# Link Load Correlation-Based Blocking Performance Analysis for Tree-Type Data Center Networks
## 4 Simulation Results  
We conducted Monte Carlo simulations to validate the performance of the proposed LLC-based network blocking analysis method. The simulation scenario is a symmetric two-layer tree DCN with tree factor \(K\) and over-subscription ratio \(F_{OS}\). A uniform traffic distribution was deployed, and the load of all connection types was \(\rho_{0}\). The blocking performance of four analysis methods were compared, including the Monte Carlo approach and three analytical approaches: [PERSON]'s method [11], the classical EFPA method [14], and the proposed LLC-based method. Cases with different link capacities and over-subscription ratios were considered. For cases with tree factor \(K=4\) and over-subscription ratio \(F_{OS}=1\), the simulated and analytical blocking rates under different sub-channel numbers are compared in Figures 2-4, respectively. The curves of the EFPA and LLC-based methods both match well with the simulation curves, while the curves of the [PERSON]'s method are always above them. When \(F_{OS}=1\), for each edge switch, the aggregated bandwidth of its \(L_{0}\) layer links is equal to the bandwidth of its \(L_{0}\) layer link. Thus, if one \(L_{1}\) layer link of some CC connection is blocked, we can infer that at least one \(L_{0}\) layer link of this CC connection must be blocked at the same time. In this case, we only need to consider the blockings in the edge links, and the load correlation between edge and core links is not necessarily considered. Therefore, for over-subscription ratio \(F_{OS}=1\), which is also called the fat-tree network case, the EFPA and LLC-based methods can both obtain satisfactory accuracy in blocking probability analysis. Figure 3: Network blocking performance of different methods with \(K=4\), \(F_{OS}=1\), and \(M_{0}=20\). Figure 2: Network blocking performance of different methods with tree factor \(K=4\), over-subscription ratio \(F_{OS}=1\), and sub-channel number in edge links \(M_{0}=10\). For cases with tree factor \(K=4\) and over-subscription ratio \(F_{\text{OS}}=1.5\), the simulated and analytical blocking rates under different sub-channel numbers are compared in Figures 5-7, respectively. As shown in these figures, the LLC-based method curve is closer to the simulation curve than both [PERSON]'s method and the EFPA method. The LLC-based method curve agrees well with the simulation curve, especially in the blocking rate range below 4%. When the blocking rate is higher than 4%, the approximation errors of the LLC-based method and EFPA method both gradually increase with the traffic load and channel number. From the perspective of network deployment, blocking probabilities above 5% are generally unacceptable for users, and network operators should update the capacity or limit the subscribed users to reduce the network blocking rate. Therefore, the low blocking rate regime is much more important in blocking performance evaluations. Figure 4: Network blocking performance of different methods with \(K=4\), \(F_{\text{OS}}=1\), and \(M_{0}=30\). Figure 5: Network blocking performance of different methods with \(K=4\), \(F_{\text{OS}}=1.5\), and \(M_{0}=9\). For cases with tree factor \(K=4\) and over-subscription ratio \(F_{\text{OS}}=2\), the simulated and analytical blocking rates under different sub-channel numbers are compared in Figures 8-10, respectively. As shown in these figures, the LLC-based method curve is closer to the simulation curve than the other two in the blocking rate range of 1% to 4%. The EFPA method matches best with the simulation curve in the blocking rate range below 1%. For higher blocking rates, the curves of the LLC-based method and EFPA method gradually converge. Figure 6: Network blocking performance of different methods with \(K=4\), \(F_{\text{OS}}=1.5\), and \(M_{0}=18\). Figure 7: Network blocking performance of different methods with \(K=4\), \(F_{\text{OS}}=1.5\), and \(M_{0}=30\). Figure 8: Network blocking performance of different methods with \(K=4\), \(F_{\text{OS}}=2\), and \(M_{0}=10\).