## 5 Evaluations  
### Experiment Setup  
We implemented our proposed mechanism in a Python-based framework. We used networkx for the modeling of network topologies and generating different traffic patterns. All of the ILP instances were solved in Gurobi Optimizer (version 9.5.1, 64-bit Windows), which offers a Python API for the integration. The hardware platform is a PC with an Intel Core i7-8750H CPU running at 2.20 GHz and 8 GB RAM. The network topologies in test cases consist of four synthetic network topologies shown in Figure 4.  
We define three flow groups depicted in Tables 2-4. Each flow is generated by a random and independent selection of groups with their probabilities. The source and destination of each flow were randomly selected from end systems to cover the unicast and multicast broadcast.  
We evaluated the performance of the MPFRS-FC algorithm, APCRS-FC algorithm, and baseline algorithms. SPRF-EtoED is the shortest path first routing and time slots scheduling by minimizing end-to-end delay. The TSN standard on path control and reservation recommends the constrained shortest path first routing scheme for the transmission of time-sensitive flows in 2016 [38]. LBF-EtoED is the load balance-based first routing and  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Flow Index** & **F.prd/\(\mathrm{\SIUnitSymbolMicro s}\)** & **F.td/\(\mathrm{\SIUnitSymbolMicro s}\)** & **Probability** \\ \hline F1 & 143 & 1 & 1/40 \\ F2 & 130 & 1 & 1/40 \\ F3 & 500 & 2 & 1/4 \\ F4 & 100 & 2 & 9/20 \\ F5 & 250 & 1 & 1/4 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Test group 3 of candidate flows.  
Figure 4: Synthetic topology test cases. (**a**) A small ring topology. (**b**) A small mesh topology. (**c**) A medium ring topology. (**d**) A medium mesh topology.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Flow Index** & **F.prd/\(\mathrm{\SIUnitSymbolMicro s}\)** & **F.td/\(\mathrm{\SIUnitSymbolMicro s}\)** & **Probability** \\ \hline F1 & 100 & 1 & 1/5 \\ F2 & 200 & 2 & 1/5 \\ F3 & 40 & 4 & 1/5 \\ F4 & 80 & 8 & 1/5 \\ F5 & 160 & 8 & 1/5 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Test group 1 of candidate flows.  
time slots scheduling by minimizing end-to-end delay [39,40]. All the above are solved by the ILP model. These techniques are adapted to support no-waiting scheduling. Under the conditions of the same number of flows, network topology, and the same flow group, we generated 100 patterns on the three routing and scheduling algorithms and conducted statistics on the scheduling results separately. The comparison is conducted in terms of runtime and the success scheduling rate. SR, SM, and MR model instances were run in Gurobi Optimizer with a time limit of 1 min, MM model instance was set at a time limit of 3 min, and the solution number limit of each model was set to one. Each model finishes with one of the following three possible results:  
* Optimal: One suboptimal schedule was found (scheduling as much flow as possible is the pursuit of this paper rather than the optimal solution).
* Timeout: The time limit was reached before the schedule was found.
* Infeasible: No schedule was found with the given flows.  
As shown in Figure 5, we simulate four algorithms followed by this workflow. The input is network topology and flow requirements selected from four networks and three flow groups. The flow set generation module generates K random flows based on our given generation rules. The topology and flow resolution module generates the related network graph matrix, vertex association matrix, edge association matrix, and several other data structures that aid in the computation. The routing and scheduling algorithm selection module chooses a specific simulated algorithm to generate route and schedule constraints in the constraint generation module. The constraint addition and solver module calls the Gurobi optimizer to obtain routing and scheduling results. Each simulation scenario is iterated 100 times to measure the performance of the scheduling results.