## 5 Description of the Heuristics
### Second Phase: Maximum Edge Disjoint Path Problem (MEDPP)--Greedy Heuristics
\begin{table}
\begin{tabular}{c c c c c c c} \hline  & \multicolumn{3}{c}{**ALG 1**} & \multicolumn{3}{c}{**ALG 2**} \\ \hline \(\mathbf{U}\) & **Flows** & **N\_Saving\_F** & **Time** & **Flows** & **N\_Saving\_F** & **Time** \\ \hline
0.2 & 9956.00 & 61.85 & 686.42 & 9960.40 & 62.01 & 700.76 \\
0.4 & 7928.00 & 31.85 & 1195.02 & 8271.60 & 35.43 & 1724.03 \\
0.6 & 5715.60 & 18.40 & 1417.64 & 6272.80 & 23.03 & 2543.02 \\
0.8 & 3916.20 & 12.52 & 1422.73 & 4477.60 & 17.89 & 3064.09 \\
1 & 3584.20 & 10.01 & 1540.72 & 4382.80 & 13.43 & 3514.58 \\ \hline \end{tabular}
\end{table} 
Table 10: 8192 servers-10,000 flows-Flow scheduling with simple heuristics in phase 2. \begin{table}
\begin{tabular}{c c c c c c c c} \hline  & \multicolumn{3}{c}{**ALG 1**} & \multicolumn{3}{c}{**ALG 2**} \\ \hline \(\mathbf{U}\) & **\# Act** & **Act** & **N\_Saving\_A** & **Time** & **Act** & **N\_Saving\_A** & **Time** \\ \hline
0.2 & 24960.00 & 24914.00 & 61.96 & 685.80 & 24915.20 & 62.10 & 698.79 \\
0.4 & 25033.40 & 22252.40 & 34.11 & 1243.09 & 22587.60 & 37.10 & 1715.35 \\
0.6 & 24970.80 & 18468.40 & 21.18 & 1523.18 & 18979.00 & 25.20 & 2485.95 \\
0.8 & 25091.00 & 13277.40 & 13.21 & 1648.33 & 13419.40 & 21.67 & 3142.50 \\
1 & 25022.00 & 13206.40 & 11.74 & 1740.84 & 13337.40 & 16.22 & 3303.28 \\ \hline \end{tabular}
\end{table} 
Table 11: 8192 servers-10,000 flows-Activation scheduling with Simple heuristic in phase 2. ## 8 Conclusions and Future Work

In the present work, the time and path scheduling of real-time flows in DCNs was modelled and implemented using an ILP exact resolution model and through ad hoc heuristics. The problem is NP-hard, and the ILP model can not scale up when the number of servers and/or flows grows from only a few. The proposed heuristic approach consists in dividing the scheduling algorithm into three phases. In the first phase, the flow activation slots were defined using well-known scheduling policies, such as EDF or RM. In the second phase, the choice of path was performed in each slot for each flow. Finally, in the third phase, the schedule was revised and incomplete flows or activations were scheduled, when possible, or completely removed in order to gain time slots for previously left out flows/activations. Our results show that the scheduling and path selection problems can be solved simultaneously by means of the proposed heuristics in a short period of time with the advantage of reducing the energy demand of the network infrastructure while satisfying the real-time requirements. Different alternatives consisting of simple heuristics were provided that showed good overall performance. The solutions obtained by means of the proposed heuristics improved in terms of quality as more iterations were performed; however, they did so at the expense of increasing the computation time. For a first-path solution, the best results were not far from those obtained with a higher number of iterations, which shows that, depending on the generated flows on the network, a system administrator can choose between different scheduling approaches. As future work, an attempt will be made to analyse different solutions when the size of the DCN is scaled up to thousands of servers and including different DCN topologies. For this case, the possibility of dividing the network into subnetworks will be considered, as one of the main challenges of the proposed approach relies on the memory usage of the optimization algorithms. [PERSON] conducted the research of the state of the art and proposed and programmed the heuristics associated with phase 2. He participated in the manuscript preparation as well. [PERSON] proposed and programmed the first phase heuristics, prepared figures for presentation and contributed to the manuscript. [PERSON] contributed to the infrastructure network model and characteristics and revised the manuscript. [PERSON] proposed the research topic, revised previous work, analysed the real-time characteristics of the problem and prepared the manuscript.