# TFBN: A Cost Effective High Performance Hierarchical Interconnection Network
## 5 Static Cost Effectiveness Analysis of a TFBN
### Time Cost Effective Factor  
The main purpose of the use of an MPC system is to solve the computational intensive problem or a grand challenge problem in a reasonably short period of time. This time depends on four factors, viz., the processing speed of an individual node, the dividing of the tasks among the nodes an MPC system and coordination of these tasks between nodes, the underlying routing algorithm to transfer the flits between source and destination nodes, and after the execution of each individual node task merging these individual execution outcomes to conclude with the final result. The fast time implies that the MPC system fast individual node and the underlying interconnection network has good performance in terms of less blocking, delay, and congestion. Thus, time is also an important factor in measuring the efficiency of an MPC system. Cost-effective factors considered along with time factors to determine  
Figure 4: Comparison of static cost-effective factor of various interconnection networks considering different values of \(\rho\) (**a**) network size 256 nodes; (**b**) network size 4096 nodes.  
the time cost-effective factor (TCEF) of an interconnection network. Therefore, the TCEF is a very important and essential parameter to statically assess the feasibility of an MPC system, especially its underlying interconnection network topology. TCEF of an interconnection network is expressed by Equation (3):  
\[TCEF(p,T_{p})=\frac{1+\sigma T_{1}^{\alpha-1}}{1+\rho G(p)+\frac{ \sigma}{p}T_{p}^{\alpha-1}} \tag{3}\]  
TCEF of a network denotes as \(TCEF(p,T_{p})\) where \(p\) denotes the number of processor in an MPC system, and \(T_{p}\) denotes time taken to solve a problem using that MPC system. \(T_{1}\) denotes the time taken to solve the same problem using a single processing element. Here, \(G(p)=\frac{(Total\#of\ links)}{(Total\#of\ nodes)}\) and \(\rho=\frac{C_{1}}{C_{p}}\), whereby \(0\leq\rho\leq 1\). The values of \(\alpha\) and \(\sigma\) are constants and considered equal to 1 for considering linear time penalty in \(T_{p}\). Including all of these values, Equation (3) is simplified as below:  
\[TCEF(p,T_{p})=\frac{2}{1+\rho G(p)+\frac{1}{p}} \tag{4}\]  
We have assessed the time cost-effective factor of TFBN along with TTN, TESH network, 2D-torus, and 2D-mesh networks for different \(\rho\) ranging from 0.1 to 1.0 and plotted in Figure 5. Like CEF, here, we have also considered two different size networks such as 256 nodes and 4096 nodes. Figure 5a depicted the TCEF of different networks for 256 nodes and Figure 5b depicted the same for 4096 nodes. For all the values of \(\rho\), (\(0.1\leq\rho\leq 1.0\)), the TCEF of TFBN is significantly lower than that of TTN, TESH, torus, and mesh networks. From \(\rho=0.3\)) and onward, the TCEF of the TFBN is expressively lower than those networks for both 256 node and 4096 node networks. This proves that TFBN is a promising hierarchical interconnection network as compared to these networks. This qualifies TFBN to be a plausible alternative choice compared to many conventional and hierarchical interconnection networks to build next-generation supercomputers or MPC systems.