## 5 Analysis of Trade-Off between Traffic Congestion and Packing Density
### Trade-Off between Traffic Congestion and Packing Density  
The main contribution of this work is the statistical analysis of the density parameters, especially the trade-off between packing density and MTD. The packing density reveals the number of nodes per unit static cost, without considering the number of communication links in the packaging. On the other hand, in an MPC system, messages passes via communication links among all the nodes. So, more links mean more ways for a message to arrive at its intended destination. Moreover, the results from individual nodes can be merged to obtain the final results of computational intensive problems. This alternative path gives alternative choices in a routing algorithm to send the message to its destination. The number of links grows as well with an increment of node number.  
After manufacturing an MPC system, the nodes and their associated communication links become fixed and cannot be altered. Only the board level and system level nodes can be altered; however, the efficient use of communication links will lessen the MTD; therefore, after realization of a MPC system, the distribution and coordination of the packets among the nodes by the efficient use of communication links reveal the success of a MPC system. Since evaluation of traffic congestion effect on packing density after practical realization of a MPC system is quite expensive, the static trade-off analysis between traffic congestion and packing density is important to assess if the interconnection network will be suitable or not for the future generation MPC system.  
The trade-off factor between message congestion and packing density, called TCPDTF, is defined as the ratio between MTD and packing density. It is represented as in Equation (5), as the product of the average distance and static cost per communication link.  
\[\begin{array}{rcl}TCPDTF&=&\dfrac{\textit{Message Traffic Density}}{\textit{Packing Density}}\\ &=&\left\{\dfrac{\textit{Avg. Dist.}\times\#\textit{of Nodes}}{\#\textit{of  Links}}\right\}\times\left\{\dfrac{\textit{Static Cost}}{\#\textit{of Nodes}}\right\}\\ &=&\left\{\dfrac{\textit{Avg. Dist.}\times Static\textit{Cost}}{\#\textit{of Links}}\right\}\end{array} \tag{5}\]  
\begin{table}
\begin{tabular}{l l l l} \hline \hline
**Network** & **Packing Density** & **Message Traffic Density** & **Traffic Congestion vs. Packing Density Trade-off Factor (TCPDTF)** \\ \hline \hline \multicolumn{4}{c}{256 Nodes} \\ \hline
2D-Mesh & 2.13 & 5.69 & 2.668 \\
2D-Torus & 4.00 & 4.00 & 1.00 \\ \%hline TESH & 3.05 & 6.44 & 2.114 \\ TTN & 2.84 & 3.50 & 1.231 \\ TFBN & 3.20 & 1.84 & 0.575 \\ \hline \hline \multicolumn{4}{c}{4096 Nodes} \\ \hline
2D-Mesh & 8.13 & 21.67 & 2.667 \\
2D-Torus & 16.00 & 16.00 & 1.00 \\
3D-Mesh & 15.17 & 5.69 & 0.375 \\
3D-Torus & 28.44 & 4.00 & 0.141 \\ TESH & 32.00 & 10.91 & 0.341 \\ TTN & 28.40 & 5.91 & 0.208 \\ TFBN & 26.90 & 3.39 & 0.125 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Performance comparison of the packing density, message traffic density, and the TCPDTF of various networks.  
The traffic congestion and packing density trade-off factor (TCPDTF) of various networks were calculated according to Equation (5) and then enlisted in Table 2. For 256 node networks, the TCPDTF of the proposed TFBN is much lower than 2D- and 3D-Torus, as well as 2D- and 3D-Mesh networks. For 4096 node networks, the TCPDTF of the proposed TFBN is also lower than both 2D- and 3D-Mesh, 2D- and 3D-Torus, and TESH and TTN networks. The portrayed results also reveal that the proposed TFBN has less traffic congestion compared to the other networks. For the TFBN, less traffic congestion means lower latency and higher throughput. These insights can also be observed from the graphical representation of Table 2 in Figure 5. The graph shows the performance comparison for both 256 nodes and 4096 nodes.