## 2 System Model
### Baseline Scheduling Algorithm with Sub-Flow Division
#### 2.4.1 Stream Preprocessing, Filtering, and Sub-Flow Division  
The initial phase focuses on preparing the input data streams and determining parameters for sub-flow division if necessary. This involves several steps:  
1. Input Stream Characterization: The algorithm takes as input a set of all data streams \(\mathcal{F}_{all}\), where each stream \(F_{i}\in\mathcal{F}_{all}\) is characterized by its length \(L_{i}\), transmission time \(t_{i}\) (derived from \(L_{i}\) and link speed), period \(T_{i}\), deadline \(D_{i}\), transmission path \(P_{i}\), and Priority Code Point \(PCP_{i}\). 2. Schedulability Filtering: Not all requested streams may be schedulable given network constraints. Streams are first grouped by their unique transmission paths. For each path, the algorithm checks if the sum of bandwidth utilization ratios (\(\sum(t_{j}/T_{j})\) for streams \(F_{j}\) on that path) is less than 1 (as per Equation (7)). If a path is oversubscribed, a culling process is initiated:* Streams with the lowest \(PCP_{i}\) values are removed first. * If still oversubscribed, streams with the largest bandwidth utilization ratios \((t_{j}/T_{j})\) are considered for removal. This filtering results in a set of preliminarily schedulable streams \(\mathcal{S}_{sched}\). If \(\mathcal{S}_{sched}\) is empty, no schedule can be generated. * Hyperperiod Calculation: The hyperperiod \(\mathcal{T}\), which defines the repeating cycle length of the GCL, is calculated as the least common multiple (LCM) of the periods \(T_{i}\) of all streams in \(\mathcal{S}_{sched}\). * Sub-Flow Parameter Calculation: To handle high-bandwidth streams that might otherwise block shorter, more urgent streams or violate deadlines, a sub-flow division mechanism is employed. The parameters for this division (available time \(A_{i}\), number of sub-streams \(N_{i}\), and individual sub-stream transmission times \(a_{i,x}\)) are calculated for each stream \(F_{i}\in\mathcal{S}_{sched}\). The rationale behind this sub-flow division heuristic is to increase scheduling flexibility by breaking down long-duration streams. The number of sub-flows, \(N_{i}\), is determined by comparing the stream's original transmission time, \(t_{i}\), with a dynamically calculated target_substream_duration. This target duration is derived from the available time on the path, \(A_{i}\), normalized by the ratio of the stream's period to that of the fastest periodic stream in the set. This heuristic aims to segment longer streams into pieces whose durations are more comparable to those of the base-rate streams in the network, thereby preventing the blocking of shorter, deadline-critical flows and improving overall schedulability. This entire preprocessing, filtering, and sub-flow parameter calculation process is detailed in Algorithm 1. The output of this phase includes the hyperperiod \(\mathcal{T}\), the set of schedulable streams \(\mathcal{S}_{sched}\), and for each stream \(F_{i}\in\mathcal{S}_{sched}\), its sub-flow parameters (\(N_{i}\) and \(a_{i,x}\)). If \(N_{i}=1\), the stream is not divided. These parameters are then used in the subsequent GCL generation phase. The mathematical constraints outlined in Section 2.3 will apply to these streams or their constituent sub-streams. Illustrative Example: The Necessity of Sub-Flow Division  
To illustrate the critical role of the sub-flow division mechanism, consider a common scheduling conflict scenario detailed in Table 2. This scenario involves a high-bandwidth stream (Data Stream 1) with a long transmission time and another stream (Data Stream 0) with a tight, deadline-critical requirement. As shown in Figure 1, attempting to schedule these streams without sub-flow division leads to a failure. The long, contiguous transmission of Data Stream 1 occupies a critical time window, making it impossible for the scheduler to find a valid slot for Data Stream 0 to meet its deadline in a subsequent cycle.