\mu}<\varphi_{i,N_{i},\mu}+a_{i,N_{i}}<(\mu-1)T_{i}+D_{i}\] (1) This fundamental constraint enforces two critical properties for each stream instance. First, it guarantees the sequential, non-overlapping transmission of all constituent sub-streams. Second, it imposes a hard real-time deadline, \(D_{i}\), relative to the start of the period, by which the entire stream transmission must be completed. Similar deadline-based constraints are foundational in the formal modeling of Time-Aware Shaping schedules [25]. 2. Inter-Cycle Start Time Variation (Interval Fairness): To maintain a predictable transmission pattern and limit jitter introduced by scheduling variations across different cycles of the same stream, the start time of any given sub-stream \(f_{i,j}\) in any cycle instance \(\mu\) (where \(\mu>1\)) should not deviate excessively from its start time in the first cycle instance (\(\mu=1\)), relative to their respective cycle start times. This deviation is bounded by \(\delta_{i}\).