## 2 System Model
### Gate Control Constraints
\[\varphi_{i,j,\mu}-(\mu-1)T_{i}-\varphi_{i,j,1}\leq\delta_{i},\quad\text{for }\mu>1\] (2) This helps in ensuring that a stream does not drift uncontrollably within its allocated periodic windows over the hyperperiod. This type of constraint is crucial for bounding the scheduling jitter, which is the variation in a stream's latency over time. By limiting this deviation with \(\delta_{i}\), the algorithm provides a more predictable transmission pattern, a key requirement for many real-time control applications [7]. 3. Inter-Hop Scheduling Delay (Node Timing): For any sub-stream \(f_{i,j}\) traversing its path \(P_{i}\), let \(\varphi_{i,j,\mu}^{(h)}\) denote its scheduled start of transmission time from hop \(h\). For any two consecutive hops \(h\) and \(h+1\) along \(P_{i}\), the start time at hop \(h+1\) must account for the transmission time \(a_{i,j}\), the propagation delay \(\text{PropDelay}^{(h,h+1)}\), and the per-hop processing time \(\text{StreamSchedulingTime}\) at node \(h+1\). \[\varphi_{i,j,\mu}^{(h+1)}\geq\varphi_{i,j,\mu}^{(h)}+a_{i,j}+\text{PropDelay} ^{(h,h+1)}+\text{StreamSchedulingTime}\] (3) This formulation models the per-hop delay accumulation. The end-to-end latency of a stream is composed of fixed delays (serialization and propagation) and variable delays (queuing and processing). This constraint explicitly accounts for the propagation delay over the physical link and allocates a budget, StreamSchedulingTime, for the variable processing delay within the switch (e.g., for GCL lookup and gate operations), ensuring a valid schedule across multiple hops [6]. 4. End-to-End Latency and Jitter: For each cycle instance \(\mu\) of an original stream \(F_{i}\), the total time \(B_{i,\mu}\) from the start of its first sub-stream (\(f_{i,1}\)) at the source node to the completion of its last sub-stream (\(f_{i,N_{i}}\)) at the final destination node must not exceed the stream's end-to-end latency requirement. \[B_{i,\mu}=(\varphi_{i,N_{i},\mu}^{(\text{dest})}+a_{i,N_{i}})-\varphi_{i,1, \mu}^{(\text{src})}\leq\text{LatencyRequirements}_{i}\] (4) Furthermore, the variation in this end-to-end latency across all \(U_{i}\) cycle instances of stream \(F_{i}\) within the hyperperiod must be bounded by its jitter requirement. \[\max_{\mu\in[1,U_{i}]}\{B_{i,\mu}\}-\min_{\mu\in[1,U_{i}]}\{B_{i,\mu}\}\leq \text{JitterRequirements}_{i}\] (5) Equation (4) bounds the total end-to-end latency for each stream instance, measured from the start of the first sub-stream's transmission at the source to the completion of the last sub-stream's transmission at the destination. Equation (5) further constrains the end-to-end jitter by limiting the difference between the maximum and minimum observed latencies across all cycle instances. Guaranteeing both latency and jitter bounds is critical for automotive applications [25]. * No-Conflict (Resource Exclusivity): On any given transmission link or switch output port, only one sub-stream can be transmitted at any point in time. Thus, for any two distinct sub-stream transmissions that are scheduled on the same output port, their transmission windows must not overlap. \[(\varphi^{(p)}_{x,k,\mu_{x}}+a_{x,k}\leq\varphi^{(p)}_{y,l,\mu_{y}})\vee( \varphi^{(p)}_{y,l,\mu_{y}}+a_{y,l}\leq\varphi^{(p)}_{x,k,\mu_{x}})\] (6) This constraint is the core of conflict-free scheduling on any shared resource, typically an egress port of a switch. It enforces mutual exclusion in the time domain, ensuring that no two frames are scheduled for transmission at the same time on the same link. This formulation is a fundamental requirement in nearly all TAS scheduling problem definitions [18]. * Link Bandwidth Utilization: The sum of the bandwidth utilized by all original data streams \(F_{i}\) routed over any particular communication link \(l\) must not exceed the link's capacity. This is typically ensured during an initial schedulability analysis phase. \[\text{For any link }l,\quad\sum_{F_{i}\text{ traversing link }l}\frac{t_{i}}{T_{i}}<1\] (7) This constraint represents a fundamental schedulability test.